text: SVM SVM(2) SVM(3) 1.0 SVM SVM(2) SVM(3) 1.0 OPIDe OPID OPIDe OPID 0.9 0.9 0.8 0.8 0.7 0.6 0.7 20 40 60 80 20 40 60 80 100 120 nubmber of points per batch nubmber of points per batch (a) USPS0vs5 (b) USPS0vs3vs5 SVM SVM(2) SVM(3) 1.00 SVM SVM(2) SVM(3) OPIDe OPID OPIDe OPID 0.6 0.98 0.96 0.5 0.94 0.92 0.4 0.90 60 150 300 450 30 60 90 120 nubmber of points per batch nubmber of points per batch (c) Protein (d) Satimage Figure 2: Testing accuracy of diﬀerent methods on diﬀerent data sets with diﬀerent numbers of training and testing instances. (1) When we use the classiﬁer trained in C-stage to assist learning in E-stage, the testing per- formance will increase signiﬁcantly, especially when the training points in E-stage are rare. This is consistent with intuition since the assistance from C-stage will be weaker with the increase of training points. (2) Compared with the accuracy of SVM(a), our results have a remarkable improvement. This validates that our methods could, to some extent, inherit the metric from C-stage. (3) It seems that the improvement of our method with respect to other approaches is much larger in multi-class scenario. The reason may be that the binary classiﬁcation accuracy is high enough 18 ycarucca gnitset ycarucca gnitset ycarucca gnitset ycarucca gnitset ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: There are at least two observations from Fig. (3). (1) Our proposed methods outperform traditional methods, no matter what the percentage of survived features is. It validates the eﬀectiveness of our method in dealing the problem in this setting. (2) With the increase number of survived features, OPID and OPIDe achieve higher accuracies on SensIT Vehicle data. Nevertheless, their performances have a little turbulence on Gisette data. The reason may be that compared with SensIT Vehicle data whose total feature number is 100, the feature number of Gisette is 4955. It trends to be suﬀered from the curse of dimensionality [Don00]. 5. Conclusion In this paper, we study the problem of learning with incremental and decremental features, and propose the one-pass learning approach that does not need to keep the whole data for optimization. Our approach is particularly useful when both the data and features are evolving, while robust learning performance are needed, and is well scalable because it only needs to scan each instance once. In this paper we focus on one-shot feature change where the survived and augmented features do not vanish. It will be interesting to extend to multi-shot feature change where the survived and augmented features can vanish later. References [Bre96] Leo Breiman. Stacked regressions. Machine Learning, 24(1):49–64, 1996. [CBL06] Nicolo Cesa-Bianchi and Gabor Lugosi. Prediction, Learning, and Games. Cambridge University Press, New York, NY, USA, 2006. [CP00] Gert Cauwenberghs and Tomaso A. Poggio. Incremental and decremental support vector machine learning. In NIPS 2000, pages 409–415, 2000. [Dem06] Janez Demsar. Statistical comparisons of classiﬁers over multiple data sets. Journal of Machine Learning Research, 7:1–30, 2006. [Don00] David L. Donoho. High-dimensional data analysis: The curses and blessings of di- mensionality. In AMS Conference on Math Challenges of the 21st Century, 2000. 20

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text: the

text: Century,


text: The previous statement makes it possible to derive the dimension formula of π(K) by using an adapted Mandelbrot measure, whilst in [22] Falconer builds statistically self- similar subsets of π(K) of Hausdorﬀ dimension smaller than but arbitrarily close to the value given by (1.2). The new point is the variational principle invoking Mandelbrot measures in (3.6) and the related uniqueness property. Corollary 3.4 (Dekking-Grimmett-Falconer formula revisited). Let (cid:32) (cid:33) (cid:88) − m 1 ≥ (cid:55)→ E (3.5) ϕ : h 0 log (N )h / log(m). i i=0 (cid:54) ∅ With probability 1, conditionally on K = , we have dim π(K) = dim (π(K)) H B (3.6) = ≤in≤f ϕ(h) 0 h 1 { (cid:48) (cid:48) } = max dim(π∗µ ) : µ is a Mandelbrot measure supported on K . (cid:80) (cid:48) ≤ Moreover, the maximum in (3.6) is attained at a unique point if and only if ϕ (0) 0, − E ≤ i.e. m 1 log( (N )) 0. i=0 i (cid:80) − E E ≤ Remark 3.5. One has dim π(K) = dim K if and only if m 1 (N ) log (N ) 0, H H i=0 i i i.e. in (3.6) the inﬁmum is attained at h = 1. (cid:40) Assume (2.4) and deﬁne (cid:101) ∞ ≥ q if q < and τ (q ) T (q ) c c ν c c q = { ≥ } , c inf q > q : τ (q) T (q) otherwise  (cid:40) c ν (cid:41)  (cid:88)∅ with the convention that inf = qc. Let  − − (cid:40) m 1 E ≤ ≤ (cid:41) inf log (cid:88) (N )h : 0 h 1 if q = 0,  m i  (cid:55)→  i=−0 τ : q − m 1 − ≤ ≤ ≤  inf log pqm qTi(s)/s : q s 1 if 0 < q 1(cid:101), (cid:101) m i i=0 ∞ min(τ (q), T (q)) if 1 < q < q or q = q < . ν c c 12

text: The

text: previous

text: statement

text: makes

text: it

text: possible

text: to

text: derive

text: the

text: dimension

text: formula

text: of

text: π(K)

text: by

text: using

text: an

text: adapted

text: Mandelbrot

text: measure,

text: whilst

text: in

text: Falconer

text: builds

text: statistically

text: self-

text: similar

text: subsets

text: of

text: π(K)

text: of

text: Hausdorﬀ

text: dimension

text: smaller

text: than

text: but

text: arbitrarily

text: close

text: to

text: the

text: value

text: given

text: by

text: The

text: new

text: point

text: is

text: the

text: variational

text: principle

text: invoking

text: Mandelbrot

text: measures

text: in

text: and

text: the

text: related

text: uniqueness

text: property.

text: Corollary

text: (Dekking-Grimmett-Falconer

text: formula

text: revisited).

text: Let

label: equation

text: m

text: E

text: ϕ

text: h

text: log

text: (N

text: )h

text: log(m).

text: i

text: With

text: probability

text: conditionally

text: on

text: K

text: we

text: have

text: dim

text: π(K)

text: dim

text: (π(K))

text: H

text: B

text: ≤in≤f

text: ϕ(h)

text: h

text: max

text: dim(π∗µ

text: µ

text: is

text: a

text: Mandelbrot

text: measure

text: supported

text: on

text: K

text: Moreover,

text: the

text: maximum

text: in

text: is

text: attained

text: at

text: a

text: unique

text: point

text: if

text: and

text: only

text: if

text: ϕ

text: E

text: i.e.

text: m

text: log(

text: (N

text: i

text: E

text: E

text: Remark

text: One

text: has

text: dim

text: π(K)

text: dim

text: K

text: if

text: and

text: only

text: if

text: m

text: (N

text: log

text: (N

text: H

text: H

text: i

text: i

text: i.e.

text: in

text: the

text: inﬁmum

text: is

text: attained

text: at

text: h

text: Assume

text: and

text: deﬁne

text: q

text: if

text: q

text: and

text: τ

text: (q

text: T

text: (q

text: c

text: c

text: ν

text: c

text: c

text: q

text: c

text: inf

text: q

text: q

text: τ

text: (q)

text: T

text: (q)

text: otherwise

text: c

text: ν

text: with

text: the

text: convention

text: that

text: inf

text: qc.

text: Let

text: m

text: E

text: inf

text: log

text: (N

text: )h

text: h

text: if

text: q

text: m

text: i

text: τ

text: q

text: m

text: inf

text: log

text: pqm

text: qTi(s)/s

text: q

text: s

text: if

text: q

text: m

text: i

text: min(τ

text: (q),

text: T

text: (q))

text: if

text: q

text: q

text: or

text: q

text: q

text: ν

text: c

text: c


text: (cid:101) Theorem 3.6. The function τ is diﬀerentiable everywhere except at the possible points in (1, q ) at which the graphs of T and τ cross each other transversa(cid:101)lly. Moreover, c ν { (cid:54) } ∈ (1) with probability 1, conditionally on(cid:88)µ = 0 , for all q [0, q ) we have c − 1 (3.7) τ(q) = nl→im∞ n logm | | 1(cid:101){π∗µ([u])>0}π∗µ([u])q. u =n ∞ (cid:48) − In partic(cid:101)ular τπ∗µ(q) = τ(q). Also, if qc = qc < , we have τπ∗µ(q) = qT (qc ) for q > q . c ∈ (cid:48) − (cid:48) { (cid:54) } (2) If α (τ (q ), τ (0+)], with probability 1, conditionally on µ = 0 , the multi- c fractal formalism holds at α. ∞ (cid:48) ≤ Remark 3.7. Notice that when q < , the equality τ (q ) = T (q ) cannot hold if τ (q ) (cid:48) − c ∗ (cid:48) ≤ ∗ ν(cid:48) c− c ∗ ◦ (cid:48) ν c T (q ), for this would imply that τ (τ (q )) T (T (q )) = 0, while τ τ is always c ν ν c c ν ν R positive over . Remark 3.8. The same conclusion as in (3.7) and Theorem 3.6(2) holds if we replace π∗µ([x| ]) by π∗µ ([x| ]) in the deﬁnition of the level sets E(π∗µ, α). n n n ≤ ≤ − Remark 3.9. (1) One has τ = T = τ over [0, q ) if and only if for all 0 i m 1 ν c E ∈ { } ≤ ≤ − such that p > 0 one has (N ) = 1 and W 0, p almost surely for all 0 j m 1, i i i,j i i.e. T is identically equal to 0 (see Section ). i R (2) On the other hand, a suﬃcient condition to have τ = τ o(cid:81)ver and τ > T over ν + ν ∞ P { } { ≤ ≤ (0, 1) and τ < T over (1, ) is ( ): there exists a partition I , . . . , I of 0 i ν 1 L − } ∈ E m 1 : p > 0 such that (i) p does not depend on i I and ∈ (N ) = 1 for each ≤ ≤ i ≤i ≤ ≥ k E i Ik(cid:54) i 1 k L; (ii) there exists 1 k L such that #I 2 and (N ) = 1 for at least two k i ∈ ≤ ≤ − ∈ { E } values of i I ; (iii) for all 0 i m 1 such that p > 0, one has W 0, p / (N ) k i i,j i i ≤ ≤ − almost surely for all 0 j m 1. See the proof of Lemma 8.6. Remark 3.10. In all the examples we have examined numerically and for which we do not have τ = T , the functions τ and T coincide at three points at most. We do not know ν ν whether this is a general fact. 13 ##LTLine##

