text: A SKEW-PRODUCT WITHOUT FIXED-CURVES 11 The next crucial deﬁnition ﬁxes in detail all quantities and objects mentioned above. R ∗ R ∗ Deﬁnition 3.4. We start by deﬁning (0 ) := (0 , n0, α0, δ0, 0,−0, 0) and ϕ0∗ := ϕ(0∗,n0,α0,δ0,0,0,0) by choosing (Deﬁnition 3∗.1) n0 = 1∗, α0 < 12 ∗= 2 n0 and −δ0 <∗ α0 small enough so that the intervals B [0 ] = B [0 ] , B [1 ] and B− [( 1) ] = − ∗ 0 − ∗ ∗α0∈ α−0 ∗ 1 B [( 1) ] are pairwise disjoint; and ( 2) , 2 / B− [( 1) ] and, additionally, α0 ∗ ∩ ∗ ∅ 1 Bd (B [0 ]) O (ω) = . (cid:40) α0 − S \{ } −→ I We also set a+ = a = a = 0, and we deﬁne the map γ : 1 0 by 0 0 0 0 ∈ ∗ \{ } γ (θ) = ϕ0∗(θ) if θ ∈ Bα0[0∗] 0 , 0 0 if θ / B [0 ]. α0 ≥ For consistency with the deﬁnition of γ in the case j 1, we deﬁne the map S \{ } −→ I j ∈ S ∗ γ−±1 : 1 0 ∗ ± by γ−1(θ∗) ±= 0 for every θ 1. Then, notice tha∈t, a0 = γ∗−1(0 ), a0 = ϕ0∗(0 α0) = γ∈−1N(0 α0), andRγ0∗(θ) R= γ−−1(θ∗) for every θS /\B∗α0[0 ] . Next, for every j we deﬁne (j ), (( j) ) and (γ , 1 Z ) from the R ∗ ∗ ⊂ ∗ ∈ j j S \ ∗ corresponding boxes (i ) and Bα|i|[i ] Bi [i ] for i Zj−1, and (γj−1, 1 Zj−1) as follows. We ta(cid:16)ke nj(cid:0), δj and αj (cid:1)such th(cid:0)at (see F(cid:1)(cid:17)igure 4 to ﬁx ideas): − (R.1) nj > nj−1, δj < αj < 2 nj < δj−1 < αj−1 and − ∗ ∪ ∗ ∩ ∗ ∅ Bd B [( j) ] Bd B [j ] O (ω) = . αj αj (R.2) The intervals (cid:0) (cid:1) ∗ ∗ B [j ] = B [j ] , j αj ∗ ∗ R B [j ] = B [(j + 1) ] , ω −αj ∗ αj − ∗ B−j [( j)−] = Bαj−∗1 [( j) ] a−nd ∗ B− [( (j + 1)) ] = B [( (j + 1)) ] (cid:0) (j+1)(cid:1) (cid:2) αj (cid:3) are pairwise disjoint, ∗ ⊂ ∗ − − ∗ − γj−1 Bαj [(cid:96) ] γj−1((cid:96) ) 2 nj, γj−1((cid:96) ) + 2 nj ∈ { − } for every (cid:96) j + 1, (j + 1) , ∗ ∩ ∗ { ∗} ∈ { − } B [(cid:96) ] Z = (cid:96) for (cid:96) j, (j + 1) and (cid:96) j+1 ∗ ∩ ∗ { ∗} (cid:0) Bαj [(j(cid:1)+ 1(cid:0)) ] Zj+1 = (j + 1)(cid:1) , − ∗ ∗ ∈ − ∗ − ∗ and ( (j + 2)) , (j + 2) / B− [( (j + 1)) ] = B [( (j + 1)) ] . ∗ ∩ ∗ (∪j+1) − ∗ ∅ αj ∈ (R.3) Bd B [(k + 1) ] B [j ] B [( j) ] = for every k Z − . α|k| αj ∈ αj ∗ ∩ j∗1(cid:54) ∅ (R.4) Assume that there exists k Z − such that B [(j + 1) ] B [k ] = and | | j 1 αj ∗ k k is maximal verifying these conditions. Then, B [(j + 1) ] is contained in ∗ \α{j ∗} ∗ ∩ one of the two connected components of B (k ) k when B [(j + 1) ] ∗ (cid:54) ∅ ∗ α|k| αj B [k ] = , and B [(j + 1) ] is contained in one of the two connected α|k| ∗α\j ∗ ∗ ∩ ∗ ∅ components of B (k ) B [k ] if B [(j + 1) ] B [k ] = (note that, k α|k| αj α|k| in this case, k must be negative). ∈ { − } (cid:83) ∗ (R.5) Let (cid:96) j, (j + 1) (recall that the ball B [(cid:96) ] has diameter α for these (cid:96) j two values of (cid:96) and only for them). ∗ ∈ ∗ ∗ ∩ ∗ ∅ [(R.5.i)]If (cid:96) / ∈ B [i ] then, B [(cid:96) ] B [i ] = for ∈ i Z∗j∈−1 i ∗ (cid:96) ∈ i every i Z − . If (cid:96) B [m ] for some m Z − such that | | j 1 m j 1 m is maximal with these properties, then ∗ ∩ ∗ ∅ ∈ | | ≥ | | (cid:54) ((ba)) (R.5.ii.1) B [(cid:96) ] B [i ] = for every i Z − such that i m , i = (cid:96) i j 1 m, and ##LTLine##

text: A

label: paragraph

text: SKEW-PRODUCT

text: WITHOUT

text: FIXED-CURVES

text: The

text: next

text: crucial

text: deﬁnition

text: ﬁxes

text: in

text: detail

text: all

text: quantities

text: and

text: objects

text: mentioned

text: above.

text: R

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text: and

text: ϕ0∗

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text: B

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text: α−0

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text: ##LTLine##


text: A SKEW-PRODUCT WITHOUT FIXED-CURVES 17 | | ≤ | | ≤ | | We have k r m < j and ∗ ∩ ∗ ⊃ ∗ ∩ ∗ (cid:54) ∅ Br [r ] Bk [k ] Bα|r| (r ) (cid:0)Bα|(cid:0)k| (k ) = (cid:1) (cid:1) ∗ ⊂ ∗ ∩ ∗ | | | | ∗ because B [(cid:96) ] B (r ) B (k ) . Then, by the claim, k < r and B [r ] is (cid:96) α|r| α|k| ∗ \ ∗ ∪ { ∗} r contained in a connected component of B (k ) Bd B [k ] k . The fact ∗ ⊂ ∗ \{ ∗} k (cid:16) ∗ ⊂ α(cid:17)|k| ∗ \{ ∗} that B [(cid:96) ] B (k ) k implies that B [r ] B (k ) k . By the part (cid:96) α|k| | |r α|k| R ∗ already proven and Deﬁnition 3.4(R.6.ii) for r < j(cid:16)we get that (cid:17)((cid:96) ) is contained in R ∗ \ (cid:20)(cid:20)(cid:20) ∗ R ∗ one of the two connected components of Int (r ) r and (r ) is contained R ∗ \ (cid:20)(cid:20)(cid:20) ∗ in one of the two connected components of Int (k ) k . This shows that (cid:12) Deﬁnition 3.4(R.6.ii) can be guaranteed. (cid:12) S \ ∗ Let us prove that (a) holds for j. Since the set 1 Z is residual, to prove that S \ ∗ ∈ PCG j (γj(cid:12)(cid:12), 1 Zj ) we ±have to sh∗o±w that γj S1\Zj∗∗ ±is continuous. Note that, from Deﬁnition 3.4(R.6.ii), a(cid:96) = ϕ(cid:96)∗((cid:96) αj) = γj−1((cid:96) αj)S. H\en∗ce, ⊃theS c\on∗tinuity of γj S1\Z∗ follows from the fact that γj−1 is continuous on 1 Zj−1 1 Zj and the j continuity of ϕ ∗ and ϕ − ∗ (Deﬁnition 3.1). This(cid:0)ends thje(cid:1) pro(cid:2)of o(fjt)he ﬁrst statement of the le(cid:3)mma and the ﬁrst statement ∈ { − } of (a). For every (cid:96) j + 1, (j + 1) , from By Deﬁnition 3.4(R.1,2) we get: ∗ ⊂ ∗ − − ∗ − γj−1 Bαj [(cid:96) ∗] γj−1((cid:96) ) 2 nj, γj−∗1((cid:96) ) + 2 nj − ∗ ⊃ − ∗ Bαj{[(cid:96)∗}] ∈is disjoin∗t ∩from∗ Bα⊂j [j ] an∗d∩Bαj∗−1 [( {j)∗}] Bαj [( j)) ], and (cid:96) / B [(cid:96) ] Z − B [(cid:96) ] Z = (cid:96) . αj j(cid:12)(cid:12) 1 αj (cid:12)(cid:12) j+1 So, from the deﬁnition of γ it follows that j γ ∗ = γ − ∗ j Bαj[(cid:96) ] j 1 Bαj[(cid:96) ] and, thus, (a) holds. Statement (c) follows immediately from Deﬁnition 3.4(R.6) and Remark 3.5(R.6). Next we pr(cid:16)ove(cid:12)(b,d,e,f,g)(cid:17). (cid:12) ∗ \ ∗ ∗ \{ ∗} (d) When n = j, we get B [(cid:96) ] Z = B [(cid:96) ] (cid:96) from Deﬁnition 3.4(R.2). ⊂αj R ∗ j αj Hence, Graph γ ∗ \ ∗ ((cid:96) ) by the deﬁnition of γ (Deﬁnition 3.4) and j Bαj[(cid:96) ] Zj j the deﬁnition of ϕ(cid:96)∗ (Deﬁnition 3.1)∈. ∗ \ ∗ Now assume that n > j and ﬁx θ B [(cid:96) ] Z . We have to show that the point ∈ R ∗ ∈ ∗ αj n | | ≤ (θ, γ (θ)) ((cid:96) ). If θ / B [m ] for every m such that j < m n then, by the n α|m| iterative use of the deﬁnition of γ for i = j + 1, j + 2, . . . , n (Deﬁnition 3.4) and i Deﬁnition 3.1, · · · ∈ R ∗ (θ, γ (θ)) = (θ, γ − (θ)) = = (θ, γ (θ)) = (θ, γ (θ)) = (θ, ϕ ∗(θ)) ((cid:96) ). n n 1 j+1 ∈j Z (cid:96) | | | | ≤ Otherwise, by Deﬁnition 3.4(R.2), there exists m such that (cid:96) < m n, ∈ ∗ \ ∗ ∈ ∗ | | | | ≤ θ B [m ] Z , and θ / B [s ] for every s such that m < s n. This im- α|m| ∗ (cid:0)∩n (cid:0) ∗ ⊃ (cid:1)α|s| ∗ ∩(cid:1) ∗ (cid:54) ∅ | | plies that B [(cid:96) ] B [m ] B [(cid:96) ] B [m ] = and m is maximal with these (cid:96) m αj | | α|m| ∗ properties. So, by the claim for j = m , B [m ] is contained in a connected compo- ∗ \ ∗ ∪ { ∗} m ∈ ∗ ∩ ∗ (cid:54) ∅ nent of B ((cid:96) ) Bd B [(cid:96) ] (cid:96) . Moreover, since θ B (m ) B [(cid:96) ] = , ∗ ⊂(cid:96) ∗ \{ ∗α}|(cid:96)| m αj B [m ] B [(cid:96) ] (cid:96) . Thus, by Deﬁnition 3.4(R.6.ii) and Remark 3.5(R.6.ii) m | | α|(cid:96)| R ∗ ⊂ R ∗ for j = m , (cid:96) replaced by m and k replaced by (cid:96), (m ) ((cid:96) ) and (d) follows | | from the part already proven by replacing (cid:96) by m and j by m . (cid:0) (cid:0) ∈ (cid:1)Z (cid:1) | | ≥ | | (cid:54) (g) By the claim we have that for every (cid:96), m such that (cid:96) m , (cid:96) = m ∗ ∩ ∗ (cid:54) ∅ | | | | ∗ and B [(cid:96) ] B [m ] = , it follows that (cid:96) > m , and B [(cid:96) ] is contained in a (cid:96) m ∗ \ ∗ ∪ { ∗}(cid:96) connected component of B (m ) Bd B [m ] m . Only it remains to ∗ ⊂ m ∗ \{ ∗} α|m| R ∗ show that if B [(cid:96) ] B (m ) m , then the region ((cid:96) ) is contained in one (cid:96) α|m|

text: A

text: SKEW-PRODUCT

text: WITHOUT

text: FIXED-CURVES

text: We

text: have

text: k

text: r

text: m

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text: αj

text: α|m|

text: properties.

text: So,

text: by

text: the

text: claim

text: for

text: j

text: m

text: B

text: [m

text: is

text: contained

text: in

text: a

text: connected

text: compo-

text: m

text: nent

text: of

text: B

text: Bd

text: B

text: Moreover,

text: since

text: θ

text: B

text: (m

text: B

text: m

text: αj

text: B

text: [m

text: B

text: Thus,

text: by

text: Deﬁnition

text: and

text: Remark

text: m

text: R

text: R

text: for

text: j

text: m

text: replaced

text: by

text: m

text: and

text: k

text: replaced

text: by

text: (m

text: and

text: (d)

text: follows

text: from

text: the

text: part

text: already

text: proven

text: by

text: replacing

text: by

text: m

text: and

text: j

text: by

text: m

text: (g)