text: Theorem

text: The

text: function

text: τ

text: is

text: diﬀerentiable

text: everywhere

text: except

text: at

text: the

text: possible

text: points

text: in

text: q

text: at

text: which

text: the

text: graphs

text: of

text: T

text: and

text: τ

text: cross

text: each

text: other

label: list

text: Moreover,

text: c

text: ν

text: with

text: probability

text: conditionally

text: for

text: all

text: q

text: q

text: we

text: have

text: c

text: τ(q)

text: nl→im∞

text: n

text: logm

text: u

text: =n

text: In

text: τπ∗µ(q)

text: τ(q).

text: Also,

text: if

text: qc

text: qc

text: we

text: have

text: τπ∗µ(q)

text: qT

text: (qc

text: for

text: q

text: q

text: c

text: If

text: α

text: (τ

text: (q

text: τ

text: with

text: probability

text: conditionally

text: on

text: µ

text: the

text: multi-

text: c

text: fractal

text: formalism

text: holds

text: at

text: α.

text: Remark

text: Notice

text: that

text: when

text: q

text: the

text: equality

text: τ

text: (q

text: T

text: (q

text: cannot

text: hold

text: if

text: τ

text: (q

text: c

text: c−

text: c

text: ν

text: c

text: T

text: (q

text: for

text: this

text: would

text: imply

text: that

text: τ

text: (τ

text: (q

text: T

text: (T

text: (q

text: while

text: τ

text: τ

text: is

text: always

text: c

text: ν

text: ν

text: c

text: c

text: ν

text: ν

text: R

text: positive

text: over

text: Remark

text: The

text: same

text: conclusion

text: as

text: in

text: and

text: Theorem

text: holds

text: if

text: we

text: replace

text: π∗µ([x|

text: by

text: π∗µ

text: ([x|

text: in

text: the

text: deﬁnition

text: of

text: the

text: level

text: sets

text: E(π∗µ,

text: α).

text: n

text: n

text: n

text: Remark

text: One

text: has

text: τ

text: T

text: τ

text: over

text: q

text: if

text: and

text: only

text: if

text: for

text: all

text: i

text: m

text: ν

text: c

text: E

text: such

text: that

text: p

text: one

text: has

text: (N

text: and

text: W

text: p

text: almost

text: surely

text: for

text: all

text: j

text: m

text: i

text: i

text: i,j

text: i

text: i.e.

text: T

text: is

text: identically

text: equal

text: to

text: (see

text: Section

text: i

text: R

text: On

text: the

text: other

text: hand,

text: a

text: suﬃcient

text: condition

text: to

text: have

text: τ

text: τ

text: and

text: τ

text: T

text: over

text: ν

text: ν

text: P

text: and

text: τ

text: T

text: over

text: is

text: there

text: exists

text: a

text: partition

text: I

text: I

text: of

text: i

text: ν

text: L

text: E

text: m

text: p

text: such

text: that

text: (i)

text: p

text: does

text: not

text: depend

text: on

text: i

text: I

text: and

text: (N

text: for

text: each

text: i

text: ≤i

text: k

text: E

text: i

text: i

text: k

text: L;

text: (ii)

text: there

text: exists

text: k

text: L

text: such

text: that

text: #I

text: and

text: (N

text: for

text: at

text: least

text: two

text: k

text: i

text: E

text: values

text: of

text: i

text: I

text: (iii)

text: for

text: all

text: i

text: m

text: such

text: that

text: p

text: one

text: has

text: W

text: p

text: (N

text: k

text: i

text: i,j

text: i

text: i

text: almost

text: surely

text: for

text: all

text: j

text: m

text: See

text: the

text: proof

text: of

text: Lemma

text: Remark

text: In

text: all

text: the

text: examples

text: we

text: have

text: examined

text: numerically

text: and

text: for

text: which

text: we

text: do

text: not

text: have

text: τ

text: T

text: the

text: functions

text: τ

text: and

text: T

text: coincide

text: at

text: three

text: points

text: at

text: most.

text: We

text: do

text: not

text: know

text: ν

text: ν

text: whether

text: this

text: is

text: a

text: general

text: fact.

text: ##LTLine##


text: 0 70 65 −0.2 60 −0.4 55 50 −0.6 45 40 −0.8 35 −1 30 25 −1.2 20 −1.4 15 10 −1.6 5 0 −1.8 −5 −2 −10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 10 20 30 40 50 60 70 80 90 100 110 Figur(cid:101)e 2. Illustration of Example 4.3 with m = 2, p = .62, β = 3.22, 0 λ = .99 and V = .99 (blue curve: T ; black curve: τ ; red curve: τ). ∞ 00 ν ∈ q = q = . One second order phase transition at some q (0, 1) and c c (cid:48) ∈ ∞ 0 one ﬁrst order phase transition at (cid:48)some q0 (1, ). τπ∗µ = τ >∞max(τν, T ) over [0, q0), τπ∗µ = T = τµ over [q0, q0], and τπ∗µ = τν over [q0, ). E E (4.1) (N ) (N ) > 1, (cid:18) 0 (cid:19) 1 (cid:18) (cid:19) E E − (4.2) (N )p0 (N )1 p0 < 1, 0 1 E(N ) p0 E(N ) 1−p0 (4.3) 0 − 1 > 1. p 1 p 0 0 (cid:48) (cid:48) E E Properties (4.2) and (4.3) yield τ (1) > T (1) > 0. Also (4.1) implies (N )+ (N ) > 2 − ν 1 2 hence T (0) < 1 = τ (0). The graphs of τ and T cross each other on [0, 1]. Let G be ν ν ∈ deﬁned as in the previous example. Property (4.1) yields ∂G(q, s) < 0 for all s [q, 1] ∂s if q is close enough to 0, hence τ(q) is attained at q = 1; τ(q) = τ (q). Moreover, (4.2) ν ∈ implies that ∂G(q, s) > 0 for all s [q, 1] if q is close to 1, hence τ(q) is attained at s = q: ∂s τ(q) = T (q). Then our study of τ in Section implies that on a non trivial interval we have τ(q) > max(τ (q), T (q)), i.e. τ is given by a third analytic expression. ν (cid:48) − (cid:48) ∞ It is also possible to choose the parameters so that T (1) < log (p ) = τ (+ ) and E ∞ ∞ 2 0 ν p0 > (N0), hence τν > T over (1, ) and qc < , which implies that τπ∗µ(q) = τµ(q) ∞ E E over [1, ). A concrete choice is p = .8, (N ) = .6 and (N ) = 1.8. 0 0 1 Example 4.5 (Previous example continued). We can use the same model as in Exam- ple 4.4 to get other diﬀerent behaviors. See Figures 4 to 6. 18 ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: −1

text: −5

text: −2

text: Illustration

text: of

text: Example

text: with

text: m

text: p

text: β

text: λ

text: and

text: V

text: (blue

text: curve:

text: T

text: black

text: curve:

text: τ

text: red

text: curve:

text: τ).

text: ν

text: q

text: q

text: One

text: second

text: order

text: phase

text: transition

text: at

text: some

text: q

text: and

text: c

text: c

text: one

text: ﬁrst

text: order

text: phase

text: transition

text: at

text: τπ∗µ

text: τ

text: >∞max(τν,

text: T

text: over

text: τπ∗µ

text: T

text: τµ

text: over

text: and

text: τπ∗µ

text: τν

text: over

text: E

text: E

text: (N

text: (N

text: E

text: E

text: (N

text: (N

text: E(N

text: E(N

text: p

text: p

text: E

text: E

text: Properties

text: and

text: yield

text: τ

text: T

text: Also

text: implies

text: (N

text: (N

text: ν

text: hence

text: T

text: τ

text: The

text: graphs

text: of

text: τ

text: and

text: T

text: cross

text: each

text: other

text: on

text: Let

text: G

text: be

text: ν

text: ν

text: deﬁned

text: as

text: in

text: the

text: previous

text: example.

text: Property

text: yields

text: ∂G(q,

text: s)

text: for

text: all

text: s

text: [q,

text: ∂s

text: if

text: q

text: is

text: close

text: enough

text: to

text: hence

text: τ(q)

text: is

text: attained

text: at

text: q

text: τ(q)

text: τ

text: (q).

text: Moreover,

text: ν

text: implies

text: that

text: ∂G(q,

text: s)

text: for

text: all

text: s

text: [q,

text: if

text: q

text: is

text: close

text: to

text: hence

text: τ(q)

text: is

text: attained

text: at

text: s

text: q:

text: ∂s

text: τ(q)

text: T

text: (q).

text: Then

text: our

text: study

text: of

text: τ

text: in

text: Section

text: implies

text: that

text: on

text: a

text: non

text: trivial

text: interval

text: we

text: have

text: τ(q)

text: max(τ

text: (q),

text: T

text: (q)),

text: i.e.

text: τ

text: is

text: given

text: by

text: a

text: third

text: analytic

text: expression.

text: ν

text: It

text: is

text: also

text: possible

text: to

text: choose

text: the

text: parameters

text: so

text: that

text: T

text: log

text: (p

text: τ

text: and

text: E

text: ν

text: hence

text: τν

text: T

text: over

text: and

text: qc

text: which

text: implies

text: that

text: τπ∗µ(q)

text: τµ(q)

text: E

text: E

text: over

text: A

text: concrete

text: choice

text: is

text: p

text: (N

text: and

text: (N

text: Example

text: (Previous

text: example

text: continued).

text: We

text: can

text: use

text: the

text: same

text: model

text: as

text: in

text: Exam-

text: ple

text: to

text: get

text: other

text: diﬀerent

text: behaviors.

text: See

text: Figures

text: to

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

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text: 1.4 1.2 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −1.2 −1.4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 E Figure 5. Same(cid:101)model as in Example 4.4 with p = .3, (N ) = .25 and 0 0 E (N ) = 2 (blue curve: T ; dashed blue curve: τ ; black curve: τ ; red 1 (cid:39) ∞ µ ν curve: τ). q = q 2.176 < . One second order phase transition at ∈ c c (cid:48) ∈ some q (0, 1). One ﬁrst order phase transition at some q (1, q ), and 0 0 c one second ord−er phase transition at qc. τπ−∗µ = τ >−max(τν, T ) over [0, q0) (in p(cid:48)articular τ(0) = dimH π(K) < min( τν(0), T (0))), τπ∗µ(cid:48)= τν over [q0,∞q0] and τπ∗µ = T = τµ over [q0, qc], and τπ∗µ(q) = τµ(q) = T (qc)q over [q , ). c (cid:80) Proposition 5.1. Let q > 1 such that T (q) > 0. Let η be the Bernoulli product measure (cid:48) (cid:48) { − (cid:48) − } on Σ generated by a probability vector (p , . . . , p − ). Set A := max 1, m 1 p m Ti(q) . 0 m 1 i=0 i Then there exists a polynomial f depending on W and q such that q ≤ E ≤ ∀ ∈ N (5.3) An P⊗ (Xq) f (n)An, n . η n q ∈ Furthermore, when q (1, 2], f (n) can be taken as a constant. q Corollary 5.2. Let q > 1 such that T (q) > 0. Then there exists a polynomial f depending q on W and q such that (cid:16) (cid:88) (cid:17) − { } ≤ E ≤ − { } (5.4) m nmin τν(q),T(q) π∗µ([u])q f (n)m nmin τν(q),T(q) q ∈ u Σn 21 ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: −1

text: E

text: Figure

text: as

text: in

text: Example

text: with

text: p

text: (N

text: and

text: E

text: (N

text: (blue

text: curve:

text: T

text: dashed

text: blue

text: curve:

text: τ

text: black

text: curve:

text: τ

text: red

text: µ

text: ν

text: curve:

text: τ).

text: q

text: q

text: One

text: second

text: order

text: phase

text: transition

text: at

text: c

text: c

text: some

text: q

text: One

text: ﬁrst

text: order

text: phase

text: transition

text: at

text: some

text: q

text: q

text: and

text: c

text: one

text: second

text: ord−er

text: phase

text: transition

text: at

text: qc.

text: τπ−∗µ

text: τ

text: >−max(τν,

text: T

text: over

text: (in

text: dimH

text: π(K)

text: min(

text: T

text: τν

text: over

text: and

text: τπ∗µ

text: T

text: τµ

text: over

text: qc],

text: and

text: τπ∗µ(q)

text: τµ(q)

text: T

text: (qc)q

text: over

text: [q

text: c

text: Proposition

text: Let

text: q

text: such

text: that

text: T

text: (q)

text: Let

text: η

text: be

text: the

text: Bernoulli

text: product

text: measure

text: on

text: Σ

text: generated

text: by

text: a

text: probability

text: vector

text: (p

text: p

text: Set

text: A

text: max

text: m

text: p

text: m

text: Ti(q)

text: m

text: i

text: Then

text: there

text: exists

text: a

text: polynomial

text: f

text: depending

text: on

text: W

text: and

text: q

text: such

text: that

text: q

text: E

text: N

text: An

text: P⊗

text: (Xq)

text: f

text: (n)An,

text: n

text: η

text: n

text: q

text: Furthermore,

text: when

text: q

text: f

text: (n)

text: can

text: be

text: taken

text: as

text: a

text: constant.

text: q

text: Corollary

text: Let

text: q

text: such

text: that

text: T

text: (q)

text: Then

text: there

text: exists

text: a

text: polynomial

text: f

text: depending

text: q

text: on

text: W

text: and

text: q

text: such

text: that

text: E

text: m

text: nmin

text: τν(q),T(q)

text: π∗µ([u])q

text: f

text: (n)m

text: nmin

text: τν(q),T(q)

text: q

text: u

text: Σn

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text: hence both the absolute continuity of ν with respect to π∗µ and the desired result about (cid:3) the density of ν with respect to π∗µ. ≤ Proof of Theorem 3.1(2). If dim(µ) < dim(ν), there is nothing to prove since dim (π∗µ) P dim(µ). (cid:80) (cid:101) (cid:48) P ⊗ Suppose now that dim(µ) = T (1)(cid:81)= dim(ν). This time, under ν, the martingale − (cid:48) − X (ω, x) converges to 0 almost surely since m 1 p T (1 ) = 0 (see Theorem B.1 again). n i=0 i i This implies that M (x) = max ∈ n V (x| − , v| − ) converge to 0 almost surely P⊗ n v Σn j=1 xj,vj j 1 j 1 E under ν. Using (5.6) this time yields the convergence in law to 0 for X , and P⊗ (fλ) = n ν ∈ P ⊗ 0 for all λ (0, 1). Consequently, f = 0 with ν probability 1, which is equivalent to (cid:3) the fact that π∗µ and ν are almost surely mutually singular. When dim(µ) > dim(ν), since by Theorem 3.1(1)(i) π∗µ is absolutely continuous with (cid:54) respect to ν, we already know that if µ = 0, we have dim(π∗µ) = dim(ν). However, under the assumption that T is ﬁnite in a neighborhood of 1, we give an alternative proof which works regardless of the respective positions of dim(µ) and dim(ν), and independently of absolute continuity considerations. We will use Corollary 5.2 and the following elementary lemma. (cid:80) (cid:80) ≥ Lemma 5.4. Let ρ be a positive and ﬁnite Borel measure on Σ. Let D 0. If for − − ∞ all (cid:15)(cid:80)> 0 there exists (cid:80)q > 1 such that ≥ mn(q 1)(D (cid:15)) | | ρ([u])q < , then ≥ n 1 u =n ∈ dim (ρ, x) D for ρ-almost every x. Also, if for all (cid:15) > 0 there exists q (0, 1) such loc − ∞ ≤ that ≥ mn(q 1)(D+(cid:15)) | | ρ([u])q < , then dim (ρ, x) D for ρ-almost every x. n 1 u =n loc ≥ Proof. Fix (cid:15) > 0. For all q > 1 and n 1, applying Markov’s inequality we have (cid:16)(cid:110) (cid:111)(cid:17) (cid:16)(cid:110) (cid:111)(cid:17) ∈ log(ρ([x| ])) ≤ − ∈ (cid:90) − ≥ − − − ρ x Σ : − n D (cid:15) = ρ x Σ : ρ([x| ])q 1 m n(q 1)(D (cid:15)) n log(m) n ≤ − − (cid:88) − mn(q 1)(D (cid:15)) ρ([x| ])q 1 ρ(dx) n Σ − − = mn(q 1)(D (cid:15)) ρ([u])q. | | u =n 26 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: hence

text: both

text: the

text: absolute

text: continuity

text: of

text: ν

text: with

text: respect

text: to

text: π∗µ

text: and

text: the

text: desired

text: result

text: about

text: the

text: density

text: of

text: ν

text: with

text: respect

text: to

text: π∗µ.

text: Proof

text: of

text: Theorem

text: If

text: dim(µ)

text: dim(ν),

text: there

text: is

text: nothing

text: to

text: prove

text: since

text: dim

text: (π∗µ)

text: P

text: dim(µ).

text: P

text: Suppose

text: now

text: that

text: dim(µ)

text: T

text: dim(ν).

text: This

text: time,

text: under

text: ν,

text: the

text: martingale

text: X

text: (ω,

text: x)

text: converges

text: to

text: almost

text: surely

text: since

text: m

text: p

text: T

text: (see

text: Theorem

text: again).

text: n

text: i

text: i

text: This

text: implies

text: that

text: M

text: (x)

text: max

text: n

text: V

text: (x|

text: v|

text: converge

text: to

text: almost

text: surely

text: P⊗

text: n

text: v

text: Σn

text: xj,vj

text: j

text: j

text: E

text: under

text: ν.