text: By

text: the

text: claim

text: we

text: have

text: that

text: for

text: every

text: m

text: such

text: that

text: m

text: m

text: and

text: B

text: B

text: [m

text: it

text: follows

text: that

text: m

text: and

text: B

text: is

text: contained

text: in

text: a

text: m

text: connected

text: component

text: of

text: B

text: (m

text: Bd

text: B

text: [m

text: m

text: Only

text: it

text: remains

text: to

text: m

text: α|m|

text: R

text: show

text: that

text: if

text: B

text: B

text: (m

text: m

text: then

text: the

text: region

text: is

text: contained

text: in

text: one

text: α|m|


text: 24 LL. ALSEDA`, F. MAN˜OSAS, AND L. MORALES ∈ Z Lemma 4.1. The following statements hold for every i − (cid:12)≤ ≤ (cid:12) ≤ ∈ ∗ (a) 1(cid:12) m (θ) M (θ(cid:12)) 1 for every θ B [i ]. i i i (b) m and M are continuous. i i (c) mi Bα|i|[i∗] and Mi Bα|i|[i∗] are piecewise∈linear∗. \ ∗ (d) m (θ) = M (θ) = γ (θ) if and only if θ B [i ] B (i ) . i i |i| i α|i| Proof. It follows easily from Deﬁnition 3.1, the deﬁnition of a winged region and (cid:3) Lemma 3.6(b,f). (cid:91) (cid:91) ∈ Z Notice that, for every i , R ∗ R ∗ { } × (i ) = (i )θ = θ [m (θ), M (θ)]. i i ∈ ∗ ∈ ∗ θ B [i ] θ B [i ] i i⊂ I ∈ ∗ In what follows the interval [m (θ), M (θ)] , deﬁned for every θ B [i ] , will I i ∈i ∗ R ∗ { } × I i be denoted by . Clearly, for every θ B [i ] , (i )θ = θ . i,θ i i,θ By Deﬁnition 3.4(R.2) and Remark 3.5(R.2), ∗ \{ ∗} ∗ B [i ] i is disjoint from Z| |. i i  (cid:12) Hence, Lemmas 3(cid:12).6(a,d) and 3.10(c) can be summarized as:  γ| | ∗ \{ ∗} is continuous, (cid:96) B(cid:96) [∈(cid:96) I] (cid:96) ∈ ∗ \{ ∗} (2) γ (θ) for every θ B [(cid:96) ] (cid:96) , and |(cid:96)| { (cid:96),θ } (cid:96) ∈ ∗ \{ ∗} Aθ|(cid:96)| = (θ, γ|(cid:96)|(θ) for every θ B(cid:96) [(cid:96) ] (cid:96) ∈ { } (cid:0) (cid:1) for (cid:96) i, i + 1 . R ∗ −→ ∈ Z Now we deﬁne a family of continuous maps G : (i ) Ω with i , by i G (θ, x) = R (θ), g (θ, x) i ω i ∈ ∗ · I −→ I Also, for every θ B [i ] , we will denote the map g (θ, ): by g . i i i,θ i,θ To deﬁne the functions g , for clarity, we will consider separately two diﬀerent i,θ situations: • ≥ R ∗ R ∗ ∗ ∗ R ∗ i 0, when (i ) = (i ), B [i ] = B [i ] and G ( (i )) strictly R i∗ α|i| i contains the smaller box ((i + 1) ), and • ≤ − R ∗ R ∗ i 1, when G ( (i )) is strictly contained in the bigger box ((i+1) ). i ≥ We start by deﬁning g for i 0 in three diﬀerent ways, depending on the base ∈ ∗ i,θ R ∗ point θ B [i ]. In this deﬁnition, for simplicity we will use (i ) instead of R ∗ αi ∗ ∗ (i ) and B [i ] instead of B [i ] . α|i| i Notice that, by Deﬁnition 3.4(R.1) and Lemma 3.6(c), ≥ for every i 0 ∗ ⊂ ∗ ∗ ⊂ ∗ ⊂ ∗ (3) Bδi+1 ∗[i ] Bαi+1 (i ) and B∗ αi+1 [i ] Bδi (i ) Bαi (i ) , and γi−1(i ) = ai and γi((i + 1) ) = ai+1. (cid:0) (cid:1) ≥ Deﬁnition 4.2 (Deﬁnition of g for i 0). i ∈ ∗ ∗ ∗ − θ ∈ Bδi+1[i ∗]:\ ∗ g(cid:0)i,θ(x) := γi(((cid:0)i + 1) (cid:1))(cid:1)+ 2n2ni+i1(cid:0) γi−1(i ) (cid:0) x . (cid:1)(cid:1) θ B [i ] B (i ): we deﬁne g to be the unique piecewise aﬃne map αi+1 δi+1 i,θ (cid:0) I(cid:0) (cid:1)(cid:1) with two aﬃne pieces, deﬁned on , whose graph joins i,θ (cid:0) (cid:1) (cid:0) (cid:1) m (θ), M R (θ) with γ (θ), γ R (θ) , and i i+1 ω i i+1 ω this with the point M (θ), m R (θ) (in particu- i i+1 ω lar, g γ (θ) = γ R (θ) ), i,θ i i+1 ω ##LTLine##

text: LL.

text: ALSEDA`,

text: F.

text: MAN˜OSAS,

text: AND

text: L.

text: MORALES

text: Z

text: Lemma

text: The

text: following

text: statements

text: hold

text: for

text: every

text: i

label: list

text: (a)

text: m

text: (θ)

text: M

text: for

text: every

text: θ

text: B

text: [i

text: i

text: i

text: i

text: (b)

text: m

text: and

text: M

text: are

text: continuous.

text: i

text: i

text: (c)

text: mi

text: Bα|i|[i∗]

text: and

text: Mi

text: Bα|i|[i∗]

text: are

text: piecewise∈linear∗.

text: (d)

text: m

text: (θ)

text: M

text: (θ)

text: γ

text: (θ)

text: if

text: and

text: only

text: if

text: θ

text: B

text: [i

text: B

text: (i

text: i

text: i

text: |i|

text: i

text: α|i|

text: Proof.

text: It

text: follows

text: easily

text: from

text: Deﬁnition

text: the

text: deﬁnition

text: of

text: a

text: winged

text: region

text: and

text: Lemma

text: Z

text: Notice

text: that,

text: for

text: every

text: i

text: R

text: R

text: (i

text: (i

text: )θ

text: θ

text: [m

text: (θ),

text: M

text: (θ)].

text: i

text: i

text: θ

text: B

text: [i

text: θ

text: B

text: [i

text: i

text: i⊂

text: I

text: In

text: what

text: follows

text: the

text: interval

text: [m

text: (θ),

text: M

text: (θ)]

text: deﬁned

text: for

text: every

text: θ

text: B

text: [i

text: will

text: I

text: i

text: ∈i

text: R

text: I

text: i

text: be

text: denoted

text: by

text: Clearly,

text: for

text: every

text: θ

text: B

text: [i

text: (i

text: )θ

text: θ

text: i,θ

text: i

text: i,θ

text: By

text: Deﬁnition

text: and

text: Remark

text: B

text: [i

text: i

text: is

text: disjoint

text: from

text: Z|

text: i

text: i

text: Hence,

text: Lemmas

text: and

text: can

text: be

text: summarized

text: as:

text: γ|

text: is

text: continuous,

text: I]

text: γ

text: (θ)

text: for

text: every

text: θ

text: B

text: and

text: (θ,

text: for

text: every

text: θ

text: for

text: i,

text: i

text: R

text: Z

text: Now

text: we

text: deﬁne

text: a

text: family

text: of

text: continuous

text: maps

text: G

text: (i

text: Ω

text: with

text: i

text: by

text: i

text: G

text: (θ,

text: x)

text: R

text: (θ),

text: g

text: (θ,

text: x)

text: i

text: ω

text: i

text: I

text: I

text: Also,

text: for

text: every

text: θ

text: B

text: [i

text: we

text: will

text: denote

text: the

text: map

text: g

text: (θ,

text: by

text: g

text: i

text: i

text: i,θ

text: i,θ

text: To

text: deﬁne

text: the

text: functions

text: g

text: for

text: clarity,

text: we

text: will

text: consider

text: separately

text: two

text: diﬀerent

text: i,θ

text: situations:

text: R

text: R

text: R

text: i

text: when

text: (i

text: (i

text: B

text: [i

text: B

text: [i

text: and

text: G

text: (i

text: strictly

text: R

text: i∗

text: α|i|

text: i

text: contains

text: the

text: smaller

text: box

text: ((i

text: and

text: R

text: R

text: i

text: when

text: G

text: (i

text: is

text: strictly

text: contained

text: in

text: the

text: bigger

text: box

text: i

text: We

text: start

text: by

text: deﬁning

text: g

text: for

text: i

text: in

text: three

text: diﬀerent

text: ways,

text: depending

text: on

text: the

text: base

text: i,θ

text: R

text: point

text: θ

text: B

text: [i

text: In

text: this

text: deﬁnition,

text: for

text: simplicity

text: we

text: will

text: use

text: (i

text: instead

text: of

text: R

text: αi

text: (i

text: and

text: B

text: [i

text: instead

text: of

text: B

text: [i

text: α|i|

text: i

text: Notice

text: that,

text: by

text: Deﬁnition

text: and

text: Lemma

text: for

text: every

text: i

text: ∗[i

text: (i

text: and

text: B∗

text: [i

text: Bδi

text: (i

text: Bαi

text: (i

text: and

text: ai

text: and

text: γi((i

text: Deﬁnition

text: (Deﬁnition

text: of

text: g

text: for

text: i

text: i

text: θ

text: x

text: θ

text: B

text: [i

text: B

text: (i

text: we

text: deﬁne

text: g

text: to

text: be

text: the

text: unique

text: piecewise

text: aﬃne

text: map

text: i,θ

text: with

text: two

text: aﬃne

text: pieces,

text: deﬁned

text: on

text: whose

text: graph

text: joins

text: i,θ

text: m

text: (θ),

text: M

text: R

text: (θ)

text: with

text: γ

text: (θ),

text: γ

text: R

text: (θ)

text: and

text: i

text: ω

text: i

text: ω

text: this

text: with

text: the

text: point

text: M

text: (θ),

text: m

text: R

text: (θ)

text: (in

text: particu-

text: i

text: ω

text: lar,

text: g

text: γ

text: (θ)

text: γ

text: R

text: (θ)

text: i,θ

text: i

text: ω

text: ##LTLine##


text: A SKEW-PRODUCT WITHOUT FIXED-CURVES 33 ≥ In any case, by Deﬁnition 3.4(R(cid:40).2) and Remark 3.5(R.2) with j = i when i 0 and − (cid:96) = (j + 1) = i + 1 when i < 0, and Lemma 3.6(a), ∗ ≥ ∈ Z when i 0, and Rω(θ) / i∗ Z| |− when i < 0, i+1 1 | | − ∗ ⊂ ∗ and γ (R (θ)) is well deﬁned for (cid:96) = 0, 1, . . . , i + 1 1 (recall that Z Z (cid:96) ω ≥ m m+1 for every m 0). Now, assume by way of contradiction that (cid:54) ∈ {| | | | | | − } γ(cid:96) (Rω(θ)) = γ(cid:96)−1 (Rω(θ)) for some (cid:96) k + 1 + 1, k + 1 + 2, . . . , i + 1 1 , ≥ and (cid:96) is minimal with this property (observe that (cid:96) 1). By the deﬁnition of the map γ (Deﬁnition 3.4), (cid:96) ∈ ∗ ∈ { − − } R (θ) B ((q + 1) ) with q (cid:96) 1, ((cid:96) + 1) ω α(cid:96) ∈ ∗ and, hence, θ B (q ) . | | α(cid:96)≤ | | − ≤ − Since k + 1 + 1 (cid:96) < i + 1 , when q = ((cid:96) + 1) 2, | | ≤ − ≤ | | ∗ − ∗ ∗ k + 1 + 2 q i + 1 and B (q ) = B− (( ((cid:96) + 1)) ) = B (q ) . α(cid:96) ((cid:96)+1) q − ≥ | | ≤ ≤ | | − Otherwise, when q = (cid:96) 1 0, k + 1 q i + 1 2 and ∗ ⊂ − ∗ − ∗ ∗ Bα(cid:96) (q ) Bα(cid:96)−1 (((cid:96) 1) ) = B(cid:96)−1 (((cid:96) 1) ) = Bq (q ) , by Deﬁnition 3.4(R.1). ∗ (cid:32) ∗ (cid:32) ∗ Next we want to use Lemma 3.6(g) to show that B [i ] B [q ] B [k ] . To | | | | | | i q k this end we have to compare q with i and k . ∗ ∩ ∗ (cid:54) ∅ Notice B [q ] B [k ] = because q k ∈ ∗ ∩ ∗ ⊂ ∗ ∩ ∗ θ B (q ) B [i ] B (q ) B [k ] . q i q k ≥ | | ≥ | | | | | | ≥ | | | | | | If k 0, q k + 1 > k . When k, q < 0, q k + 1 +2 = k +1 > k . If k < 0 ≥ | | ≥ | | | |− | |− − and q 0, q = q k + 1 = k 1. If q = k 1 (that is, k = (q+1)), as above, ∗ ∩ ∗ ∅ by Deﬁnition 3.4(R.2) with j = q we get B [k ] B [q ] = ; a contradiction. So, | | | | | | | | ≤ k q | | ≥ | | q > k unless q = k and k < 0 q. Summarizing, we have shown that q k (cid:54) | | | | ∗ (cid:32) ∗ and q = k. Then, from Lemma 3.6(g) we get that q > k and B [q ] B [k ] . ∗ ∗ q k Now we will study the relation of B [q ] with the box B [i ] . From above we ∗ ∩ ∗ (cid:54) ∅ | q| ≤ | | | | − i ≥ get that B [q ] B [i ] = . If i < 0, q i + 1 = i 1. When q, i 0, we have | | ≤ |q | − i | | − ≥ | | ≤ | | | | q = q i + 1 2 = i 1. If i 0 and q < 0, q i + 1 = i + 1. ≥ − − Assume that i 0 an(cid:0)d q = (cid:1)(i + 1) < 0. In this case, additionally, q = ((cid:96) + 1) ≥ and, thus, i = (cid:96) 1. Then, ∈ ∗ ∗ ∗ R (θ) R B [i ] = R (B [i ]) = B [(i + 1) ] , and ω ∈ ω i ∗ ω αi− ∗ ⊂ αi − ∗ R (θ) B ((q + 1) ) = B (( i) ) B− (( i) ) , ω α(cid:96) αi i | | | | which is a contradiction by Deﬁnition 3.4(R.2). Summarizing, q < i unless | | | | ≤ | | ≤ | | (cid:54) q = i and q < 0 i (that is, q i and q = i). Then, again by Lemma 3.6(g), | | | | ∗ (cid:32) ∗ (cid:32) ∗ q < i and B [i ] B [q ] B [k ] . So, as before, i q k ≥ ≥ m = depth(i) depth(q) + 1 depth(k) + 2 = m + 1; a contradiction. This ends the proof of (a). ∈ ∗ \ ∗ Now we assume that θ B [i ] B (i ) . By Lemmas 3.6(e) and 4.1(d), i α|i| I { } { } { } γ (θ) = γ (θ) and = γ (θ) = γ (θ) = γ (θ) . |i| |i|−1 i,θ |i| |i|−1 |k| ∗ ∩ B ∅ (cid:54) ∗ On the other hand, by Lemma 5.3(b), D = which implies that θ = k ∗ ∈ ∗ ∈ ∗ \ ∗m−⊂1 B m because k D and θ B [i ] B (i ) . So, by (2), m−1 i α|i| m I { } { } ⊂ I = γ (θ) = γ (θ) . i,θ |i| |k| k,θ

text: A

text: SKEW-PRODUCT

text: WITHOUT

text: FIXED-CURVES

text: In

text: any

text: case,

text: by

text: Deﬁnition

text: and

text: Remark

text: with

text: j

text: i

text: when

text: i

text: and

text: (j

text: i

text: when

text: i

text: and

text: Lemma

text: Z

text: when

text: i

text: and

text: Rω(θ)

text: i∗

text: Z|

text: when

text: i

text: and

text: γ

text: (R

text: (θ))

text: is

text: well

text: deﬁned

text: for

text: i

text: (recall

text: that

text: Z

text: Z

text: ω

text: m

text: for

text: every

text: m

text: Now,

text: assume

text: by

text: way

text: of

text: contradiction

text: that

text: (Rω(θ))

text: (Rω(θ))

text: for

text: some

text: k

text: k

text: i

text: and

text: is

text: minimal

text: with

text: this

text: property

text: (observe

text: that

text: By

text: the

text: deﬁnition

text: of

text: the

text: map

text: γ

text: (Deﬁnition

text: R

text: (θ)

text: B

text: ((q

text: with

text: q

text: ω

text: and,

text: hence,

text: θ

text: B

text: (q

text: Since

text: k

text: i

text: when

text: q

text: k

text: q

text: i

text: and

text: B

text: (q

text: B−

text: B

text: (q

text: q

text: Otherwise,

text: when

text: q

text: k

text: q

text: i

text: and

text: (q

text: Bq

text: (q

text: by

text: Deﬁnition

text: Next

text: we

text: want

text: to

text: use

text: Lemma

text: to

text: show

text: that

text: B

text: [i

text: B

text: [q

text: B

text: [k

text: To

text: i

text: q

text: k

text: this

text: end

text: we

text: have

text: to

text: compare

text: q

text: with

text: i

text: and

text: k

text: Notice

text: B

text: [q

text: B

text: [k

text: because

text: q

text: k

text: θ

text: B

text: (q

text: B

text: [i

text: B

text: (q

text: B

text: [k

text: q

text: i

text: q

text: k

text: If

text: k

text: q

text: k

text: k

text: When

text: k,

text: q

text: q

text: k

text: k

text: k

text: If

text: k

text: and

text: q

text: q

text: q

text: k

text: k

text: If

text: q

text: k

text: (that

text: is,

text: k

text: as

text: above,

text: by

text: Deﬁnition

text: with

text: j

text: q

text: we

text: get

text: B

text: [k

text: B

text: [q

text: a

text: contradiction.