text: Using

text: this

text: time

text: yields

text: the

text: convergence

text: in

text: law

text: to

text: for

text: X

text: and

text: P⊗

text: (fλ)

text: n

text: ν

text: P

text: for

text: all

text: λ

text: Consequently,

text: f

text: with

text: ν

text: probability

text: which

text: is

text: equivalent

text: to

text: the

text: fact

text: that

text: π∗µ

text: and

text: ν

text: are

text: almost

text: surely

text: mutually

text: singular.

text: When

text: dim(µ)

text: dim(ν),

text: since

text: by

text: Theorem

text: π∗µ

text: is

text: absolutely

text: continuous

text: with

text: respect

text: to

text: ν,

text: we

text: already

text: know

text: that

text: if

text: µ

text: we

text: have

text: dim(π∗µ)

text: dim(ν).

text: However,

text: under

text: the

text: assumption

text: that

text: T

text: is

text: ﬁnite

text: in

text: a

text: neighborhood

text: of

text: we

text: give

text: an

text: alternative

text: proof

text: which

text: works

text: regardless

text: of

text: the

text: respective

text: positions

text: of

text: dim(µ)

text: and

text: dim(ν),

text: and

text: independently

text: of

text: absolute

text: continuity

text: considerations.

text: We

text: will

text: use

text: Corollary

text: and

text: the

text: following

text: elementary

text: lemma.

text: Lemma

text: Let

text: ρ

text: be

text: a

text: positive

text: and

text: ﬁnite

text: Borel

text: measure

text: on

text: Σ.

text: Let

text: D

text: If

text: for

text: all

text: there

text: exists

text: such

text: that

text: mn(q

text: ρ([u])q

text: then

text: n

text: u

text: =n

text: dim

text: (ρ,

text: x)

text: D

text: for

text: ρ-almost

text: every

text: x.

text: Also,

text: if

text: for

text: all

text: there

text: exists

text: q

text: such

text: loc

text: that

text: mn(q

text: ρ([u])q

text: then

text: dim

text: (ρ,

text: x)

text: D

text: for

text: ρ-almost

text: every

text: x.

text: n

text: u

text: =n

text: loc

text: Proof.

text: Fix

text: For

text: all

text: q

text: and

text: n

text: applying

text: Markov’s

text: inequality

text: we

text: have

text: log(ρ([x|

text: ρ

text: x

text: Σ

text: n

text: D

text: ρ

text: x

text: Σ

text: ρ([x|

text: ])q

text: m

text: n(q

text: n

text: log(m)

text: n

text: mn(q

text: ρ([x|

text: ])q

text: ρ(dx)

text: n

text: Σ

text: mn(q

text: ρ([u])q.

text: u

text: =n

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:48) (cid:48) Let µ be the a(cid:80)ssociated Mandelbrot measure and(cid:32)ν the Berno(cid:33)ulli product associated (cid:48) (cid:88) with p . We have (cid:80) dim(ν(cid:48)) = −h0 mi=−01 Emi=(−N01i)Eh(0Nloi)ghm0(E(Ni)) + logm mi=−01 E(Ni)h0 = ϕ(h0) − h0ϕ(cid:48)(h0) and (cid:80) (cid:88) (cid:80) (5.8) mi=−01 p(cid:48)iTV(cid:48)i(cid:48)(1) = mi=−01 E(miN=−0i1)hE0(lNogim)h(0E(Ni)) = ϕ(cid:48)(h0). ≥ (cid:48) ≥ Next we show that dim π(K) dim π(µ ) ϕ(h ), by considering the scenarios H H 0 ∈ (cid:48) h = 1, h (0, 1) and h = 0, separately. First suppose that h = 1. Then µ is the 0 0 0 0 (cid:88) so-called branching measure on K, and we see that − m 1 (cid:48) (cid:48) (cid:48) E dim(ν ) + p T (cid:48)(1) = ϕ(1) = log( (N))/ log(m) > 0, i V i i=0 (cid:48) hence µ is non-degenerate with positive p(cid:80)robability (a fact that can also be directly seen (cid:48) from T (cid:48)(1)). Moreover, since on [0, 1] ϕ takes the minimum at h = 1, by smoothness of W (cid:48) ≤ − (cid:48) (cid:48) ≤ ϕ we mu(cid:48)st have ϕ (cid:48)(1) 0, consequently mi=01 piTVi(cid:48)(1) 0(cid:80), and(cid:54) th∅us by Theorem 3.2, dim(π∗µ ) = dim(µ ) and dim π(K) = dim (K) = ϕ(1) when K = . H H (cid:48) − (cid:48) (cid:48) Next suppose that 0 < h < 1. We have ϕ (h ) = 0, hence m 1 p T (cid:48)(1) = 0 and thus 0 0 i=0 i V i by Theorem 3.2, (cid:48) (cid:48) (cid:48) dim(π∗µ ) = dim(µ ) = dim(ν ) = ϕ(h ), (cid:80) 0 ≥ (cid:54) ∅ yielding dim π(K) ϕ(h ) when K = . H 0 (cid:48) ≥ − (cid:48) (cid:48) ≥ Finally suppose that h = 0. Then ϕ (0) 0, so m 1 p T (cid:48)(1) 0 and thus by (cid:48) 0 (cid:48) ≤ (cid:48) i=0 i(cid:48) (cid:54)Vi Theorem 3.2, dim(π∗µ ) = dim(ν ) = ϕ(0) dim(µ ) when µ = 0, and consequently, ≥ (cid:54) ∅ dim π(K) ϕ(0) when K = . H So far we have proved (3.6). Below we discuss the uniqueness problem regarding the last variational relation in (3.6). (cid:48) Notice that the Mandelbrot measure µ considered above has a dimension equal to (cid:48) − E dim K if and only if T (cid:48)(1) = T (cid:48)(0) = log ( (N)), that is T (cid:48) is linear. If h = 1, (cid:48) H W (cid:16) (cid:88) W (cid:17) m (cid:88) W 0 µ is the branching measure. If h < 1, since 0 − − m 1 m 1 E − E − T (cid:48)(q) = q log (N )h0 log 1{ } (N )1+q(h0 1) W m i m pi>0 i i=0 i=0 E and the second derivative of T (cid:48) vanishes, we get that (N ) = 1 for each i such that W i (cid:48) p > 0. Once again µ is the branching measure. i For the uniqueness problem, the case when dim K = dim π(K) is clear from the H H above discussion, since the same argument in fact shows that a Mandelbrot measure 33 ##LTLine## ##LTLine##

text: Let

text: µ

text: be

text: the

text: Mandelbrot

text: measure

text: the

text: product

text: associated

text: with

text: p

text: We

text: have

text: logm

text: and

text: Next

text: we

text: show

text: that

text: dim

text: π(K)

text: dim

text: π(µ

text: ϕ(h

text: by

text: considering

text: the

text: scenarios

text: H

text: H

text: h

text: h

text: and

text: h

text: separately.

text: First

text: suppose

text: that

text: h

text: Then

text: µ

text: is

text: the

text: so-called

text: branching

text: measure

text: on

text: K,

text: and

text: we

text: see

text: that

text: m

text: E

text: dim(ν

text: p

text: T

text: log(

text: (N))/

text: log(m)

text: i

text: V

text: i

text: hence

text: µ

text: is

text: non-degenerate

text: with

text: positive

text: (a

text: fact

text: that

text: can

text: also

text: be

text: directly

text: seen

text: from

text: T

text: Moreover,

text: since

text: on

text: ϕ

text: takes

text: the

text: minimum

text: at

text: h

text: by

text: smoothness

text: of

text: W

text: ϕ

text: we

text: have

text: ϕ

text: consequently

text: th∅us

text: by

text: Theorem

text: dim(π∗µ

text: dim(µ

text: and

text: dim

text: π(K)

text: dim

text: (K)

text: when

text: K

text: H

text: H

text: Next

text: suppose

text: that

text: h

text: We

text: have

text: ϕ

text: (h

text: hence

text: m

text: p

text: T

text: and

text: thus

text: i

text: V

text: i

text: by

text: Theorem

text: dim(π∗µ

text: dim(µ

text: dim(ν

text: ϕ(h

text: yielding

text: dim

text: π(K)

text: ϕ(h

text: when

text: K

text: H

text: Finally

text: suppose

text: that

text: h

text: Then

text: ϕ

text: so

text: m

text: p

text: T

text: and

text: thus

text: by

text: Theorem

text: dim(π∗µ

text: dim(ν

text: dim(µ

text: when

text: µ

text: and

text: consequently,

text: dim

text: π(K)

text: when

text: K

text: H

text: So

text: far

text: we

text: have

text: proved

text: Below

text: we

text: discuss

text: the

text: uniqueness

text: problem

text: regarding

text: the

text: last

text: variational

text: relation

text: in

text: Notice

text: that

text: the

text: Mandelbrot

text: measure

text: µ

text: considered

text: above

text: has

text: a

text: dimension

text: equal

text: to

text: E

text: dim

text: K

text: if

text: and

text: only

text: if

text: T

text: T

text: log

text: (N)),

text: that

text: is

text: T

text: is

text: linear.

text: If

text: h

text: H

text: W

text: W

text: m

text: W

text: µ

text: is

text: the

text: branching

text: measure.

text: If

text: h

text: since

text: m

text: m

text: E

text: E

text: T

text: q

text: log

text: (N

text: log

text: (N

text: W

text: m

text: i

text: m

text: i

text: E

text: and

text: the

text: second

text: derivative

text: of

text: T

text: vanishes,

text: we

text: get

text: that

text: (N

text: for

text: each

text: i

text: such

text: that

text: W

text: i

text: p

text: Once

text: again

text: µ

text: is

text: the

text: branching

text: measure.

text: i

text: For

text: the

text: uniqueness

text: problem,

text: the

text: case

text: when

text: dim

text: K

text: dim

text: π(K)

text: is

text: clear

text: from

text: the

text: H

text: H

text: above

text: discussion,

text: since

text: the

text: same

text: argument

text: in

text: fact

text: shows

text: that

text: a

text: Mandelbrot

text: measure

text: ##LTLine##

text: ##LTLine##


text: (cid:48) − ∂G(q, s(q)) ∂q (6.1) τ (q) = log(m)G(q, s(q)) (cid:48) − implies that τ (q) has a limit at 1 , hence by the mean value theorem τ is left diﬀeren- tiable at 1. ∈ − \ { } Suppose that q < 1. If s(q) (q, 1) for all q in [q η, q + η] q for some η > 0, 0 0 0 0 · then formula (6.1) and the continuity of s( ) combined with the mean value theorem yield − ∈ the fact that τ is C1 at q . If s(q) = q on [q η, q ) and s(q) (q, 1) on (q , q + η], 0 0 0 0 0 → · we ﬁrst notice that s(q)/q tends to 1 as q q + by continuity of s( ). It is then almost 0 (cid:48) (cid:48) → direct to see that τ (q) given by (6.1) converges to T (q ) as q q +. Indeed, one has 0 0 (cid:88) − m 1 ∂G − − (6.2) ∂q (q, s(q)) = (cid:88) pqi m Ti(s(q))q/s(q)(log(pi) log(m)Ti(s(q))/s(q)) i=0 − m 1 − − (cid:48) (6.3) = pqm Ti(s(q))q/s(q)(log(p ) log(m)T (s(q)), i i i i=0 due to the equality ∂G(q, s(q)) = 0. Then, letting q tend to q + and using the fact ∂s (cid:48) (cid:48) 0 that s(q)/q tends to 1, we get lim → τ (q) = T (q ). On the other hand, τ = T over − q q0+ 0 [q η, q ), hence τ is C1 at q . The other cases can be treated similarly. 0 0 0 We will show later that the diﬀerentiability of τ over [0, 1] combined with other ar- (cid:54) guments yield the equality of τ with the Lq-spectrum of π∗µ, conditionally on µ = 0. Consequently, τ is concave, and automatically diﬀerentiable at the right hand side of 0 if it is right continuous at 0. Due to the previous discussion, it is enough to prove the continuity at 0. However, we (cid:48) will examine the value of τ (0+). We distinguish two cases. At ﬁrst suppose that h(q) = q/s(q) does not tend to 0 as q tends to 0. It follows that s(q) tends to 0. Suppose that for some sequence (q ) ≥ tending to 0 we have n n 0 → ∈ h(q ) h∗ (0, 1]. The study achieved above gives g(q , s(q )) = 0 if q < s(q ) < 1 n n n n n ≤ and g(q , s(q )) 0 if(cid:88)s(q ) = q . This implies that n n n n − m 1 E(N )h∗ log (E(N )) = lim g(q , s(q )) i m i → ∞ n n n + i=0 38 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: ∂G(q,

text: s(q))

text: ∂q

text: τ

text: (q)

text: log(m)G(q,

text: s(q))

text: implies

text: that

text: τ

text: (q)

text: has

text: a

text: limit

text: at

text: hence

text: by

text: the

text: mean

text: value

text: theorem

text: τ

text: is

text: left

text: diﬀeren-

text: tiable

text: at

text: Suppose

text: that

text: q

text: If

text: s(q)

text: (q,

text: for

text: all

text: q

text: in

text: [q

text: η,

text: q

text: η]

text: q

text: for

text: some

text: η

text: then

text: formula

text: and

text: the

text: continuity

text: of

text: s(

text: combined

text: with

text: the

text: mean

text: value

text: theorem

text: yield

text: the

text: fact

text: that

text: τ

text: is

text: at

text: q

text: If

text: s(q)

text: q

text: on

text: [q

text: η,

text: q

text: and

text: s(q)

text: (q,

text: on

text: (q

text: q

text: η],

text: we

text: ﬁrst

text: notice

text: that

text: s(q)/q

text: tends

text: to

text: as

text: q

text: q

text: by

text: continuity

text: of

text: s(

text: It

text: is

text: then

text: almost

text: direct

text: to

text: see

text: that

text: τ

text: (q)

text: given

text: by

text: converges

text: to

text: T

text: (q

text: as

text: q

text: q

text: Indeed,

text: one

text: has

text: m

text: ∂G

text: ∂q

text: (q,

text: s(q))

text: pqi

text: m

text: Ti(s(q))q/s(q)(log(pi)

text: log(m)Ti(s(q))/s(q))

text: m

text: pqm

text: Ti(s(q))q/s(q)(log(p

text: log(m)T

text: (s(q)),

text: i

text: i

text: i

text: due

text: to

text: the

text: equality

text: ∂G(q,

text: s(q))

text: Then,

text: letting

text: q

text: tend

text: to

text: q

text: and

text: using

text: the

text: fact

text: ∂s

text: that

text: s(q)/q

text: tends

text: to

text: we

text: get

text: lim

text: τ

text: (q)

text: T

text: (q

text: On

text: the

text: other

text: hand,

text: τ

text: T

text: over

text: q

text: [q

text: η,

text: q

text: hence

text: τ

text: is

text: at

text: q

text: The

text: other

text: cases

text: can

text: be

text: treated

text: similarly.

text: We

text: will

text: show

text: later

text: that

text: the

text: diﬀerentiability

text: of

text: τ

text: over

text: combined

text: with

text: other

text: ar-

text: guments

text: yield

text: the

text: equality

text: of

text: τ

text: with

text: the

text: Lq-spectrum

text: of

text: π∗µ,

text: conditionally

text: on

text: µ

text: Consequently,

text: τ

text: is

text: concave,

text: and

text: automatically

text: diﬀerentiable

text: at

text: the

text: right

text: hand

text: side

text: of

text: if

text: it

text: is

text: right

text: continuous

text: at

text: Due

text: to

text: the

text: previous

text: discussion,

text: it

text: is

text: enough

text: to

text: prove

text: the

text: continuity

text: at

text: However,

text: we

text: will

text: examine

text: the

text: value

text: of

text: τ

text: We

text: distinguish

text: two

text: cases.

text: At

text: ﬁrst

text: suppose

text: that

text: h(q)

text: q/s(q)

text: does

text: not

text: tend

text: to

text: as

text: q

text: tends

text: to

text: It

text: follows

text: that

text: s(q)

text: tends

text: to

text: Suppose

text: that

text: for

text: some

text: sequence

text: (q

text: tending

text: to

text: we

text: have

text: n

text: n

text: h(q

text: h∗

text: The

text: study

text: achieved

text: above

text: gives

text: g(q

text: s(q

text: if

text: q

text: s(q

text: n

text: n

text: n

text: n

text: n

text: and

text: g(q

text: s(q

text: q

text: This

text: implies

text: that

text: n

text: n

text: n

text: n

text: m

text: E(N

text: )h∗

text: log

text: (E(N

text: lim

text: g(q

text: s(q

text: i

text: m

text: i

text: n

text: n

text: n

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:16) (cid:88) (cid:17) (cid:88) E ≥ E π∗µ([u])q (cid:88) ν([u])q ((cid:16)X(cid:88)(u)s(cid:16))q/s (cid:17) (cid:17) ∈ ∈ u Σn u Σn E µ([u, v]) s q/s = (cid:88) ν([u])q (cid:16) (cid:88) (cid:89) (cid:17) ∈ ∈ ν([u]) u Σn v Σn n E q/s = ∈ ν([u])(cid:88)q (cid:89)∈ Y (u, v)s Vuk,vk(u|k−1, v|k−1)s u Σn v Σn k=1 n E − = (cid:16)((cid:88)Y s)q/s p(cid:17)u m Tuk(s)q/s ∈ k u Σn k=1 − m 1 ≥ − n pqm qTi(s)/s , i i=0 E ≤ E since 1 = (Y ) (Y s)1/s. Consequently, due to Corollary 9.9, we have (cid:88) − m 1 − − − ≥ − (8.4) τ(q) = max( τ (q), T (q)) sup log pqm qTi(s)/s. ν ≤ ≤ m i 1 s q i=0 (cid:80) (cid:80)Since τ(q) = T (q), this implies that the supremum is reached at s = q. Diﬀeren- − − ∗ (cid:48) ≤ tiating with respect to s at s = q then yields m 1 pqm Ti(q)T (T (q)) 0, hence − − ∗ (cid:48) ≤ i=0 i i i m 1 pqmT(q) Ti(q)T (T (q)) 0. i=0 i i i In the end, suppose that 0 < q < 1. By the deﬁnition of τ, the condition τ(q) = T (q) (cid:88) also implies that the following inﬁmum − m 1 − ≤in≤f logm pqi m qTi(s)/s q s 1 i=0 (cid:88) is attained at q. Hence diﬀerentiating with respect to s at s = q yields − m 1 − ∗ (cid:48) ≤ pqm Ti(q)T (T (q)) 0. i i i i=0 (cid:3) This completes the proof of the lemma. (cid:101) ≡ ≡ In this section, we suppose that we do not have τ τ T over [0, q ), i.e. we are not ν c ≤ ≤ − in the case where for each 0 i m 1 such that p > 0 the function T is equal to 0. i i We will use the notations of Section , and we set s(q) = 1 if both q > 1 and τ(q) = τ (q) ν hold. Also we recall Remark 8.2. 46 ##LTLine##

text: E

text: E

text: π∗µ([u])q

text: ν([u])q

text: u

text: Σn

text: u

text: Σn

text: E

text: µ([u,

text: v])

text: s

text: q/s

text: ν([u])q

text: ν([u])

text: u

text: Σn

text: v

text: Σn

text: n

text: E

text: q/s

text: Y

text: (u,

text: v)s

text: u

text: Σn

text: v

text: Σn

text: n

text: E

text: s)q/s

text: m

text: Tuk(s)q/s

text: k

text: u

text: Σn

text: m

text: n

text: pqm

text: qTi(s)/s

text: i

text: E

text: E

text: since

text: (Y

text: (Y

text: Consequently,

text: due

text: to

text: Corollary

text: we

text: have

text: m

text: τ(q)

text: max(

text: τ

text: (q),

text: T

text: (q))

text: sup

text: log

text: pqm

text: qTi(s)/s.