text: So,

text: k

text: q

text: q

text: k

text: unless

text: q

text: k

text: and

text: k

text: q.

text: Summarizing,

text: we

text: have

text: shown

text: that

text: q

text: k

text: and

text: q

text: k.

text: Then,

text: from

text: Lemma

text: we

text: get

text: that

text: q

text: k

text: and

text: B

text: [q

text: B

text: [k

text: q

text: k

text: Now

text: we

text: will

text: study

text: the

text: relation

text: of

text: B

text: [q

text: with

text: the

text: box

text: B

text: [i

text: From

text: above

text: we

text: q|

text: i

text: get

text: that

text: B

text: [q

text: B

text: [i

text: If

text: i

text: q

text: i

text: i

text: When

text: q,

text: i

text: we

text: have

text: |q

text: i

text: q

text: q

text: i

text: i

text: If

text: i

text: and

text: q

text: q

text: i

text: i

text: Assume

text: that

text: i

text: q

text: In

text: this

text: case,

text: additionally,

text: q

text: and,

text: thus,

text: i

text: Then,

text: R

text: (θ)

text: R

text: B

text: [i

text: R

text: (B

text: [i

text: B

text: [(i

text: and

text: ω

text: ω

text: i

text: ω

text: αi−

text: αi

text: R

text: (θ)

text: B

text: ((q

text: B

text: i)

text: B−

text: i)

text: ω

text: αi

text: i

text: which

text: is

text: a

text: contradiction

text: by

text: Deﬁnition

text: Summarizing,

text: q

text: i

text: unless

text: q

text: i

text: and

text: q

text: i

text: (that

text: is,

text: q

text: i

text: and

text: q

text: i).

text: Then,

text: again

text: by

text: Lemma

text: q

text: i

text: and

text: B

text: [i

text: B

text: [q

text: B

text: [k

text: So,

text: as

text: before,

text: i

text: q

text: k

text: m

text: depth(i)

text: depth(q)

text: depth(k)

text: m

text: a

text: contradiction.

text: This

text: ends

text: the

text: proof

text: of

text: (a).

text: Now

text: we

text: assume

text: that

text: θ

text: B

text: [i

text: B

text: (i

text: By

text: Lemmas

text: and

text: i

text: α|i|

text: I

text: γ

text: (θ)

text: γ

text: (θ)

text: and

text: γ

text: (θ)

text: γ

text: (θ)

text: γ

text: (θ)

text: |i|

text: i,θ

text: |i|

text: |k|

text: B

text: On

text: the

text: other

text: hand,

text: by

text: Lemma

text: D

text: which

text: implies

text: that

text: θ

text: k

text: ∗m−⊂1

text: B

text: m

text: because

text: k

text: D

text: and

text: θ

text: B

text: [i

text: B

text: (i

text: So,

text: by

text: i

text: α|i|

text: m

text: I

text: I

text: γ

text: (θ)

text: γ

text: (θ)

text: i,θ

text: |i|

text: |k|

text: k,θ


text: A SKEW-PRODUCT WITHOUT FIXED-CURVES 35 ∈ ∗ Now we prove (d) iteratively. Fix ν Bα| |((i(cid:96)) ) . By (b) i(cid:96) ∈ ∗ \ ∗ ⊂ ∗ \ ∗ ν Bim+1 ((im+1) ) Bα|im+1| [(im+1) ] Bim ((im) ) Bα|im| [(im) ] provided that (cid:96) = led (θ, m) > m + 1. Hence, by Lemmas 4.1(d) and 5.4, · · · γ| |(ν) = γ| | (ν) = = γ| |(ν), and imI { im +1 } { im+1} I im,ν = γ|im|(ν) = γ|im+1|(ν) = im+1,ν. By iterating this argument we get, · · · I I γ|im|(ν) = γ|im|+1(ν) = = γ|i(cid:96)−1|(ν) and im,ν = i(cid:96)−1,ν. Again by (b) and Lemmas 4.1(d) and 5.4, (cid:16) (cid:17) · · · I I γ|im|(ν) = γ|im|+1(ν) = = γ|i(cid:96)|(ν) and im,ν = i(cid:96),ν ∈ ∗ when ν Bd Bα| |[(i(cid:96)) ] and, otherwise, i(cid:96) · · · I ⊂ I γ|im|(ν) = γ|im|+1(ν) = = γ|i(cid:96)−1|(ν) and im,ν i(cid:96),ν. (cid:3) Equipped with above results and deﬁnition we are going to deﬁne two maps, analogous to the maps m and M , on the wings of the negative boxes. i i ∈ Z Deﬁnition 6.2. For every m + we deﬁn(cid:91)e { ∈ WDB } ⊂ Z WFD := b (θ, led(θ, m)) : θ , m m WIB WDB ∗ m := In(cid:91)t( (cid:0) m) = B(cid:1)α|i| (i ) , ∈ i WFD m WB ∗ \ ∗ m := ∈(cid:91) Bi(cid:0) [i ] B(cid:1)α|i| (i ) , and i D m i<0 EB ∗ ⊂ B := Bd B [i ] . m ∈ i m i D m By Lemma 6.1(a,c), WFD is well deﬁned and m WIB ⊂ WDB ⊂ B \B ⊂ WB . m m m m m Consequently, B B ∪ WB (cid:40) = . m WmB −→ mI WB −→ I Then, we can deﬁne functions τ : and λ : as follows: m m m m ∈ WIB τ (θ) := (cid:40)Mb (θ,led(θ,m))(θ) if θ m, m γ| |(θ) otherwise, b(θ,m) ∈ WIB m (θ) if θ , λ (θ) := b (θ,led(θ,m)) m m γ| |(θ) otherwise. b(θ,m) Clearly, by Lemmas 4.1(a) and 3.6(b), − ≤ ≤ ≤ 1 λ (θ) τ (θ) 1 m m ∈ WB for every θ . So, we can deﬁne m IW ⊂ := [λ (θ), τ (θ)] [0, 1]. m,θ m m The next lemmas will help us in the deﬁnition and study of the maps T . m

text: A

text: SKEW-PRODUCT

text: WITHOUT

text: FIXED-CURVES

text: Now

text: we

text: prove

text: (d)

text: iteratively.

text: Fix

text: ν

text: Bα|

text: By

text: (b)

text: ν

text: Bim

text: ((im)

text: Bα|im|

text: [(im)

text: provided

text: that

text: led

text: (θ,

text: m)

text: m

text: Hence,

text: by

text: Lemmas

text: and

text: γ|

text: |(ν)

text: γ|

text: (ν)

text: γ|

text: |(ν),

text: and

text: imI

text: im

text: I

text: im,ν

text: γ|im|(ν)

text: By

text: iterating

text: this

text: argument

text: we

text: get,

text: I

text: I

text: γ|im|(ν)

text: and

text: im,ν

text: Again

text: by

text: (b)

text: and

text: Lemmas

text: and

text: I

text: I

text: γ|im|(ν)

text: and

text: im,ν

text: when

text: ν

text: Bd

text: Bα|

text: and,

text: otherwise,

text: I

text: I

text: γ|im|(ν)

text: and

text: im,ν

text: Equipped

text: with

text: above

text: results

text: and

text: deﬁnition

text: we

text: are

text: going

text: to

text: deﬁne

text: two

text: maps,

text: analogous

text: to

text: the

text: maps

text: m

text: and

text: M

text: on

text: the

text: wings

text: of

text: the

text: negative

text: boxes.

text: i

text: i

text: Z

text: Deﬁnition

text: For

text: every

text: m

text: we

text: WDB

text: Z

text: WFD

text: b

text: (θ,

text: led(θ,

text: m))

text: θ

text: m

text: m

text: WIB

text: WDB

text: m

text: m)

text: (i

text: i

text: WFD

text: m

text: WB

text: m

text: [i

text: (i

text: and

text: i

text: D

text: m

text: EB

text: B

text: Bd

text: B

text: [i

text: m

text: i

text: m

text: i

text: D

text: m

text: By

text: Lemma

text: WFD

text: is

text: well

text: deﬁned

text: and

text: m

text: WIB

text: WDB

text: B

text: \B

text: WB

text: m

text: m

text: m

text: m

text: m

text: Consequently,

text: B

text: B

text: WB

text: m

text: WmB

text: mI

text: WB

text: I

text: Then,

text: we

text: can

text: deﬁne

text: functions

text: τ

text: and

text: λ

text: as

text: follows:

text: m

text: m

text: m

text: m

text: WIB

text: τ

text: (θ)

text: (θ,led(θ,m))(θ)

text: if

text: θ

text: m,

text: m

text: γ|

text: |(θ)

text: otherwise,

text: b(θ,m)

text: WIB

text: m

text: (θ)

text: if

text: θ

text: λ

text: (θ)

text: b

text: (θ,led(θ,m))

text: m

text: m

text: γ|

text: |(θ)

text: otherwise.

text: b(θ,m)

text: Clearly,

text: by

text: Lemmas

text: and

text: λ

text: (θ)

text: τ

text: (θ)

text: m

text: m

text: WB

text: for

text: every

text: θ

text: So,

text: we

text: can

text: deﬁne

text: m

text: IW

text: [λ

text: (θ),

text: τ

text: (θ)]

text: m,θ

text: m

text: m

text: The

text: next

text: lemmas

text: will

text: help

text: us

text: in

text: the

text: deﬁnition

text: and

text: study

text: of

text: the

text: maps

text: T

text: m


text: 38 LL. ALSEDA`, F(cid:101). MAN˜OSAS, AND L. MORALES (cid:101) ∈ WIB(cid:101) (cid:101) Observe that, by Lemma 5.2(b), θ / . Hence, by (13) and Lemma 6.1(c,d), m (cid:12) (cid:12) (cid:101) (cid:12) (cid:101) (cid:12) λ (θ) = γ (θ) = m (θ(cid:12)). (cid:12) m |i| (cid:96) | | ≤ − If (cid:96) < N, t(cid:12)(cid:12)hen (cid:101)dS1(θ, ν) <(cid:12)(cid:12)δ δ(cid:96) and, by (15), m(cid:96)(θ) m(cid:96)(ν) < ε/2. Other- wise, by Lemm(cid:12)a 3.6(f), (cid:12) (cid:12)(cid:12)(cid:12) m(cid:96)(cid:101)(θ) − m(cid:96)(ν(cid:12)(cid:12)(cid:12)) < diam (R((cid:96)∗)) ≤ 2−|(cid:96)| ≤ 2−N < ε/2. − (cid:12) (cid:12) (cid:12) (cid:12) In any case, m(cid:96)(θ) m(cid:96)(ν) <(cid:12)(cid:12)ε/2. Thus, ag(cid:101)a(cid:12)(cid:12)in b(cid:12)(cid:12)y (13(cid:101)) and (14)(cid:12)(cid:12), (cid:12) (cid:12) (cid:12) (cid:12) | − | ≤ (cid:12) − (cid:101) (cid:12) (cid:12) (cid:101) − (cid:12) λ (θ) λ (ν) (cid:12)λ (θ) λ (θ)(cid:12) + (cid:12)λ (θ) λ (ν)(cid:12) m m m m m m − − = γ (θ) γ (θ) + m (θ) m (ν) < ε. |i| |i| (cid:96) (cid:96) (cid:3) 7. A Cauchy sequence of skew products. Proof of Theorem A In this section prove Theorem A. To do this we inductively construct a Cauchy { }∞ ⊂ S sequence T (Ω) that gives the function T from Theorem A as a limit. m m={0 }∞ ⊂ S The sequence T (Ω) is deﬁned so that m m=0 T (θ, x) = (R (θ), f (θ, x)) m ω m −→ I and f : Ω is continuous in both variables. To build these functions we will m R ∗ −→ ∈ Z use the auxiliary functions G : (i ) Ω with i from Section . The maps · i f (θ, ) will also be denoted as f , and will be deﬁned non-increasing, and such m − − m,θ ∈ S that f (2) = 2 and f ( 2) = 2 for every θ 1. m,θ m,θ To make more evident the strategy of the construction of this sequence of maps we will separate several cases, and we will state without proofs the results that study these maps. After establishing all the deﬁnitions and results related to the construc- { }∞ tion of the sequence T without having been distracted by the technicalities m m=0 involving the proofs, we will proceed to provide the missing proofs. More precisely, we will start by deﬁning the map T and stating without proof the proposition that 0 summarizes the necessary properties of this map. Next we will inductively deﬁne { }∞ ⊂ S the maps T (Ω) and state without proof the proposition that establishes m m=1 { }∞ the properties of the whole sequence T . m m=0 Then, as we have said, we prove Theorem A and in the next three sections we will provide all pending proofs. C I I I In what follows ( , ) will denote the class of all continuous maps from to itself. C I I (cid:107)·(cid:107) C I I (cid:107)·(cid:107) We endow ( , ) with the supremum metric denoted by so that ( ( , ), ) is a complete metric space. Next we deﬁne the map T . 0 ∈ B Deﬁnition 7.1 (The map T ). Assume ﬁrst that θ and let i = b (θ, 0) (that is θ ∈ Bi [i∗]). In this case w0e set: 0 ∈ I g (x) if x , f (x) = gii,,θθ(mi(θ))−2(x + 2) + 2 if x ∈ [−i,θ2, m (θ)], 0,θ mi(θ)+2 i gi,θ(Mi(θ−))+2(x − 2) − 2 if x ∈ [M (θ), 2]. Mi(θ) 2 i ∈ S \B If θ 1 then we deﬁne f to be the unique piecewise aﬃne map with two 0 0,θ − aﬃne pieces whose graph joins the point ( 2, 2) with (0, γ(R (θ))), and this with − ω the point (2, 2). ##LTLine## ##LTLine##

text: LL.