text: ν

text: m

text: i

text: s

text: q

text: τ(q)

text: T

text: (q),

text: this

text: implies

text: that

text: the

text: supremum

text: is

text: reached

text: at

text: s

text: q.

text: Diﬀeren-

text: tiating

text: with

text: respect

text: to

text: s

text: at

text: s

text: q

text: then

text: yields

text: m

text: pqm

text: Ti(q)T

text: (T

text: (q))

text: hence

text: i

text: i

text: i

text: m

text: pqmT(q)

text: Ti(q)T

text: (T

text: (q))

text: i

text: i

text: i

text: In

text: the

text: end,

text: suppose

text: that

text: q

text: By

text: the

text: deﬁnition

text: of

text: τ,

text: the

text: condition

text: τ(q)

text: T

text: (q)

text: also

text: implies

text: that

text: the

text: following

text: inﬁmum

text: m

text: ≤in≤f

text: logm

text: pqi

text: m

text: qTi(s)/s

text: q

text: s

text: is

text: attained

text: at

text: q.

text: Hence

text: diﬀerentiating

text: with

text: respect

text: to

text: s

text: at

text: s

text: q

text: yields

text: m

text: pqm

text: Ti(q)T

text: (T

text: (q))

text: i

text: i

text: i

text: This

text: completes

text: the

text: proof

text: of

text: the

text: lemma.

text: In

text: this

text: section,

text: we

text: suppose

text: that

text: we

text: do

text: not

text: have

text: τ

text: τ

text: T

text: over

text: q

text: i.e.

text: we

text: are

text: not

text: ν

text: c

text: in

text: the

text: case

text: where

text: for

text: each

text: i

text: m

text: such

text: that

text: p

text: the

text: function

text: T

text: is

text: equal

text: to

text: i

text: i

text: We

text: will

text: use

text: the

text: notations

text: of

text: Section

text: and

text: we

text: set

text: s(q)

text: if

text: both

text: q

text: and

text: τ(q)

text: τ

text: (q)

text: ν

text: hold.

text: Also

text: we

text: recall

text: Remark

text: ##LTLine##


text: topological support as µ almost surely by Lemma 8.7). The main result of this section is the following. (cid:101) ∈ \ { } (cid:54) ∈ Proposition 8.8. Let q (0, q ) 1 at which τ(q) = T (q) or q (1, q ) at which c c (cid:48) (cid:101) (cid:48) (cid:101) τ(q) = τ (q) = T (q). Set α = τ (q) if s(q) < 1 and α = τ (q) otherwise. ν ν (cid:54) With probability 1, conditionally on µ = 0, we have ν (E(π∗µ, α)) > 0, and dim(ν ) = q q ∗ ≥ ∗ τ (α). Consequently, dimH E((cid:101)π∗µ, α) τ (α). ∈ \ { } (cid:54) From now on we ﬁx q (0, q ) 1 at which τ(q) = T (q) or τ(q) = τ (q) = T (q). c ν (cid:101) (cid:80) (cid:48) Lemma 8.9. Suppose that ν = ν . Let S stand for a maximal open interval of points q − (cid:48) ∗ (cid:48) E ∞ s > 0 such that m 1 p T (T (s)) > 0 and (Y s) < . With probability 1, conditionally (cid:54) (cid:48) i=0 i i i (cid:88) (cid:16) ∈(cid:17) (cid:88) on µ = 0, for ν -almost every x in π(K), for all s S we have − − 1 µ([x| , v]) s m 1 (cid:48) n lim log = p T (s). n→∞ n m ∈ ν(x|n) i i v Σn i=0 S Proof. By convexity, we only need to check this for e(cid:80)ach s in(cid:16) a dense(cid:17)countable set of S . { (cid:54) } P (cid:54) Indeed, if this is done, there exists a subset of µ = 0 of probability (µ = 0) such that the sequence of concave functions f (s) = −1 log ∈ µ([x|n,v]) s converge pointwise S n n m v Σn ν(x|n) on , and this is enough to get the convergence over S . ∈ S ≥ (cid:48) Fix s . For n 1 and x in the topological support of ν , set (cid:16) (cid:89) (cid:17) (cid:88) (cid:16) (cid:17) n µ([x| , v]) s Zs,n(x) = (cid:88) mTxk(s) ∈(cid:89) ν([xn|n]) k=1 v Σn n · = ∈ Y (x|n, v)s mTxk(s)Vxk,vk(x|k−1, v|k−1)s. v Σn k=1 (cid:80) (cid:80) (cid:48) ≤ ≤ − E (cid:48) (cid:80)Deﬁne V , 0 i m 1, as in (8.5). Since S is open, we have (Y q s) and − (cid:48) s,i (cid:48) −∞ (cid:48) − (cid:48) (cid:48) m 1 p T (cid:48) (q ) > for some q > 1, and since m 1 p T (cid:48) (1) > 0, we also have i=−0 i(cid:48) Vs,i (cid:48) (cid:48) i=0 i Vs,i (cid:48) m 1 p T (cid:48) (q ) > 0 if q is close enough to 1. By Proposition B.2 applied with η = ν i=0 i Vs(cid:48)(cid:101),i P ⊗ (cid:48) and U = V , the sequence Z (x) converges ν almost surely to the same non degen- i s,i s,n (cid:88) (cid:89) erate limit Z (x) as th(cid:101)e Mandelbrot martingale in random environment s n Zs,n(x) = ∈ mTxk(s)Vxk,vk(x|k−1, v|k−1)s. v Σn k=1 This variable satisﬁes the equation 49 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: topological

text: support

text: as

text: µ

text: almost

text: surely

text: by

text: Lemma

text: The

text: main

text: result

text: of

text: this

text: section

text: is

text: the

text: following.

text: Proposition

text: Let

text: q

text: q

text: at

text: which

text: τ(q)

text: T

text: (q)

text: or

text: q

text: q

text: at

text: which

text: c

text: c

text: τ(q)

text: τ

text: (q)

text: T

text: (q).

text: Set

text: α

text: τ

text: (q)

text: if

text: s(q)

text: and

text: α

text: τ

text: (q)

text: otherwise.

text: ν

text: ν

text: With

text: probability

text: conditionally

text: on

text: µ

text: we

text: have

text: ν

text: (E(π∗µ,

text: α))

text: and

text: dim(ν

text: q

text: q

text: τ

text: (α).

text: Consequently,

text: dimH

text: α)

text: τ

text: (α).

text: From

text: now

text: on

text: we

text: ﬁx

text: q

text: q

text: at

text: which

text: τ(q)

text: T

text: (q)

text: or

text: τ(q)

text: τ

text: (q)

text: T

text: (q).

text: c

text: ν

text: Lemma

text: Suppose

text: that

text: ν

text: ν

text: Let

text: S

text: stand

text: for

text: a

text: maximal

text: open

text: interval

text: of

text: points

text: q

text: E

text: s

text: such

text: that

text: m

text: p

text: T

text: (T

text: (s))

text: and

text: (Y

text: s)

text: With

text: probability

text: conditionally

text: i

text: i

text: i

text: on

text: µ

text: for

text: ν

text: -almost

text: every

text: x

text: in

text: π(K),

text: for

text: all

text: s

text: S

text: we

text: have

text: µ([x|

text: v])

text: s

text: m

text: n

text: lim

text: log

text: p

text: T

text: (s).

text: n→∞

text: n

text: m

text: ν(x|n)

text: i

text: i

text: v

text: Σn

text: S

text: Proof.