text: ALSEDA`,

text: MAN˜OSAS,

text: AND

text: L.

text: MORALES

text: Observe

text: that,

text: by

text: Lemma

text: θ

text: Hence,

text: by

text: and

text: Lemma

text: m

text: λ

text: (θ)

text: γ

text: (θ)

text: m

text: m

text: |i|

text: If

text: N,

text: ν)

text: and,

text: by

text: Other-

text: wise,

text: by

text: diam

text: In

text: any

text: case,

text: Thus,

text: and

text: λ

text: (θ)

text: λ

text: (ν)

text: (θ)

text: λ

text: (θ)

text: λ

text: m

text: m

text: m

text: m

text: m

text: m

text: γ

text: (θ)

text: γ

text: (θ)

text: m

text: (θ)

text: m

text: (ν)

text: ε.

text: |i|

text: |i|

text: A

text: Cauchy

text: sequence

text: of

text: skew

text: products.

text: Proof

text: of

text: Theorem

text: A

text: In

text: this

text: section

text: prove

text: Theorem

text: A.

text: To

text: do

text: this

text: we

text: inductively

text: construct

text: a

text: Cauchy

text: S

text: sequence

text: T

text: (Ω)

text: that

text: gives

text: the

text: function

text: T

text: from

text: Theorem

text: A

text: as

text: a

text: limit.

text: m

text: S

text: The

text: sequence

text: T

text: (Ω)

text: is

text: deﬁned

text: so

text: that

text: m

text: T

text: (θ,

text: x)

text: (R

text: (θ),

text: f

text: (θ,

text: x))

text: m

text: ω

text: m

text: I

text: and

text: f

text: Ω

text: is

text: continuous

text: in

text: both

text: variables.

text: To

text: build

text: these

text: functions

text: we

text: will

text: m

text: R

text: Z

text: use

text: the

text: auxiliary

text: functions

text: G

text: (i

text: Ω

text: with

text: i

text: from

text: Section

text: The

text: maps

text: i

text: f

text: (θ,

text: will

text: also

text: be

text: denoted

text: as

text: f

text: and

text: will

text: be

text: deﬁned

text: non-increasing,

text: and

text: such

text: m

text: m,θ

text: S

text: that

text: f

text: and

text: f

text: for

text: every

text: θ

text: m,θ

text: m,θ

text: To

text: make

text: more

text: evident

text: the

text: strategy

text: of

text: the

text: construction

text: of

text: this

text: sequence

text: of

text: maps

text: we

text: will

text: separate

text: several

text: cases,

text: and

text: we

text: will

text: state

text: without

text: proofs

text: the

text: results

text: that

text: study

text: these

text: maps.

text: After

text: establishing

text: all

text: the

text: deﬁnitions

text: and

text: results

text: related

text: to

text: the

text: construc-

text: tion

text: of

text: the

text: sequence

text: T

text: without

text: having

text: been

text: distracted

text: by

text: the

text: technicalities

text: m

text: involving

text: the

text: proofs,

text: we

text: will

text: proceed

text: to

text: provide

text: the

text: missing

text: proofs.

text: More

text: precisely,

text: we

text: will

text: start

text: by

text: deﬁning

text: the

text: map

text: T

text: and

text: stating

text: without

text: proof

text: the

text: proposition

text: that

text: summarizes

text: the

text: necessary

text: properties

text: of

text: this

text: map.

text: Next

text: we

text: will

text: inductively

text: deﬁne

text: S

text: the

text: maps

text: T

text: (Ω)

text: and

text: state

text: without

text: proof

text: the

text: proposition

text: that

text: establishes

text: m

text: the

text: properties

text: of

text: the

text: whole

text: sequence

text: T

text: m

text: Then,

text: as

text: we

text: have

text: said,

text: we

text: prove

text: Theorem

text: A

text: and

text: in

text: the

text: next

text: three

text: sections

text: we

text: will

text: provide

text: all

text: pending

text: proofs.

text: C

text: I

text: I

text: I

text: In

text: what

text: follows

text: will

text: denote

text: the

text: class

text: of

text: all

text: continuous

text: maps

text: from

text: to

text: itself.

text: C

text: I

text: I

text: C

text: I

text: I

text: We

text: endow

text: with

text: the

text: supremum

text: metric

text: denoted

text: by

text: so

text: that

text: is

text: a

text: complete

text: metric

text: space.

text: Next

text: we

text: deﬁne

text: the

text: map

text: T

text: B

text: Deﬁnition

text: (The

text: map

text: T

text: Assume

text: ﬁrst

text: that

text: θ

text: and

text: let

text: i

text: b

text: (θ,

text: (that

text: is

text: θ

text: Bi

text: [i∗]).

text: In

text: this

text: case

text: set:

text: I

text: g

text: (x)

text: if

text: x

text: f

text: (x)

text: if

text: x

text: m

text: (θ)],

text: i

text: if

text: x

text: [M

text: (θ),

text: Mi(θ)

text: i

text: S

text: \B

text: If

text: θ

text: then

text: we

text: deﬁne

text: f

text: to

text: be

text: the

text: unique

text: piecewise

text: aﬃne

text: map

text: with

text: two

text: aﬃne

text: pieces

text: whose

text: graph

text: joins

text: the

text: point

text: with

text: γ(R

text: (θ))),

text: and

text: this

text: with

text: ω

text: the

text: point

text: ##LTLine##

text: ##LTLine##


text: 40 LL. ALSEDA`, F. MAN˜OSAS, AND L. MORALES ∈ S · Theorem 7.5. There exists a map T (Ω) with f(θ, ) non-increasing for every ∈ S θ 1, such that T permutes the upper and lower circles of Ω (thus having a periodic orbit of period two of curves), and there exists a connected pseudo-curve ⊂ A Ω which does not contain any arc of a curve such that T(A) = A and there does not exist any T-invariant curve. Proof. By Propositions 7.3 and 7.4, there exists a map ∈ S T(θ, x) = (R (θ), f(θ, x)) = (R (θ), lim f (θ, x)) (Ω) ω ω →∞ m m · ∈ S with f(θ, ) non-increasing for every θ 1 such that T permutes the upper and − − lower circles of Ω (that is, f(θ, 2) = 2 and f(θ, 2) = 2). As the connected set A we take the one given by Proposition 3.12 (and Deﬁnition 3.9). To end the proof of the theorem we need to show that T(A) = A, since this already implies that there does not exist any T-invariant curve. To see it, assume by way of contradiction that there exists an invariant curve and denote its graph by B. Since B is the graph of a (continuous) curve, it is compact and connected. On \ the other hand, let Ω and Ω− be the two connected components of Ω A from the + · proof of Proposition 3.12. The facts that T(A) = A, f(θ, ) is decreasing for every ∈ S θ 1, and T permutes the upper and lower circles of Ω imply that T(Ω ) = Ω− (cid:42) (cid:42) + and T(Ω−) = Ω . Hence, by the invariance of B, B Ω and B Ω−. The + ∈ ∩ + connectivity of A and B imply that there exists (θ, x) A B. Consequently, { ∈ Z } ⊂ B = Tn(θ, x) : n + A; (cid:0) (cid:1) (cid:12) (cid:12) a contradiction because A does not contain any arc of a cu(cid:12)rve. (cid:12) So, only it remains to prove that T(A) = A. By using Proposition 7.3(c) and ∗ ∗ (cid:0) (cid:1) Lemma 3.11(b) we get that T Ai = A(i+1) , and T ∗ = T ∗ for every ∈ Z ≥ ∈ m k Ai m Ai k, m +, k m and i D . Consequently, by the deﬁnition of the map T we ∗ ∗ m ∈ Z (cid:20)(cid:20)(cid:20) ∗ (cid:20)(cid:20)(cid:20) ∗ have, T(Ai ) = A(i+1) for every i or, equivalently, T A O (ω) = A O (ω). ∈ S \ ∗ ∗ S Now we consider Aθ with θ 1 O (ω). Since O (ω) is dense in 1, there exists a { }∞ ⊂ (cid:20)(cid:20)(cid:20) ∗ sequence (θ , x ) A O (ω) such that lim →∞ θ = θ. By the compacity of A n n n=0 n n { }∞ we can assume (by taking a convergent subsequence, if necessary) that (θ , x ) ∈ n n n=0 is convergent to a point (θ, x) A. By Lemma 3.11(c), Aθ = (θ, x) (and x = γ(θ)). ∈ On the other hand, by the part of the statement already proven, T(θ , x ) A for n n every n. Hence, by the continuity of T and the compacity of A, ∈ T(θ, x) = (Rω(θ), f(θ, x)) = l→im∞ T(θn, xn) ARω(θ). n ∈ ∗ (cid:16) (cid:0) (cid:1)(cid:17) (cid:0) ∈ ∗(cid:1) Since θ / O (ω) we have that R (θ) / O (ω) and, again by Lemma 3.11(c), ω ∈ S \ ∗ ARω(θ) consists of a unique point. Hence, T(Aθ) = ARω(θ) for every θ 1 O (ω). (cid:20)(cid:20)(cid:20) S \ ∗ (cid:20)(cid:20)(cid:20) S \ ∗ Equivalently, T A 1 O (ω) = A 1 O (ω) . This ends the proof of the theorem. (cid:3) 8. Proof of Proposition 7.3 in the case m = 0 This section is devoted to prove Proposition 7.3 for m = 0; that is, to study the map T . It is the ﬁrst technical counterpart of Section . 0 To prove Proposition 7.3 for T we will need some more notation and a technical 0 lemma. S × I Given a skew product F(θ, x) = (R (θ), ζ(θ, x) from Ω = 1 to itself we ω S −→ C I I · deﬁne the ﬁbre map function of F, ﬁb(F): 1 ( , ) by ﬁb(F)(θ) := ζ(θ, ). A · simple exercise shows that F is continuous if and only if ζ(θ, ) is continuous for ∈ S every θ 1, and ﬁb(F) is continuous. ##LTLine##

text: LL.

text: ALSEDA`,

text: F.

text: MAN˜OSAS,

text: AND

text: L.

text: MORALES

text: S

text: Theorem

text: There

text: exists

text: a

text: map

text: T

text: (Ω)

text: with

text: f(θ,

text: non-increasing

text: for

text: every

text: S

text: θ

text: such

text: that

text: T

text: permutes

text: the

text: upper

text: and

text: lower

text: circles

text: of

text: Ω

text: (thus

text: having

text: a

text: periodic

text: orbit

text: of

text: period

text: two

text: of

text: curves),

text: and

text: there

text: exists

text: a

text: connected

text: pseudo-curve

text: A

text: Ω

text: which

text: does

text: not

text: contain

text: any

text: arc

text: of

text: a

text: curve

text: such

text: that

text: T(A)

text: A

text: and

text: there

text: does

text: not

text: exist

text: any

text: T-invariant

text: curve.

text: Proof.

text: By

text: Propositions

text: and

text: there

text: exists

text: a

text: map

text: S

text: T(θ,

text: x)

text: (R

text: (θ),

text: f(θ,

text: x))

text: (R

text: (θ),

text: lim

text: f

text: (θ,

text: x))

text: (Ω)

text: ω

text: ω

text: m

text: m

text: S

text: with

text: f(θ,

text: non-increasing

text: for

text: every

text: θ

text: such

text: that

text: T

text: permutes

text: the

text: upper

text: and

text: lower

text: circles

text: of

text: Ω

text: (that

text: is,

text: f(θ,

text: and

text: f(θ,

text: As

text: the

text: connected

text: set

text: A

text: we

text: take

text: the

text: one

text: given

text: by

text: Proposition

text: (and

text: Deﬁnition

text: To

text: end

text: the

text: proof

text: of

text: the

text: theorem

text: we

text: need

text: to

text: show

text: that

text: T(A)

text: A,

text: since

text: this

text: already

text: implies

text: that

text: there

text: does

text: not

text: exist

text: any

text: T-invariant

text: curve.

text: To

text: see

text: it,

text: assume

text: by

text: way

text: of

text: contradiction

text: that

text: there

text: exists

text: an

text: invariant

text: curve

text: and

text: denote

text: its

text: graph

text: by

text: B.

text: Since

text: B

text: is

text: the

text: graph

text: of

text: a

text: (continuous)

text: curve,

text: it

text: is

text: compact

text: and

text: connected.

text: On

text: the

text: other

text: hand,

text: let

text: Ω

text: and

text: Ω−

text: be

text: the

text: two

text: connected

text: components

text: of

text: Ω

text: A

text: from

text: the

text: proof

text: of

text: Proposition

text: The

text: facts

text: that

text: T(A)

text: A,

text: f(θ,

text: is

text: decreasing

text: for

text: every

text: S

text: θ

text: and

text: T

text: permutes

text: the

text: upper

text: and

text: lower

text: circles

text: of

text: Ω

text: imply

text: that

text: T(Ω

text: Ω−

text: and

text: T(Ω−)

text: Ω

text: Hence,

text: by

text: the

text: invariance

text: of

text: B,

text: B

text: Ω

text: and

text: B

text: Ω−.

text: The

text: connectivity

text: of

text: A

text: and

text: B

text: imply

text: that

text: there

text: exists

text: (θ,

text: x)

text: A

text: B.

text: Consequently,

text: Z

text: B

text: Tn(θ,

text: x)

text: n

text: A;

text: a

text: contradiction

text: because

text: A

text: does

text: not

text: contain

text: any

text: arc

text: of

text: a

text: So,

text: only

text: it

text: remains

text: to

text: prove

text: that

text: T(A)

text: A.

text: By

text: using

text: Proposition

text: and

text: Lemma

text: we

text: get

text: that

text: T

text: Ai

text: and

text: T

text: T

text: for

text: every

text: Z

text: m

text: k

text: Ai

text: m

text: Ai

text: k,

text: m

text: k

text: m

text: and

text: i

text: D

text: Consequently,

text: by

text: the

text: deﬁnition

text: of

text: the

text: map

text: T

text: we

text: m

text: Z

text: have,

text: T(Ai

text: for

text: every

text: i

text: or,

text: equivalently,

text: T

text: A

text: O

text: (ω)

text: A

text: O

text: (ω).

text: S

text: S

text: Now

text: we

text: consider

text: Aθ

text: with

text: θ

text: O

text: (ω).

text: Since

text: O

text: (ω)

text: is

text: dense

text: in

text: there

text: exists

text: a

text: sequence

text: (θ

text: x

text: A

text: O

text: (ω)

text: such

text: that

text: lim

text: θ

text: θ.

text: By

text: the

text: compacity

text: of

text: A

text: n

text: n

text: n

text: n

text: we

text: can

text: assume

text: (by

text: taking

text: a

text: convergent

text: subsequence,

text: if

text: necessary)

text: that

text: (θ

text: x

text: n

text: n

text: is

text: convergent

text: to

text: a

text: point

text: (θ,

text: x)

text: A.

text: By

text: Lemma

text: Aθ

text: (θ,

text: x)

text: (and

text: x

text: γ(θ)).

text: On

text: the

text: other

text: hand,

text: by

text: the

text: part

text: of

text: the

text: statement

text: already

text: proven,

text: T(θ

text: x

text: A

text: for

text: n

text: n

text: every

text: n.

text: Hence,

text: by

text: the

text: continuity

text: of

text: T

text: and

text: the

text: compacity

text: of

text: A,

text: T(θ,

text: x)

text: (Rω(θ),

text: f(θ,

text: x))

text: l→im∞

text: T(θn,

text: xn)

text: ARω(θ).

text: n

text: Since

text: θ

text: O

text: (ω)

text: we

text: have

text: that

text: R

text: (θ)

text: O

text: (ω)

text: and,

text: again

text: by

text: Lemma

text: ω

text: S

text: ARω(θ)

text: consists

text: of

text: a

text: unique

text: point.

text: Hence,

text: T(Aθ)

text: ARω(θ)

text: for

text: every

text: θ

text: O

text: (ω).

text: S

text: S

text: Equivalently,

text: T

text: A

text: O

text: (ω)

text: A

text: O

text: (ω)

text: This

text: ends

text: the

text: proof

text: of

text: the

text: theorem.

text: Proof

text: of

text: Proposition

text: in

text: the

text: case

text: m

text: This

text: section

text: is

text: devoted

text: to

text: prove

text: Proposition

text: for

text: m

text: that

text: is,

text: to

text: study

text: the

text: map

text: T

text: It

text: is

text: the

text: ﬁrst

text: technical

text: counterpart

text: of

text: Section

text: To

text: prove

text: Proposition

text: for

text: T

text: we

text: will

text: need

text: some

text: more

text: notation

text: and

text: a

text: technical

text: lemma.

text: S

text: I

text: Given

text: a

text: skew

text: product

text: F(θ,

text: x)

text: (R

text: (θ),

text: ζ(θ,

text: x)

text: from

text: Ω

text: to

text: itself

text: we

text: ω

text: S

text: C

text: I

text: I

text: deﬁne

text: the

text: ﬁbre

text: map

text: function

text: of

text: F,

text: ﬁb(F):

text: by

text: ﬁb(F)(θ)

text: ζ(θ,

text: A

text: simple

text: exercise

text: shows

text: that

text: F

text: is

text: continuous

text: if

text: and

text: only

text: if

text: ζ(θ,

text: is

text: continuous

text: for

text: S

text: every

text: θ

text: and

text: ﬁb(F)

text: is

text: continuous.

text: ##LTLine##


text: (cid:98) (cid:98) A SKEW-PRODUCT WITHOUT FIXED-CURVES 45 { }∞ ⊂ S \ ∗ ⊂ S \ ∗ Rω(cid:12)(cid:12)(θj(cid:96)) (cid:96)=≥1 1 O ∈(ωN) (cid:98) 1(cid:12)(cid:12)Z|ij+(cid:12)(cid:12)1|, γ|ij+1| is deﬁned(cid:98) for(cid:12)(cid:12) every Rω(θj(cid:96)). Then, for e(cid:12)very j N and (cid:96) , we h(cid:12)ave (cid:12) (cid:12) (cid:12) (cid:12) − ≤ −(cid:12) (cid:98) (cid:98) (cid:12) γ(Rω(θ)) γ|ij+1|(Rω(θj(cid:96))) γ(Rω(θ(cid:16))) (cid:12) γ(R(cid:17)ω(θj(cid:96))) + (cid:12) − γ(Rω(θj(cid:96))) γ|ij+1|(Rω(θj(cid:96))) < ε + d∞ γ, γ| | < ε. (cid:12) (cid:12) 2 (cid:12) ij+1 (cid:12) (cid:12) (cid:12) (cid:12) (cid:98) (cid:12) Consequ(cid:12) ently, (cid:12) (cid:12) (cid:12) − − ≤ γ(Rω(θ)) γ|ij+1|(Rω(θj)) = (cid:96)l→im∞ γ(Rω(θ)) γ|ij+1|(Rω(θj(cid:96))) ε This ends the proof of the ﬁrst equality of (21). The second and third equalities of (cid:40) (21) follow as above by replacing γ| | by γ (respectively γ| |), and noting that ij+1 ij ij+2 ∗ ∈ ∗ ≥ ∗ ((i + 1)) / Z if i 0, and Rω(θj) = Rω((ij) ) = −j | | − ∗ij∈ ∗ j (( ( i 1))) / Z| |− if i < 0. j ij 2 j This ends the proof of the continuity of T , and the proposition for the case m = 0 (cid:3) 0. 9. Proof of Proposition 7.3 for m > 0 This section is the second technical counterpart of Section and is devoted to prove Proposition 7.3 for every map T with m > 0. To do this we will need some m (cid:12) more technical results. Also we will use the notion of ﬁbre map function int(cid:12)roduced in the previous section. The next two lemmas establish some basic properties of the maps Tm V and m clarify some aspects of Deﬁnition 7.2(cid:12). (cid:12) ∈ N (cid:12) (cid:12) ∈ B Lemma 9.1. For every m and for every θ , m fm,θ Ii,θ = gi,θ Ii,θ,  ∈ WB \WIB where i = b (θ, m). Moreover, assume that θ m m. Then, ∈ I g (x) if x , f (x) = 2i−−,θgi,θ(mi(θ)) (f − (x) − 2) + 2 if x ∈ [−i,θ2, m (θ)], m,θ 2 fm−1,θ(mi(θ)) m 1,θ i 2+gi,θ(Mi(θ)) (f − (x) + 2) − 2 if x ∈ [M (θ), 2]. 2+fm−1,θ(Mi(θ)) m 1,θ i ∈ B Proof. We start by proving the ﬁrst statement. When θ there is nothing ∈ B \B ∈m WB to prove. So, assume that θ . By Deﬁnition 6.2, θ , i < 0 and ∈ ∗ \ ∗ m m m θ B [i ] B (i ) . By Lemma 6.3(b), i α|i| I { } ⊂ IW = γ (θ) . i,θ |i| m,θ Consequently, by Deﬁnit(cid:0)ion 7.2(cid:1) and the deﬁnition of t(cid:0)he map(cid:1)s g for i < 0 (Deﬁ- I ⊂ R ∗ i,θ nition 4.4 — notice that (i ) by deﬁnition), i,θ f γ (θ) = γ (R (θ)) = g γ (θ) . m,θ |i| |i+1| ω i,θ |i| So, the ﬁrst statement holds. Now we prove the second one. By Lemma 6.3(b), I { } { } { } { } { } IW = m (θ) = M (θ) = γ (θ) = λ (θ) = τ (θ) = . i,θ i i |i| m m m,θ ##LTLine## ##LTLine## ##LTLine##

text: A

text: SKEW-PRODUCT

text: WITHOUT

text: FIXED-CURVES

text: S

text: S

text: O

text: ∈(ωN)

text: is

text: every

text: Then,

text: for

text: j

text: N

text: and

text: we

text: γ(Rω(θ))

text: ε

text: d∞

text: γ,

text: γ|

text: ε.

text: ently,

text: γ(Rω(θ))

text: γ(Rω(θ))

text: ε

text: This

text: ends

text: the

text: proof

text: of

text: the

text: ﬁrst

text: equality

text: of

text: The

text: second

text: and

text: third

text: equalities

text: of

text: follow

text: as

text: above

text: by

text: replacing

text: γ|

text: by

text: γ

text: (respectively

text: γ|

text: and

text: noting

text: that

text: ij

text: ((i

text: Z

text: if

text: i

text: and

text: Rω(θj)

text: Rω((ij)

text: −j

text: ∗ij∈

text: j

text: i

text: Z|

text: if

text: i

text: j

text: ij

text: j

text: This

text: ends

text: the

text: proof

text: of

text: the

text: continuity

text: of

text: T

text: and

text: the

text: proposition

text: for

text: the

text: case

text: m

text: Proof

text: of

text: Proposition

text: for

text: m

text: This

text: section

text: is

text: the

text: second

text: technical

text: counterpart

text: of

text: Section

text: and

text: is

text: devoted

text: to

text: prove

text: Proposition

text: for

text: every

text: map

text: T

text: with

text: m

text: To

text: do

text: this

text: we

text: will

text: need

text: some

text: m

text: more

text: technical

text: results.

text: Also

text: we

text: will

text: use

text: the

text: notion

text: of

text: ﬁbre

text: map

text: function

text: in

text: the

text: previous

text: section.

text: The

text: next

text: two

text: lemmas

text: establish

text: some

text: basic

text: properties

text: of

text: the

text: maps

text: Tm

text: V

text: and

text: m

text: clarify

text: some

text: aspects

text: of

text: Deﬁnition

text: N

text: B

text: Lemma

text: For

text: every

text: m

text: and

text: for

text: every

text: θ

text: m

text: fm,θ

text: Ii,θ

text: gi,θ

text: Ii,θ,

text: WB

text: \WIB

text: where

text: i

text: b

text: (θ,

text: m).

text: Moreover,

text: assume

text: that

text: θ

text: m

text: m.

text: Then,

text: I

text: g

text: (x)

text: if

text: x

text: f

text: (x)

text: (f

text: (x)

text: if

text: x

text: m

text: (θ)],

text: m,θ

text: m

text: i

text: (f

text: (x)

text: if

text: x

text: [M

text: (θ),

text: m

text: i

text: B

text: Proof.

text: We

text: start

text: by

text: proving

text: the

text: ﬁrst

text: statement.

text: When

text: θ

text: there

text: is

text: nothing

text: B

text: \B

text: ∈m

text: WB

text: to

text: prove.

text: So,

text: assume

text: that

text: θ

text: By

text: Deﬁnition

text: θ

text: i

text: and

text: m

text: m

text: m

text: θ

text: B

text: [i

text: B

text: (i

text: By

text: Lemma

text: i

text: α|i|

text: I

text: IW

text: γ

text: (θ)

text: i,θ

text: |i|

text: m,θ

text: Consequently,

text: by

text: and

text: the

text: deﬁnition

text: of

text: g

text: for

text: i

text: (Deﬁ-

text: I

text: R

text: i,θ

text: nition

text: notice

text: that

text: (i

text: by

text: deﬁnition),

text: i,θ

text: f

text: γ

text: (θ)

text: γ

text: (R

text: (θ))

text: g

text: γ

text: (θ)

text: m,θ

text: |i|

text: ω

text: i,θ

text: |i|

text: So,

text: the

text: ﬁrst

text: statement

text: holds.

text: Now

text: we

text: prove

text: the

text: second

text: one.

text: By

text: Lemma

text: I

text: IW

text: m

text: (θ)

text: M

text: (θ)

text: γ

text: (θ)

text: λ

text: (θ)

text: τ

text: (θ)

text: i,θ

text: i

text: i

text: |i|

text: m

text: m

text: m,θ

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 46 LL. ALSEDA`, F. MAN˜OSAS, AND L. MORALES  Thus, by thepart already proven, the formulas ∈ I g (x) if x , 2i−−,θgi,θ(mi(θ)) (f − (x) − 2) + 2 if x ∈ [−i,θ2, m (θ)], 2 fm−1,θ(mi(θ)) m 1,θ i 2+gi,θ(Mi(θ)) (f − (x) + 2) − 2 if x ∈ [M (θ), 2], 2+fm−1,θ(Mi(θ)) m 1,θ i and   ∈ IW γ (R (θ)) if x , 2|−−i+γ1||i+1|(ωRω(θ)) (f − (x) − 2) + 2 if x ∈ [−2m, λ,θ (θ)], 2 fm−1,θ(λm(θ)) m 1,θ m 2+γ|i+1|(Rω(θ)) (f − (x) + 2) − 2 if x ∈ [τ (θ), 2], 2+fm−1,θ(τm(θ)) m 1,θ m (cid:3) coincide. (cid:12) (cid:12) ∈ N ∈ Lemma 9.2. The following statements hold for every m and i D : m (a) The map T is well deﬁned and continuous. m∈Vi∗ ∗ (b) For every θ B [i ] , i − − (b.i) f (2) = 2 and f ( 2) = 2, (cid:12) m,θ (cid:16) m,(cid:17)θ (b.(cid:12)ii) f−m,θ≤is piecewise aﬃ≤ne and non-in≤creasing, and (b.iii) 1 f (M (θ)) f (m (θ)) 1. m(cid:12) ,θ i ∗ m,θ i ∗ (cid:12) (c) Tm R (i∗) = G(cid:12)i and Tm Ai|i| = A(|ii++11|) . (cid:12) Proof. Clear(cid:12)ly, T is wel(cid:12)l deﬁned and continuous if and only if so is f . (cid:12) m Vi∗ (cid:12) ∈ Z m Vi∗ We will prove by induction on m + that, (a), (b) and − − (b.iv) fm,θ [−2,−1] and fm,θ [1,2] are aﬃne, fm,θ( 1) < 2 and fm,θ(1) > 2 ∈ ∗ hold for every θ B [i ] . i (cid:12) ∈ First we will show that (a), (b) and (b.iv) h(cid:12)old for m = 0 and i D (we are (cid:12) 0(cid:12) including the map f0 studied earlier to correctly start t(cid:12)he induction proce(cid:12)ss). By Proposition 7.3(a,b) for m = 0 we have that T is well deﬁned and continuous 0 V ∗ i and (b) holds. By Deﬁnition 7.1, we a−lso ≤know tha≤t fm,θ [−2≤,mi(θ)] and fm,θ [Mi(θ),2] are aﬃne. Then, (b.iv) follows from 1 m (θ) M (θ) 1 (see Lemma 4.1(a)) i i and (b.iii). (cid:12) − ∈ Z Assume now that (a), (b) and ((cid:12)b.iv) hold for some m 1 + and prove it ∈ ∈ ∗ (cid:32) ∗ ∈ for m and i Dm⊂. By Lemma 5.2(a), θ Bi [i ] Bk [k ] for some k Dm−1. Consequently, Vi∗ Vk∗ and fm−1 V ∗ is well deﬁned and continuous. i By Lemma 4.1(a) and Deﬁnition 6.2, − ≤ ≤ ≤ ∈ ∗ 1 m (θ) M (θ) 1 for θ B [i ], and i i i (22) − ≤ ≤ ≤ ∈ ∗ \ ∗ ⊂ WB 1 λ (θ) τ (θ) 1 for θ B [i ] B (i ) (i < 0). m m i α|i| m − Consequently, by (b.ii) and (b.iv) for m 1, − ≤ ≤ ≤ − 2 < f − (1) f − (M (θ)) f − (m (θ)) f ( 1) < 2 m 1,θ m 1,θ i m 1,θ i m,θ ∈ ∗ for every θ B [i ], and i − ≤ ≤ ≤ − 2 < f − (1) f − (τ (θ)) f − (λ (θ)) f ( 1) < 2 m 1,θ m 1,θ m m 1,θ m m,θ ∈ ∗ \ ∗ ⊂ WB for θ B [i ] B (i ) when i < 0. i α|i| m ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: LL.