text: By

text: convexity,

text: we

text: only

text: need

text: to

text: check

text: this

text: for

text: s

text: a

text: set

text: of

text: S

text: P

text: Indeed,

text: if

text: this

text: is

text: done,

text: there

text: exists

text: a

text: subset

text: of

text: µ

text: of

text: probability

text: (µ

text: such

text: that

text: the

text: sequence

text: of

text: concave

text: functions

text: f

text: (s)

text: −1

text: log

text: µ([x|n,v])

text: s

text: converge

text: pointwise

text: S

text: n

text: n

text: m

text: v

text: Σn

text: ν(x|n)

text: on

text: and

text: this

text: is

text: enough

text: to

text: get

text: the

text: convergence

text: over

text: S

text: S

text: Fix

text: s

text: For

text: n

text: and

text: x

text: in

text: the

text: topological

text: support

text: of

text: ν

text: set

text: n

text: µ([x|

text: v])

text: s

text: Zs,n(x)

text: mTxk(s)

text: ν([xn|n])

text: v

text: Σn

text: n

text: Y

text: (x|n,

text: v)s

text: v

text: Σn

text: E

text: V

text: i

text: m

text: as

text: in

text: Since

text: S

text: is

text: open,

text: we

text: have

text: (Y

text: q

text: s)

text: and

text: s,i

text: m

text: p

text: T

text: (q

text: for

text: some

text: q

text: and

text: since

text: m

text: p

text: T

text: we

text: also

text: have

text: Vs,i

text: i

text: Vs,i

text: m

text: p

text: T

text: (q

text: if

text: q

text: is

text: close

text: enough

text: to

text: By

text: Proposition

text: applied

text: with

text: η

text: ν

text: i

text: P

text: and

text: U

text: V

text: the

text: sequence

text: Z

text: (x)

text: converges

text: ν

text: almost

text: surely

text: to

text: the

text: same

text: non

text: degen-

text: i

text: s,i

text: s,n

text: erate

text: limit

text: Z

text: (x)

text: as

text: Mandelbrot

text: martingale

text: in

text: random

text: environment

text: s

text: n

text: Zs,n(x)

text: v

text: Σn

text: This

text: variable

text: satisﬁes

text: the

text: equation

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ∪ × ∈ operation on Σ∗ (Σ Σ). The length of a word w Σ∗, i.e. its number of letters, is | | denoted as w . · · · · · · ∈ · · · ≥ ∈ For x = x x Σ, set x| = x x if n 1 and (cid:15) if n = 0. For u Σ∗, set 1 p n 1 n { ∈ } [u] = x Σ : x|| | = u . u (cid:110) (cid:111) The set Σ is endowed with the standard metric distance (cid:48) −sup n:x| =x(cid:48)| d(x, x ) = m n n , × (cid:48) (cid:48) (cid:48) (cid:48) and Σ Σ is endowed with the distance d((x, y), (x , y )) = max(d(x, x ), d(y, y )). × Given a positive and ﬁnite Borel measure ρ on Σ or Σ Σ, its topological support, i.e. the smallest closed set carrying the whole mass of µ is denoted as supp(ρ), and its lower ∈ and upper local dimensions at x supp(ρ) are deﬁned as log(ρ([x| ])) log(ρ([x| ])) n n dim (ρ, x) = lim inf − and dim (ρ, x) = lim sup − loc n→∞ ( n log(m)) loc n→∞ ( n log(m)) respectively. Let { } dim (ρ) = inf dim E : ρ(E) > 0 and E is a Borel set and H H { (cid:107) (cid:107) } dim (ρ) = inf dim E : ρ(E) = ρ and E is a Borel set , P P (cid:107) (cid:107) where dim E stands for the packing dimension of E (see e.g. [46]) and ρ stands for the P total mass of ρ . It is well known that (see e.g. [15, 16]) { ≥ } dim (ρ) = sup s : dim (ρ, x) s for ρ-almost every x and H loc { ≤ } dim (ρ) = inf s : dim (ρ, x) s for ρ–almost every x ; P loc when these two dimensions coincide, one says that ρ is exactly dimensional and writes dim(ρ) for the common value. (cid:88) R → R ∪ {−∞} The Lq-spectrum of ρ is the mapping τ : given by ρ − 1 ∈ R τ (q) = lim inf log 1{ }ρ([w])q (q ), ρ n→∞ n m ∈ ρ([w])>0 w Sn × where S stands for Σ or Σ Σ . It is well known that (cf. [49]) n n n n (cid:48) ≤ ≤ ≤ (cid:48) − τ (1+) dim (ρ) dim (ρ) τ (1 ). ρ H P ρ ∈ R For all α , set { ∈ } E(ρ, α) = x supp(ρ) : dim (ρ, x) = α , loc { ∈ } E(ρ, α) = x supp(µ) : dim (ρ, x) = α loc and ∩ E(ρ, α) = E(ρ, α) E(ρ, α). 6 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: operation

text: on

text: Σ∗

text: (Σ

text: Σ).

text: The

text: length

text: of

text: a

text: word

text: w

text: Σ∗,

text: i.e.

text: its

text: number

text: of

text: letters,

text: is

text: denoted

text: as

text: w

text: For

text: x

text: x

text: x

text: Σ,

text: set

text: x|

text: x

text: x

text: if

text: n

text: and

text: if

text: n

text: For

text: u

text: Σ∗,

text: set

text: p

text: n

text: n

text: [u]

text: x

text: Σ

text: x||

text: u

text: u

text: The

text: set

text: Σ

text: is

text: endowed

text: with

text: the

text: standard

text: metric

text: distance

text: −sup

text: n:x|

text: d(x,

text: x

text: m

text: n

text: n

text: and

text: Σ

text: Σ

text: is

text: endowed

text: with

text: the

text: distance

text: d((x,

text: y),

text: (x

text: y

text: max(d(x,

text: x

text: d(y,

text: y

text: Given

text: a

text: positive

text: and

text: ﬁnite

text: Borel

text: measure

text: ρ

text: on

text: Σ

text: or

text: Σ

text: Σ,

text: its

text: topological

text: support,

text: i.e.

text: the

text: smallest

text: closed

text: set

text: carrying

text: the

text: whole

text: mass

text: of

text: µ

text: is

text: denoted

text: as

text: supp(ρ),

text: and

text: its

text: lower

text: and

text: upper

text: local

text: dimensions

text: at

text: x

text: supp(ρ)

text: are

text: deﬁned

text: as

text: log(ρ([x|

text: log(ρ([x|

text: n

text: n

text: dim

text: (ρ,

text: x)

text: lim

text: inf

text: and

text: dim

text: (ρ,

text: x)

text: lim

text: sup

text: loc

text: n→∞

text: n

text: log(m))

text: loc

text: n→∞

text: n

text: log(m))

text: respectively.

text: Let

text: dim

text: (ρ)

text: inf

text: dim

text: E

text: ρ(E)

text: and

text: E

text: is

text: a

text: Borel

text: set

text: and

text: H

text: H

text: dim

text: (ρ)

text: inf

text: dim

text: E

text: ρ(E)

text: ρ

text: and

text: E

text: is

text: a

text: Borel

text: set

text: P

text: P

text: where

text: dim

text: E

text: stands

text: for

text: the

text: packing

text: dimension

text: of

text: E

text: (see

text: e.g.

text: and

text: ρ

text: stands

text: for

text: the

text: P

text: total

text: mass

text: of

text: ρ

text: It

text: is

text: well

text: known

text: that

text: (see

text: e.g.

text: dim

text: (ρ)

text: sup

text: s

text: dim

text: (ρ,

text: x)

text: s

text: for

text: ρ-almost

text: every

text: x

text: and

text: H

text: loc

text: dim

text: (ρ)

text: inf

text: s

text: dim

text: (ρ,

text: x)

text: s

text: for

text: ρ–almost

text: every

text: x

text: P

text: loc

text: when

text: these

text: two

text: dimensions

text: coincide,

text: one

text: says

text: that

text: ρ

text: is

text: exactly

text: dimensional

text: and

text: writes

text: dim(ρ)

text: for

text: the

text: common

text: value.

text: R

text: R

text: The

text: Lq-spectrum

text: of

text: ρ

text: is

text: the

text: mapping

text: τ

text: given

text: by

text: ρ

text: R

text: τ

text: (q)

text: lim

text: inf

text: log

text: }ρ([w])q

text: (q

text: ρ

text: n→∞

text: n

text: m

text: w

text: Sn

text: where

text: S

text: stands

text: for

text: Σ

text: or

text: Σ

text: Σ

text: It

text: is

text: well

text: known

text: that

text: (cf.

text: n

text: n

text: n

text: n

text: τ

text: dim

text: (ρ)

text: dim

text: (ρ)

text: τ

text: ρ

text: H

text: P

text: ρ

text: R

text: For

text: all

text: α

text: set

text: E(ρ,

text: α)

text: x

text: supp(ρ)

text: dim

text: (ρ,

text: x)

text: α

text: loc

text: E(ρ,

text: α)

text: x

text: supp(µ)

text: dim

text: (ρ,

text: x)

text: α

text: loc

text: and

text: E(ρ,

text: α)

text: E(ρ,

text: α)

text: E(ρ,

text: α).