text: ALSEDA`,

text: F.

text: MAN˜OSAS,

text: AND

text: L.

text: MORALES

text: Thus,

text: by

text: thepart

text: already

text: proven,

text: the

text: formulas

text: I

text: g

text: (x)

text: if

text: x

text: (f

text: (x)

text: if

text: x

text: m

text: (θ)],

text: m

text: i

text: (f

text: (x)

text: if

text: x

text: [M

text: (θ),

text: m

text: i

text: and

text: IW

text: γ

text: (R

text: (θ))

text: if

text: x

text: (f

text: (x)

text: if

text: x

text: λ,θ

text: (θ)],

text: m

text: m

text: (f

text: (x)

text: if

text: x

text: [τ

text: (θ),

text: m

text: m

text: coincide.

text: N

text: Lemma

text: The

text: following

text: statements

text: hold

text: for

text: every

text: m

text: and

text: i

text: D

text: m

text: (a)

text: The

text: map

text: T

text: is

text: well

text: deﬁned

text: and

text: continuous.

text: m∈Vi∗

text: (b)

text: For

text: every

text: θ

text: B

text: [i

text: i

text: (b.i)

text: f

text: and

text: f

text: m,θ

text: f−m,θ≤is

text: piecewise

text: aﬃ≤ne

text: and

text: non-in≤creasing,

text: and

text: (b.iii)

text: f

text: (M

text: (θ))

text: f

text: (m

text: (θ))

text: ,θ

text: i

text: m,θ

text: i

text: (c)

text: Tm

text: R

text: (i∗)

text: and

text: Tm

text: Ai|i|

text: Proof.

text: T

text: is

text: deﬁned

text: and

text: continuous

text: if

text: and

text: only

text: if

text: so

text: is

text: f

text: m

text: Vi∗

text: Z

text: m

text: Vi∗

text: We

text: will

text: prove

text: by

text: induction

text: on

text: m

text: that,

text: (a),

text: (b)

text: and

text: (b.iv)

text: fm,θ

text: and

text: fm,θ

text: are

text: aﬃne,

text: fm,θ(

text: and

text: hold

text: for

text: every

text: θ

text: B

text: [i

text: i

text: First

text: we

text: will

text: show

text: that

text: (a),

text: (b)

text: and

text: (b.iv)

text: for

text: m

text: and

text: i

text: D

text: (we

text: are

text: including

text: the

text: map

text: studied

text: earlier

text: to

text: correctly

text: start

text: induction

text: By

text: Proposition

text: for

text: m

text: we

text: have

text: that

text: T

text: is

text: well

text: deﬁned

text: and

text: continuous

text: V

text: i

text: and

text: (b)

text: holds.

text: By

text: Deﬁnition

text: we

text: a−lso

text: ≤know

text: tha≤t

text: fm,θ

text: and

text: fm,θ

text: are

text: aﬃne.

text: Then,

text: (b.iv)

text: follows

text: from

text: m

text: (θ)

text: M

text: (θ)

text: (see

text: Lemma

text: i

text: i

text: and

text: (b.iii).

text: Z

text: Assume

text: now

text: that

text: (a),

text: (b)

text: and

text: hold

text: for

text: some

text: m

text: and

text: prove

text: it

text: for

text: m

text: and

text: i

text: Dm⊂.

text: By

text: Lemma

text: θ

text: Bi

text: [i

text: Bk

text: [k

text: for

text: some

text: k

text: Consequently,

text: Vi∗

text: Vk∗

text: and

text: V

text: is

text: well

text: deﬁned

text: and

text: continuous.

text: i

text: By

text: Lemma

text: and

text: Deﬁnition

text: m

text: (θ)

text: M

text: (θ)

text: for

text: θ

text: B

text: [i

text: and

text: i

text: i

text: i

text: WB

text: λ

text: (θ)

text: τ

text: (θ)

text: for

text: θ

text: B

text: [i

text: B

text: (i

text: (i

text: m

text: m

text: i

text: α|i|

text: m

text: Consequently,

text: by

text: (b.ii)

text: and

text: (b.iv)

text: for

text: m

text: f

text: f

text: (M

text: (θ))

text: f

text: (m

text: (θ))

text: f

text: m

text: m

text: i

text: m

text: i

text: m,θ

text: for

text: every

text: θ

text: B

text: [i

text: and

text: i

text: f

text: f

text: (τ

text: (θ))

text: f

text: (λ

text: (θ))

text: f

text: m

text: m

text: m

text: m

text: m

text: m,θ

text: WB

text: for

text: θ

text: B

text: [i

text: B

text: (i

text: when

text: i

text: i

text: α|i|

text: m

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 48 LL. ALSEDA`, F. MAN˜OSAS, AND L. MORALES and 4.5(c) and (2) (alternatively, for the last e(cid:0)quality(cid:1)check directly the proofs of the Lemmas 4.3(c) and 4.5(c)), f − (m (θ)) = g (m (θ)) = g γ (θ) = γ (R (θ)) . m 1,θ i k,θ i k,θ |k| |k+1| ω (cid:3) ≥ The following lemma is the analogue of Lemma 8.1 for m 1. To state it we will use the set (cid:12) (cid:12) (cid:12) (cid:20)(cid:20)(cid:20)EB(cid:12) EB × I ⊂ V = . m m m ∈ N Lemma 9.4. Tm (cid:20)(cid:20)(cid:20)EB = Tm−1 (cid:20)(cid:20)(cid:20)EB for every m . Equivalently, fm,θ = ∈ Nm ∈ EB m f − for every m and θ . m 1,θ m ∈ N ∈ EB ⊂ B ∈ Proof. Fix m and θ . By Lemma 5.2(a,b), there exist i D ∈ ∈ m m∗ ⊂ ∗ (cid:32) ∗ m and k Dm−1 such that θ Bd (Bi [i ]) Bi [i ] Bk [k ] . So, we are in the assumptions of Lemmas 5.4 and 9.3 and, hence, I { } { } { } ⊂ I = m (θ) = γ (θ) = γ (θ) , i,θ i |i| |k| k,θ f (m (θ)) = g (m (θ)) = γ (R (θ)) , and m,θ i i,θ i |i+1| ω f − (m (θ)) = g (m (θ)) = γ (R (θ)) . m 1,θ i k,θ i |k+1| ω ≥ ∈ B Thus, if i 0, θ and, by Deﬁnition 7.2 and Lemma 9.2(a), to prove that m f = f − we only have to show that m,θ m 1,θ g (m (θ)) = γ (R (θ)) = γ (R (θ)) = f − (m (θ)) . i,θ i |i+1| ω |k+1| ω m 1,θ i ∈ WB ∩ EB ∈ WIB When i < 0, θ and, by Lemma 6.3(a), θ / . Then, by m m m Lemma 9.1, we get again that g (m (θ)) = γ (R (θ)) = γ (R (θ)) = f − (m (θ)) . i,θ i |i+1| ω |k+1| ω m 1,θ i implies f = f − . | m,|θ | m 1|,θ If k + 1 = i + 1 there is nothing to prove. So, by Lemma 5.4, we can assume | | | | that k + 1 < i + 1 and we have · · · γ (R (θ)) = γ (R (θ)) = = γ (R (θ)) . |k+1| ω |k+1|+1 ω |i+1|−1 ω ≥ Hence, we have to show that γ (R (θ)) = γ (R (θ)) . If i 0 we get |i+1| ω |i+1|−1 ω (cid:0) (cid:1) γ|i+1| (Rω(θ)) =(cid:0) γi+1 (Rω(θ)) =(cid:1)γi (Rω(θ)) = γ|i+1|−1 (Rω(θ)) ∈ ∗ ∗ by Lemma 3.6(e). Otherwise we have i < 0, θ Bd (B [i ]) = Bd B [i ] and, ∈ ∗ i α|i+1| | | consequently, R (θ) Bd B [(i + 1) ] . Again by Lemma 3.6(e) for j = i + 1 , ω α|i+1| γ (R (θ)) = γ (R (θ)) . |i+1| ω |i+1|−1 ω (cid:3) This ends the proof of the lemma. (cid:107) − (cid:107) Now we aim at computing two diﬀerent kind of upper bounds for f f − m,θ m 1,θ (Lemma 9.6 and Proposition 7.4). This will be a key tool in the proof of Propo- sitions 7.3 for m > 0 and 7.4. The next two lemmas and remark will be useful to automate and simplify(cid:13)the proofs of these two result(cid:13)s. (cid:13) (cid:12) (cid:12) (cid:13) Lemma 9.5. (cid:13)(cid:13) (cid:12) (cid:12) (cid:13)(cid:13) (cid:107)f − f − (cid:107) = (cid:13)(cid:13)(cid:13)(cid:13)fm,θ(cid:12)(cid:12)Ib(θ,m),θ − fm−1(cid:12)(cid:12),θ Ib(θ,m(cid:13)(cid:13)(cid:13)(cid:13)),θ when θ ∈ Bm\WIBm, and m,θ m 1,θ − ∈ WIB fm,θ IW fm−1,θ IW when θ m, m,θ m,θ ≥ ∈ B for every m 2 and θ . m

text: LL.

text: ALSEDA`,

text: F.

text: MAN˜OSAS,

text: AND

text: L.

text: MORALES

text: and

text: and

text: (alternatively,

text: for

text: the

text: last

text: directly

text: the

text: proofs

text: of

text: the

text: Lemmas

text: and

text: f

text: (m

text: (θ))

text: g

text: (m

text: (θ))

text: g

text: γ

text: (θ)

text: γ

text: (R

text: (θ))

text: m

text: i

text: k,θ

text: i

text: k,θ

text: |k|

text: ω

text: The

text: following

text: lemma

text: is

text: the

text: analogue

text: of

text: Lemma

text: for

text: m

text: To

text: state

text: it

text: we

text: will

text: use

text: the

text: set

text: EB

text: I

text: V

text: m

text: m

text: m

text: N

text: Lemma

text: Tm

text: for

text: every

text: m

text: Equivalently,

text: fm,θ

text: Nm

text: EB

text: m

text: f

text: for

text: every

text: m

text: and

text: θ

text: m

text: m

text: N

text: EB

text: B

text: Proof.

text: Fix

text: m

text: and

text: θ

text: By

text: Lemma

text: there

text: exist

text: i

text: D

text: m

text: m∗

text: m

text: and

text: k

text: such

text: that

text: θ

text: Bd

text: (Bi

text: [i

text: Bi

text: [i

text: Bk

text: [k

text: So,

text: we

text: are

text: in

text: the

text: assumptions

text: of

text: Lemmas

text: and

text: and,

text: hence,

text: I

text: I

text: m

text: (θ)

text: γ

text: (θ)

text: γ

text: (θ)

text: i,θ

text: i

text: |i|

text: |k|

text: k,θ

text: f

text: (m

text: (θ))

text: g

text: (m

text: (θ))

text: γ

text: (R

text: (θ))

text: and

text: m,θ

text: i

text: i,θ

text: i

text: ω

text: f

text: (m

text: (θ))

text: g

text: (m

text: (θ))

text: γ

text: (R

text: (θ))

text: m

text: i

text: k,θ

text: i

text: ω

text: B

text: Thus,

text: if

text: i

text: θ

text: and,

text: by

text: Deﬁnition

text: and

text: Lemma

text: to

text: prove

text: that

text: m

text: f

text: f

text: we

text: only

text: have

text: to

text: show

text: that

text: m,θ

text: m

text: g

text: (m

text: (θ))

text: γ

text: (R

text: (θ))

text: γ

text: (R

text: (θ))

text: f

text: (m

text: (θ))

text: i,θ

text: i

text: ω

text: ω

text: m

text: i

text: WB

text: EB

text: WIB

text: When

text: i

text: θ

text: and,

text: by

text: Lemma

text: θ

text: Then,

text: by

text: m

text: m

text: m

text: Lemma

text: we

text: get

text: again

text: that

text: g

text: (m

text: (θ))

text: γ

text: (R

text: (θ))

text: γ

text: (R

text: (θ))

text: f

text: (m

text: (θ))

text: i,θ

text: i

text: ω

text: ω

text: m

text: i

text: implies

text: f

text: f

text: m,|θ

text: m

text: If

text: k

text: i

text: there

text: is

text: nothing

text: to

text: prove.

text: So,

text: by

text: Lemma

text: we

text: can

text: assume

text: that

text: k

text: i

text: and

text: we

text: have

text: γ

text: (R

text: (θ))

text: γ

text: (R

text: (θ))

text: γ

text: (R

text: (θ))

text: ω

text: ω

text: ω

text: Hence,

text: we

text: have

text: to

text: show

text: that

text: γ

text: (R

text: (θ))

text: γ

text: (R

text: (θ))

text: If

text: i

text: we

text: get

text: ω

text: ω

text: (Rω(θ))

text: (Rω(θ))

text: (Rω(θ))

text: (Rω(θ))

text: by

text: Lemma

text: Otherwise

text: we

text: have

text: i

text: θ

text: Bd

text: (B

text: [i

text: Bd

text: B

text: [i

text: and,

text: i

text: consequently,

text: R

text: (θ)

text: Bd

text: B

text: [(i

text: Again

text: by

text: Lemma

text: for

text: j

text: i

text: ω

text: γ

text: (R

text: (θ))

text: γ

text: (R

text: (θ))

text: ω

text: ω

text: This

text: ends

text: the

text: proof

text: of

text: the

text: lemma.

text: Now

text: we

text: aim

text: at

text: computing

text: two

text: diﬀerent

text: kind

text: of

text: upper

text: bounds

text: for

text: f

text: f

text: m,θ

text: m

text: (Lemma

text: and

text: Proposition

text: This

text: will

text: be

text: a

text: key

text: tool

text: in

text: the

text: proof

text: of

text: Propo-

text: sitions

text: for

text: m

text: and

text: The

text: next

text: two

text: lemmas

text: and

text: remark

text: will

text: be

text: useful

text: to

text: automate

text: and

text: proofs

text: of

text: these

text: two

text: Lemma

text: f

text: when

text: θ

text: Bm\WIBm,

text: and

text: m,θ

text: m

text: WIB

text: fm,θ

text: IW

text: IW

text: when

text: θ

text: m,

text: m,θ

text: m,θ

text: B

text: for

text: every

text: m

text: and

text: θ

text: m


text: 6 LL. ALSEDA`, F. MAN˜OSAS, AND L. MORALES PC Proof. The fact that d∞PiCs a metric in follows from{Remark 2}.∞7. Now we prove that is complete. Assume that [ϕ , G ] is a Cauchy PC ∈ PCn n n=1 sequence in . We have to see that lim →∞[ϕ , G ] . ∩∞ n n n Set, G := G . Since this intersection is countable, G is still a residual set. i=1 n { }∞ ⊂ I ThIe deﬁnition of∈d∞ implies that the sequence ϕn(θ) n=1 is a Cauchy s−e→queInce in for every θ G. So, it is convergent and we can deﬁne a map ϕ: G by ϕ(θ) := lim →∞ ϕ (θ). n∈ PCGn ∈ PC If (ϕ, G) we have [ϕ, G] and, from the deﬁnition of ϕ it follows that | − | l→im∞ d∞([ϕ, G], [ϕn, Gn]) = su∈p∩ l→im∞ ϕ(θ) ϕn(θ) = 0. n θ G Gn n Consequently, [ϕ, G] = lim →∞[ϕ , G ]. Since ϕ is the uniform limit of a sequence n n n ∈ PCG (cid:3) of continuous functions on G, it is continuous on G. That is, (ϕ, G) . A In what follows we want to look at the space as a metric space and relate this PC metric space with ( , d∞). Let ρ denote the euclidean metric in Ω. Then, the space (Ω, ρ) is a compact metric space. We recall that th(cid:40)e Hausdorﬀ metric is deﬁned in the sp(cid:41)ace of compact subsets of (Ω, ρ), by H (A, B) = max max ρ((θ, x), B), max ρ((θ, x), A) . ρ ∈ ∈ (θ,x) A (θ,x) B A PC TAhen, ( , Hρ) is a metric space. To study the relation between ( , d∞) and ( , H ) we need a couple of simple technical results. ρ (cid:0) (cid:1) ⊂ Lemma 2.10. Let A, B Ω be compact circular sets. Then, ≤ H (A, B) max H Aθ, Bθ . ρ ∈S ρ (cid:40) θ 1 (cid:41) (cid:16) (cid:17) (cid:16) (cid:17) Proof. It follows directly from the deﬁnitions: ≤ (cid:40) H (A, B) max sup ρ (θ, x), Bθ (cid:16), sup ρ (cid:17)(θ, x), Aθ ρ ∈ ∈ (θ,x) A (θ,x) B (cid:41) = max sup max ρ (θ, x), Bθ , (cid:16) (cid:17) ∈S { ∈I ∈ } θ 1 x : (θ,x) A (cid:40) (cid:41) (cid:16) su(cid:17)p max ρ(cid:16)(θ, x), Aθ(cid:17) ∈S { ∈I ∈ } θ 1 x : (θ,x) B (cid:0) (cid:1) = sup max max ρ (θ, x), Bθ , max ρ (θ, x), Aθ ∈S { ∈I ∈ } { ∈I ∈ } θ 1 x : (θ,x) A x : (θ,x) B = sup H Aθ, Bθ . ∈S ρ θ 1 (cid:101) (cid:101) (cid:3) Propositio(cid:16)n 2.11. Let ((cid:17)ϕ, G), (ϕ, G)(cid:16)∈ PCG. Then(cid:17), (cid:0) (cid:101) (cid:101) (cid:1) Hρ A(ϕ,G), A(ϕ(cid:101),G(cid:101)) ≤ θs∈uSp1 Hρ Aθ(ϕ,G), Aθ(ϕ(cid:101),G(cid:101)) = d∞ (ϕ, G), (ϕ, G) . Proof. Th(cid:0)e ﬁrst inequal(cid:1)ity follows from Lemma 2.10. (cid:16) (cid:17) Now we prove t(cid:101)he (cid:101)second equality. By L(cid:101)emma 2.2(a), d∞ (ϕ, G), (ϕ, G) = θsu∈Gp∩G(cid:101) |ϕ(θ) − ϕ(θ)| = θsu∈Gp∩G(cid:101)Hρ Aθ(ϕ,G), Aθ(ϕ(cid:101),G(cid:101)) .

text: LL.

text: ALSEDA`,

text: F.

text: MAN˜OSAS,

text: AND

text: L.

text: MORALES

text: PC

text: Proof.

text: The

text: fact

text: that

text: d∞PiCs

text: a

text: metric

text: in

text: follows

text: from{Remark

text: Now

text: we

text: prove

text: that

text: is

text: complete.

text: Assume

text: that

text: [ϕ

text: G

text: is

text: a

text: Cauchy

text: PC

text: PCn

text: n

text: sequence

text: in

text: We

text: have

text: to

text: see

text: that

text: lim

text: →∞[ϕ

text: G

text: n

text: n

text: n

text: Set,

text: G

text: G

text: Since

text: this

text: intersection

text: is

text: countable,

text: G

text: is

text: still

text: a

text: residual

text: set.

text: n

text: I

text: ThIe

text: deﬁnition

text: of∈d∞

text: implies

text: that

text: the

text: sequence

text: ϕn(θ)

text: is

text: a

text: Cauchy

text: s−e→queInce

text: in

text: for

text: every

text: θ

text: G.

text: So,

text: it

text: is

text: convergent

text: and

text: we

text: can

text: deﬁne

text: a

text: map

text: ϕ:

text: G

text: by

text: ϕ(θ)

text: lim

text: ϕ

text: (θ).

text: n∈

text: PCGn

text: PC

text: If

text: (ϕ,

text: G)

text: we

text: have

text: [ϕ,

text: G]

text: and,

text: from

text: the

text: deﬁnition

text: of

text: ϕ

text: it

text: follows

text: that

text: l→im∞

text: d∞([ϕ,

text: G],

text: [ϕn,

text: Gn])

text: su∈p∩

text: l→im∞

text: ϕ(θ)

text: ϕn(θ)

text: n

text: θ

text: G

text: Gn

text: n

text: Consequently,

text: [ϕ,

text: G]

text: lim

text: →∞[ϕ

text: G

text: Since

text: ϕ

text: is

text: the

text: uniform

text: limit

text: of

text: a

text: sequence

text: n

text: n

text: n

text: PCG

text: of

text: continuous

text: functions

text: on

text: G,

text: it

text: is

text: continuous

text: on

text: G.

text: That

text: is,

text: (ϕ,

text: G)

text: A

text: In

text: what

text: follows

text: we

text: want

text: to

text: look

text: at

text: the

text: space

text: as

text: a

text: metric

text: space

text: and

text: relate

text: this

text: PC

text: metric

text: space

text: with

text: d∞).

text: Let

text: ρ

text: denote

text: the

text: euclidean

text: metric

text: in

text: Ω.

text: Then,

text: the

text: space

text: (Ω,

text: ρ)

text: is

text: a

text: compact

text: metric

text: space.

text: We

text: recall

text: that

text: Hausdorﬀ

text: metric

text: is

text: deﬁned

text: in

text: the

text: of

text: compact

text: subsets

text: of

text: (Ω,

text: ρ),

text: by

text: H

text: (A,

text: B)

text: max

text: max

text: ρ((θ,

text: x),

text: B),

text: max

text: ρ((θ,

text: x),

text: A)

text: ρ

text: (θ,x)

text: A

text: (θ,x)

text: B

text: A

text: PC

text: TAhen,

text: Hρ)

text: is

text: a

text: metric

text: space.

text: To

text: study

text: the

text: relation

text: between

text: d∞)

text: and

text: H

text: we

text: need

text: a

text: couple

text: of

text: simple

text: technical

text: results.

text: ρ

text: Lemma

text: Let

text: A,

text: B

text: Ω

text: be

text: compact

text: circular

text: sets.

text: Then,

text: H

text: (A,

text: B)

text: max

text: H

text: Aθ,

text: Bθ

text: ρ

text: ∈S

text: ρ

text: θ

text: Proof.

text: It

text: follows

text: directly

text: from

text: the

text: deﬁnitions:

text: H

text: (A,

text: B)

text: max

text: sup

text: ρ

text: (θ,

text: x),

text: Bθ

text: sup

text: ρ

text: x),

text: Aθ

text: ρ

text: (θ,x)

text: A

text: (θ,x)

text: B

text: max

text: sup

text: max

text: ρ

text: (θ,

text: x),

text: Bθ

text: ∈S

text: ∈I

text: θ

text: x

text: (θ,x)

text: A

text: max

text: x),

text: ∈S

text: ∈I

text: θ

text: x

text: (θ,x)

text: B

text: sup

text: max

text: max

text: ρ

text: (θ,

text: x),

text: Bθ

text: max

text: ρ

text: (θ,

text: x),

text: Aθ

text: ∈S

text: ∈I

text: ∈I

text: θ

text: x

text: (θ,x)

text: A

text: x

text: (θ,x)

text: B

text: sup

text: H

text: Aθ,

text: Bθ

text: ∈S

text: ρ

text: θ

text: Let

text: G),

text: (ϕ,

text: PCG.

text: Hρ

text: A(ϕ,G),

text: Hρ

text: Aθ(ϕ,G),

text: d∞

text: (ϕ,

text: G),

text: (ϕ,

text: G)

text: Proof.

text: ﬁrst

text: follows

text: from

text: Lemma

text: Now

text: we

text: prove

text: equality.

text: By

text: d∞

text: (ϕ,

text: G),

text: (ϕ,

text: G)

text: |ϕ(θ)

text: ϕ(θ)|

text: Aθ(ϕ,G),


text: A SKEW-PRODUCT WITHOUT FIXED-CURVES 51 ∈ ∗ \ ∗ ∗ ∈ for every ν B [(i + 1) ] B ((i + 1) ) . On the other hand, γ ((i + 1) ) I αi αi+1 i ∗ (cid:18)by Lemma 3.6(c). Hence, b(cid:19)y Remark 3.2(2), Deﬁnition 3.4(R.1) and i+1,(i+1) Remark 3.5(R.1), (cid:20)(cid:20)(cid:20) (cid:8) ∗(cid:0)\ ∗ (cid:1) (cid:9) diam A (Bαi[(i+1) ] Bαi+1((i+1) )) i+1 (cid:8) (cid:9) ≤ ∗ \ ∗ · − − max diam B [(i + 1) ] B ((i + 1) ) , 2 (2 ni + 2 ni+1) ≤ · −αi − αi+1· − − 2 max α , 2 ni + 2 ni+1 = 2 (2 ni + 2 ni+1) i · − ≤ · − < 4 2 ni 2 2 i. (cid:0) (cid:0) (cid:1)(cid:1) ≥ Summarizing, when i 0, R ∗ ≤ − · − · − diam G (i ) 2 (i+1) + 2 2 i < 3 2 i (cid:0) i (cid:0) (cid:1)(cid:1) and, from (23), R ∗ · −| | ≤ · − diam G (i ) < 3 2 i 3 2 (cid:37)m < ε/2 i (cid:0) (cid:0) (cid:1)(cid:1) ∈ Z ∈ I for every i +. Thus, for every x , i,θ | − | R ∗ f (x) f − (x) < diam G (i ) + ε/2 < ε. m,θ m 1,θ i ∈ ∗ ∩ WIB IW Now assume that θ B [i ] . We can use Lemma 9.5 with to (cid:107) − (cid:107) i m | − | m,θ compute f f − . We have to show that f (x) f − (x) < ε for ∈ ImW,θ m 1,θ ∈ W(cid:8)IB (cid:9) m,θ m 1,θ every x . Since θ , by Deﬁnition 6.2 and Lemma 6.3(b), i < 0, ∈ WB m,θ m θ and m I ⊂ IW I (cid:51) = γ (θ) = x i,θ ∈ |i| m,θ (cid:96),θ with (cid:96) = b (θ, led (θ, m)) WFD . In this case we will consider the points ∈ R ∗ m ∈ R ∗ ∈ ∗ ⊂ EB (θ, x) ((cid:96) ) and (ν, m (ν)), (θ, γ (θ)) (i ) with ν Bd (B [i ]) . i |i| | | | | i m By Lemma 6.1(b), Remark (cid:12)3.2(2) and (cid:12)Lemma 3(cid:0).6(f), i(cid:1) < (cid:96) and ≤ (cid:12) (cid:12) dΩ((θ, x), (ν, mi(ν)) ≤ dΩ(−(θ, x), (θ, γ|i|(θ)) +(cid:0)RdΩ(∗(θ,(cid:1)γ|i|(θ)), (ν, mi(ν)) x γ (θ) + diam (i ) | | ≤ Ri ∗ R ∗ diam ( ((cid:96) )) + diam (i ) ≤ −| | · −| | · −| | ≤ · − 2 (cid:96) + 2 2 i < 3 2 i 3 2 (cid:37)m < δ − (ε/2). m 1 Thus, d (T − (θ, x), T − (ν, m (ν)) < ε/2. Ω m 1 m 1 i On the other hand, by Lemma 9.2(c), Deﬁnition 7.2 and (23), d (T (θ, x)(cid:12), T (ν, m (ν))) (cid:12) Ω m ≤ (cid:12) m i (cid:12) ≤ dΩ(Tm(θ,−x), Tm(θ, γ|i|(θ))) + dΩ(T(cid:0)m(θ(cid:0), γ|i|(θ))(cid:1),(cid:1)Tm(ν, mi(ν))) f (x) f (γ (θ)) + d (G (θ, γ (θ)), G (ν, m (ν))) m,θ m,θ |i| Ω i |i| i i ≤ R ∗ · −| | = d (G (θ, x), G (ν, m (ν))) diam G (i ) < 2 2 i ≤ Ω· −i i i i 3 2 (cid:37)m < ε/2. So, in a similar way as before, Lemma 9.4 gives | − | f (x) f − (x) = d (T (θ, x), T − (θ, x)) m,θ m 1,θ ≤ Ω m m 1 d (T (θ, x), T − (ν, m (ν))) + Ω m m 1 i d (T − (ν, m (ν)), T − (θ, x)) Ω m 1 i m 1 < ε. (cid:3)

text: A

text: SKEW-PRODUCT

text: WITHOUT

text: FIXED-CURVES

text: for

text: every

text: ν

text: B

text: [(i

text: B

text: ((i

text: On

text: the

text: other

text: hand,

text: γ

text: ((i

text: I

text: αi

text: i

text: Lemma

text: Hence,

text: Remark

text: Deﬁnition

text: and

text: Remark

text: diam

text: A

text: max

text: diam

text: B

text: [(i

text: B

text: ((i

text: ni

text: −αi

text: max

text: α

text: ni

text: ni

text: i

text: ni

text: i.

text: Summarizing,

text: when

text: i

text: R

text: diam

text: G

text: (i

text: i

text: i

text: i

text: and,

text: from

text: R

text: diam

text: G

text: (i

text: i

text: i

text: Z

text: I

text: for

text: every

text: i

text: Thus,

text: for

text: every

text: x

text: i,θ

text: R

text: f

text: (x)

text: f

text: (x)

text: diam

text: G

text: (i

text: ε.

text: m,θ

text: m

text: i

text: WIB

text: IW

text: Now

text: assume

text: that

text: θ

text: B

text: [i

text: We

text: can

text: use

text: Lemma

text: with

text: to

text: i

text: m

text: m,θ

text: compute

text: f

text: f

text: We

text: have

text: to

text: show

text: that

text: f

text: (x)

text: f

text: (x)

text: ε

text: for

text: ImW,θ

text: m

text: m,θ

text: m

text: every

text: x

text: Since

text: θ

text: by

text: Deﬁnition

text: and

text: Lemma

text: i

text: WB

text: m,θ

text: m

text: θ

text: and

text: m

text: I

text: IW

text: I

text: γ

text: (θ)

text: x

text: i,θ

text: |i|

text: m,θ

text: with

text: b

text: (θ,

text: led

text: (θ,

text: m))

text: WFD

text: In

text: this

text: case

text: we

text: will

text: consider

text: the

text: points

text: R

text: m

text: R

text: EB

text: (θ,

text: x)

text: and

text: (ν,

text: m

text: (ν)),

text: (θ,

text: γ

text: (θ))

text: (i

text: with

text: ν

text: Bd

text: (B

text: [i

text: i

text: |i|

text: i

text: m

text: By

text: Lemma

text: Remark

text: and

text: and

text: dΩ((θ,

text: x),

text: (ν,

text: mi(ν))

text: dΩ(−(θ,

text: x),

text: (θ,

text: γ|i|(θ))

text: (ν,

text: mi(ν))

text: x

text: γ

text: (θ)

text: diam

text: (i

text: Ri

text: R

text: diam

text: diam

text: (i

text: i

text: i

text: δ

text: m

text: Thus,

text: d

text: (T

text: (θ,

text: x),

text: T

text: (ν,

text: m

text: (ν))

text: Ω

text: m

text: m

text: i

text: On

text: the

text: other

text: hand,

text: by

text: Lemma

text: Deﬁnition

text: and

text: d

text: (T

text: (θ,

text: T

text: (ν,

text: m

text: (ν)))

text: Ω

text: m

text: m

text: i

text: dΩ(Tm(θ,−x),

text: Tm(θ,

text: γ|i|(θ)))

text: mi(ν)))

text: f

text: (x)

text: f

text: (γ

text: (θ))

text: d

text: (G

text: (θ,

text: γ

text: (θ)),

text: G

text: (ν,

text: m

text: (ν)))

text: m,θ

text: m,θ

text: |i|

text: Ω

text: i

text: |i|

text: i

text: i

text: R

text: d

text: (G

text: (θ,

text: x),

text: G

text: (ν,

text: m

text: (ν)))

text: diam

text: G

text: (i

text: i

text: Ω·

text: −i

text: i

text: i

text: i

text: So,

text: in

text: a

text: similar

text: way

text: as

text: before,

text: Lemma

text: gives

text: f

text: (x)

text: f

text: (x)

text: d

text: (T

text: (θ,

text: x),

text: T

text: (θ,

text: x))

text: m,θ

text: m

text: Ω

text: m

text: m

text: d

text: (T

text: (θ,

text: x),

text: T

text: (ν,

text: m

text: (ν)))

text: Ω

text: m

text: m

text: i

text: d

text: (T

text: (ν,

text: m

text: (ν)),

text: T

text: (θ,

text: x))

text: Ω

text: m

text: i

text: m

text: ε.


text: A SKEW-PRODUCT WITHOUT FIXED-CURVES 53 ∈ V ∈ ∈ Next we assume that (θ, x), (ν, y) but (θ, x), (ν, y) / V ∗ for every (cid:96) D . m ∈ (cid:96) (cid:54) m By Lemma 5.2(a,b),(cid:0)there e(cid:1)xist i = b (θ, m) , k = b (ν, m(cid:0) ) D (cid:1), i = k, such that ∈ ∗(cid:101) ∈ ∈ ∗ (cid:101)∈ m θ B [i ] , (θ, x) V ∗, ν B [k ] and (ν, y) V ∗. Then, there exist i i k k ∈ ∩ ∗ ⊂ EB ∈ ∩ ∗ ⊂ EB θ A Bd B [i ] and ν A Bd B [k ] , i m k m S where A denotes the closed arc of 1 such that (cid:0) (cid:1) (cid:0) (cid:1) (cid:101) (cid:101) { } diam(A) = d (θ, ν) and Bd(A) = θ, ν . (cid:16) (cid:0)(cid:101) (cid:1)(cid:17) ∈(cid:0)S1 (cid:101)(cid:1) ∈ Clearly we have, (θ, x), θ, x V ∗, (ν, y), ν, y V ∗ and, by the previous case, i k (cid:16) ≤ (cid:0) (cid:1)(cid:17)≤ (cid:101) d (θ, x), θ, x = d θ, θ d (θ, ν) d ((θ, x), (ν, y)) < δ, Ω S1 S1 Ω (cid:0) (cid:0) (cid:1)(cid:1) (cid:0) (cid:1) (cid:101) d T (θ(cid:101), x), T θ, x < 3ε Ω (cid:0) m m(cid:0) (cid:1)(cid:1) 7 ≤ (cid:101) ≤ d (ν, y), ν, y = d ν, ν d (θ, ν) d ((θ, x), (ν, y)) < δ, and Ω (cid:0) (cid:1) (cid:0) S1 (cid:1) S1 Ω (cid:101) d (cid:101)T (ν, y), T ν, y < 3ε. Ω m m 7 ∈ (cid:20)(cid:20)(cid:20)EB ⊂ V ⊂ V On the ot(cid:16)h(cid:0)er h(cid:1)an(cid:0)d, θ(cid:1),(cid:17)x , ν, y(cid:110) (cid:0) (cid:1) (cid:111) − a(cid:8)nd, by Lemma 9(cid:9).4 and (cid:101) (cid:101) (cid:101) (cid:101)m m m 1 (24), | − | ≤ | − | d θ, x , ν, y = max d θ, ν , x y max d (θ, ν), x y Ω (cid:16) S1 (cid:0) (cid:1)(cid:17) (cid:16) (cid:0)S1 (cid:1) (cid:0) (cid:1)(cid:17) (cid:101) ≤ (cid:101) (cid:101) = d ((θ, x), (ν, y)) < δ δ (ε/7), and Ω (cid:0) (cid:0) (cid:1) m,i (cid:1) ≤ (cid:101) d (T (θ, x), T (ν, y)) d T (θ, x), T θ, x + d T θ, x , T ν, y + Ω m m Ω m (cid:16) m(cid:0) (cid:1) (cid:0)Ω (cid:1)m(cid:17) m (cid:101) (cid:101) d T ν, y , T (ν, y) Ω m m < 3ε + d T − θ, x , T − ν, y + 3ε = ε. 7 Ω m 1 m 1 7 ∈ V If (θ, x), (ν, y) / then, by Deﬁnition 7.2 and (24) , m d (T (θ, x), T (ν, y)) = d (T − (θ, x), T − (ν, y)) < ε/7 < ε Ω m m Ω m 1 m 1 ≤ because d ((θ, x), (ν, y)) < δ δ − (ε/7). (cid:101) ∈Lastly∗,Ωassume that (ν, y) ∈/ Vmm b1ut (θ(cid:0),(cid:101)x) ∈(cid:1) Vi∗ ⊂ Vm∈, for some ∗i ∈⊂DmEB(that is, θ Bi [i ]).(cid:16)In this(cid:0)s(cid:101)itu(cid:1)a(cid:17)tion, as before, there∈ex(cid:20)(cid:20)(cid:20)isEtsBθ ⊂BVd (B⊂i V[i ]) m such that, by Lemma 9.4 and Deﬁnition 7.2 ( θ, x ), and (24), − (cid:16)(cid:0) (cid:1) (cid:1)(cid:17) m m m 1 (cid:101) d (θ, x), θ, x < δ, Ω (cid:16) (cid:0) (cid:1)(cid:17) ≤ d θ, x , (ν(cid:101), y < δ δ − (ε/7), Ω m 1 (cid:16) (cid:0) (cid:1)(cid:17) (cid:16) (cid:0) (cid:1) (cid:17) d T (θ, x), T θ, x < 3ε, and (cid:101) (cid:101) Ω m m 7 ≤ (cid:16) (cid:0) (cid:1) (cid:17) (cid:101) d (T (θ, x), T (ν, y)) d T (θ, x), T θ, x + d T θ, x , T (ν, y) Ω m m Ω m m Ω m m < 3ε + d T − θ, x , T − (ν, y) < ε. 7 Ω m 1 m 1 This ends the proof of the continuity of T and, hence, of (a). m ∈ B ∈ S \B (b) When θ the statement follows from Lemma 9.2(b). When θ 1 , it m m follows from the part already proven and the continuity of T . m (c) The ﬁrst two statements follow from Lemma 9.2(c) and statement (a). On the other hand, as in the proof of Proposition 7.3(c) for m = 0, Lemma 5.3(b) ∗ ∈ B ∗ ∈ B implies that i but i / for every k > m. Then, we get f ∗ = f ∗ from m k k,i m,i (cid:3) Deﬁnition 7.2. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: A

text: SKEW-PRODUCT

text: WITHOUT

text: FIXED-CURVES

text: V

text: Next

text: we

text: assume

text: that

text: (θ,

text: x),

text: (ν,

text: y)

text: but

text: (θ,

text: x),

text: (ν,

text: y)

text: V

text: for

text: every

text: D

text: m

text: m

text: By

text: Lemma

text: i

text: b

text: (θ,

text: m)

text: k

text: b

text: (ν,

text: D

text: i

text: k,

text: such

text: that

text: m

text: θ

text: B

text: [i

text: (θ,

text: x)

text: V

text: ν

text: B

text: [k

text: and

text: (ν,

text: y)

text: V

text: Then,

text: there

text: exist

text: i

text: i

text: k

text: k

text: EB

text: EB

text: θ

text: A

text: Bd

text: B

text: [i

text: and

text: ν

text: A

text: Bd

text: B

text: [k

text: i

text: m

text: k

text: m

text: S

text: where

text: A

text: denotes

text: the

text: closed

text: arc

text: of

text: such

text: that

text: diam(A)

text: d

text: (θ,

text: ν)

text: and

text: Bd(A)

text: θ,

text: ν

text: Clearly

text: we

text: have,

text: (θ,

text: x),

text: θ,

text: x

text: V

text: (ν,

text: y),

text: ν,

text: y

text: V

text: and,

text: by

text: the

text: previous

text: case,

text: i

text: k

text: d

text: (θ,

text: x),

text: θ,

text: x

text: d

text: θ,

text: θ

text: d

text: (θ,

text: ν)

text: d

text: ((θ,

text: x),

text: (ν,

text: y))

text: δ,

text: Ω

text: Ω

text: d

text: T

text: x),

text: T

text: θ,

text: x

text: 3ε

text: Ω

text: m

text: d

text: (ν,

text: y),

text: ν,

text: y

text: d

text: ν,

text: ν

text: d

text: (θ,

text: ν)

text: d

text: ((θ,

text: x),

text: (ν,

text: y))

text: δ,

text: and

text: Ω

text: Ω

text: d

text: (ν,

text: y),

text: T

text: ν,

text: y

text: Ω

text: m

text: m

text: V

text: V

text: On

text: the

text: ν,

text: by

text: Lemma

text: and

text: m

text: m

text: d

text: θ,

text: x

text: ν,

text: y

text: max

text: d

text: θ,

text: ν

text: x

text: y

text: max

text: d

text: (θ,

text: ν),

text: x

text: y

text: Ω

text: d

text: ((θ,

text: x),

text: (ν,

text: y))

text: δ

text: δ

text: and

text: Ω

text: m,i

text: d

text: (T

text: (θ,

text: x),

text: T

text: (ν,

text: y))

text: d

text: T

text: (θ,

text: x),

text: T

text: θ,

text: x

text: d

text: T

text: θ,

text: x

text: T

text: ν,

text: y

text: Ω

text: m

text: m

text: Ω

text: m

text: m

text: d

text: T

text: ν,

text: y

text: T

text: (ν,

text: y)

text: Ω

text: m

text: m

text: 3ε

text: d

text: T

text: θ,

text: x

text: T

text: ν,

text: y

text: 3ε

text: ε.

text: Ω

text: m

text: m

text: V

text: If

text: (θ,

text: x),

text: (ν,

text: y)

text: then,

text: by

text: Deﬁnition

text: and

text: m

text: d

text: (T

text: (θ,

text: x),

text: T

text: (ν,

text: y))

text: d

text: (T

text: (θ,

text: x),

text: T

text: (ν,

text: y))

text: ε

text: Ω

text: m

text: m

text: Ω

text: m

text: m

text: because

text: d

text: ((θ,

text: x),

text: (ν,

text: y))

text: δ

text: δ

text: ∈Lastly∗,Ωassume

text: that

text: (ν,

text: y)

text: Vmm

text: Vi∗

text: Vm∈,

text: for

text: some

text: ∗i

text: ∈⊂DmEB(that

text: is,

text: θ

text: Bi

text: [i

text: as

text: before,

text: ⊂BVd

text: (B⊂i

text: V[i

text: m

text: such

text: that,

text: by

text: Lemma

text: and

text: Deﬁnition

text: θ,

text: x

text: and

text: m

text: m

text: m

text: d

text: (θ,

text: x),

text: θ,

text: x

text: δ,

text: Ω

text: d

text: θ,

text: x

text: y

text: δ

text: δ

text: Ω

text: m

text: d

text: T

text: (θ,

text: x),

text: T

text: θ,

text: x

text: and

text: Ω

text: m

text: m

text: d

text: (T

text: (θ,

text: x),

text: T

text: (ν,

text: y))

text: d

text: T

text: (θ,

text: x),

text: T

text: θ,

text: x

text: d

text: T

text: θ,

text: x

text: T

text: (ν,

text: y)

text: Ω

text: m

text: m

text: Ω

text: m

text: m

text: Ω

text: m

text: m

text: 3ε

text: d

text: T

text: θ,

text: x

text: T

text: (ν,

text: y)

text: ε.

text: Ω

text: m

text: m

text: This

text: ends

text: the

text: proof

text: of

text: the

text: continuity

text: of

text: T

text: and,

text: hence,

text: of

text: (a).

text: m

text: B

text: S

text: \B

text: (b)

text: When

text: θ

text: the

text: statement

text: follows

text: from

text: Lemma

text: When

text: θ

text: it

text: m

text: m

text: follows

text: from

text: the

text: part

text: already

text: proven

text: and

text: the

text: continuity

text: of

text: T

text: m

text: (c)

text: The

text: ﬁrst

text: two

text: statements

text: follow

text: from

text: Lemma

text: and

text: statement

text: (a).

text: On

text: the

text: other

text: hand,

text: as

text: in

text: the

text: proof

text: of

text: Proposition

text: for

text: m

text: Lemma

text: B

text: B

text: implies

text: that

text: i

text: but

text: i

text: for

text: every

text: k

text: m.

text: Then,

text: we

text: get

text: f

text: f

text: from

text: m

text: k

text: k,i

text: m,i

text: Deﬁnition

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##