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:101) (cid:101) (cid:101) On the other hand, due to (8.9), Gartner-Ellis theorem (see e.g. [21]) ensures that for ∈ (cid:48) (cid:101) ∗ (cid:101)(cid:48) (cid:101) all s (0, min(s , 1)) one has lim → lim inf →∞ f(n, T (s), (cid:15)) = T (T (s)). 0 (cid:15) 0 n (cid:101) − ≤ (cid:48) − ∗ (cid:48) This immediately yields lim sup →∞ 1 log(X (x)) T (s)) T (T (s)) for all s < n n n − ≤ − ∗ min(s , 1) close enough to min(s , 1), hence lim sup →∞ 1 log (X (x)) α T (α ). 0 0 n n m n c c(cid:3) (cid:48) (cid:48) Proof of Proposition 8.8. Recall that α stands for τ (q) if s(q) < 1 and τ (q) if s(q) = 1. ν We use the writing π∗µ([x| (cid:101)]) = ν([x| ])X (x). n n n (cid:48) At ﬁrst we suppose that ν = ν . q ∈ (cid:48) If q (0, 1) and s(q) < 1, applying ergodic theorem to ν to control the local dimension of ν, and Lemma 8.10 to X (x) after setting s = s(q), we obtain that conditionally on n 0 (cid:54) (cid:48) ∈ π∗µ = 0, for ν -almost every x π(K), (cid:16) (cid:88) (cid:17) (cid:88) (cid:88) (cid:101) − − − m 1 m 1 m 1 − (cid:48) (cid:48) − (cid:48) (cid:48) (cid:48) dim (π∗µ, x) = p log (p ) + T (s ) = p log (p ) + p T (s(q)) loc i m i 0 i m i i i i=0 i=0 i=0 (cid:48) = τ (q) = α, by using (6.1) and (6.3). On the other hand, (cid:88) (cid:88) − − m 1 m 1 (cid:48) − (cid:48) (cid:48) − (cid:48) − dim(ν ) = p log (p ) = p (q log (p ) + τ(q) qT (q)/s(q)) i m i i m i i i=0 i=0 (cid:48) − ∗ = qτ (q) τ(q) = τ (α) by using (6.1) and (6.2). (cid:48) If s(q) = 1, then ν = ν , and this time one applies Lemma 8.10 with s = s(q) = 1 to q 0 (cid:54) (cid:48) ∈ control X (x) . This yields that conditionally on π∗µ = 0, for ν -almost every x π(K), n (cid:16) (cid:88) (cid:17) − m 1 − (cid:48) (cid:48) dim (π∗µ, x) = p log (p ) + 0 = τ (q). loc i m i ν i=0 ∗ (cid:48) (cid:48) − − ∗ Moreover, dim(ν ) = τ (τ (q)) = τ (q)q τ (q) = αq τ(q) = τ (α) since we have q∈ { (cid:48) ν ν (cid:48) − } ν ν τ (q) = τ(q) and α τ (q+), τ (q ) . ν 54 ##LTLine## ##LTLine##

text: On

text: the

text: other

text: hand,

text: due

text: to

text: Gartner-Ellis

text: theorem

text: (see

text: e.g.

text: ensures

text: that

text: for

text: all

text: s

text: min(s

text: one

text: has

text: lim

text: lim

text: inf

text: f(n,

text: T

text: (s),

text: T

text: (T

text: (s)).

text: n

text: This

text: immediately

text: yields

text: lim

text: sup

text: log(X

text: (x))

text: T

text: (s))

text: T

text: (T

text: (s))

text: for

text: all

text: s

text: n

text: n

text: n

text: min(s

text: close

text: enough

text: to

text: min(s

text: hence

text: lim

text: sup

text: log

text: (X

text: (x))

text: α

text: T

text: (α

text: n

text: n

text: m

text: n

text: c

text: Proof

text: of

text: Proposition

text: Recall

text: that

text: α

text: stands

text: for

text: τ

text: (q)

text: if

text: s(q)

text: and

text: τ

text: (q)

text: if

text: s(q)

text: ν

text: We

text: use

text: the

text: writing

text: π∗µ([x|

text: ν([x|

text: ])X

text: (x).

text: n

text: n

text: n

text: At

text: ﬁrst

text: we

text: suppose

text: that

text: ν

text: ν

text: q

text: If

text: q

text: and

text: s(q)

text: applying

text: ergodic

text: theorem

text: to

text: ν

text: to

text: control

text: the

text: local

text: dimension

text: of

text: ν,

text: and

text: Lemma

text: to

text: X

text: (x)

text: after

text: setting

text: s

text: s(q),

text: we

text: obtain

text: that

text: conditionally

text: on

text: n

text: π∗µ

text: for

text: ν

text: -almost

text: every

text: x

text: π(K),

text: m

text: m

text: m

text: dim

text: (π∗µ,

text: x)

text: p

text: log

text: (p

text: T

text: (s

text: p

text: log

text: (p

text: p

text: T

text: (s(q))

text: loc

text: i

text: m

text: i

text: i

text: m

text: i

text: i

text: i

text: τ

text: (q)

text: α,

text: by

text: using

text: and

text: On

text: the

text: other

text: hand,

text: m

text: m

text: dim(ν

text: p

text: log

text: (p

text: p

text: (q

text: log

text: (p

text: τ(q)

text: qT

text: (q)/s(q))

text: i

text: m

text: i

text: i

text: m

text: i

text: i

text: qτ

text: (q)

text: τ(q)

text: τ

text: (α)

text: by

text: using

text: and

text: If

text: s(q)

text: then

text: ν

text: ν

text: and

text: this

text: time

text: one

text: applies

text: Lemma

text: with

text: s

text: s(q)

text: to

text: q

text: control

text: X

text: (x)

text: This

text: yields

text: that

text: conditionally

text: on

text: π∗µ

text: for

text: ν

text: -almost

text: every

text: x

text: π(K),

text: n

text: m

text: dim

text: (π∗µ,

text: x)

text: p

text: log

text: (p

text: τ

text: (q).

text: loc

text: i

text: m

text: i

text: ν

text: Moreover,

text: dim(ν

text: τ

text: (τ

text: (q))

text: τ

text: (q)q

text: τ

text: (q)

text: αq

text: τ(q)

text: τ

text: (α)

text: since

text: we

text: have

text: q∈

text: ν

text: ν

text: ν

text: ν

text: τ

text: (q)

text: τ(q)

text: and

text: α

text: τ

text: (q+),

text: τ

text: (q

text: ν

text: ##LTLine##

text: ##LTLine##


text: Then one also always has (see e.g. [38, 51]) that ≤ ≤ ∗ dim E(ρ, α) max(dim E(ρ, α), dim E(ρ, α)) τ (α), H H H ρ R → R ∪ {−∞} where the Legendre transform of f : is deﬁned as ∗ ∈ R (cid:55)→ − f : α inf (αq f(q)), ∈R q and a negative dimension means that the set is empty. One says that the multifractal formalism holds at α if ∗ dim E(ρ, α) = τ (α). H ρ ≤ (cid:48) Also, if α τ (0+), one has (see e.g. [38, 51]) ρ ≤ ≤ ∗ (2.1) dim E (ρ, α) τ (α), H ρ where ≤ { ∈ ≤ } E (ρ, α) = x supp(ρ) : dim (ρ, x) α . loc × Now let us formally deﬁne the Mandelbrot measures on Σ Σ. We consider a non- negative random vector W = (W ) ∈ × i,j (i,j) Σ1 Σ1 ∈ R whose entries are integrable. For q we deﬁne (cid:88) (2.2) T (q) = T (q) = − log E(1{ }W q ). (cid:80) W m ∈ × Wi,j>0 i,j (i,j) Σ1 Σ1 P ∈ { } We denote N = ∈ × 1{ }, and assume that (N 0, 1 ) < 1. (i,j) Σ1 Σ1 Wi,j>0 × (cid:83) To build a Mandelbrot measure on Σ Σ we assume that T (1) = 0 and consider a sequence (W (u, v)) ∈ × of independent copies of W , deﬁned on a probability A P (u,v) n≥0 Σn Σn space (Ω, , ). ≥ × For each n 1 let µ = µ be the measure on Σ Σ whose density with respect to the n W,n × measure of maximal entropy is constant over each cylinder [u, v] := [u] [v] of generation (cid:89) n and given by m2nQ(u, v), where n Q(u, v) = W (u| − , v| − ). uj,vj j 1 j 1 j=1 (cid:88) Denote the total mass of µ as Y , i.e. n n Y = Q(u, v). n | | | | u = v =n 7 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Then

text: one

text: also

text: always

text: has

text: (see

text: e.g.

text: that

text: dim

text: E(ρ,

text: α)

text: max(dim

text: E(ρ,

text: α),

text: dim

text: E(ρ,

text: α))

text: τ

text: (α),

text: H

text: H

text: H

text: ρ

text: R

text: R

text: where

text: the

text: Legendre

text: transform

text: of

text: f

text: is

text: deﬁned

text: as

text: R

text: f

text: α

text: inf

text: (αq

text: f(q)),

text: ∈R

text: q

text: and

text: a

text: negative

text: dimension

text: means

text: that

text: the

text: set

text: is

text: empty.

text: One

text: says

text: that

text: the

text: multifractal

text: formalism

text: holds

text: at

text: α

text: if

text: dim

text: E(ρ,

text: α)

text: τ

text: (α).

text: H

text: ρ

text: Also,

text: if

text: α

text: τ

text: one

text: has

text: (see

text: e.g.

text: ρ

text: dim

text: E

text: (ρ,

text: α)

text: τ

text: (α),

text: H

text: ρ

text: where

text: E

text: (ρ,

text: α)

text: x

text: supp(ρ)

text: dim

text: (ρ,

text: x)

text: α

text: loc

text: Now

text: let

text: us

text: formally

text: deﬁne

text: the

text: Mandelbrot

text: measures

text: on

text: Σ

text: Σ.

text: We

text: consider

text: a

text: non-

text: negative

text: random

text: vector

text: W

text: (W

text: i,j

text: (i,j)

text: Σ1

text: Σ1

text: R

text: whose

text: entries

text: are

text: integrable.

text: For

text: q

text: we

text: deﬁne

text: T

text: (q)

text: T

text: (q)

text: log

text: }W

text: q

text: W

text: m

text: i,j

text: (i,j)

text: Σ1

text: Σ1

text: P

text: We

text: denote

text: N

text: and

text: assume

text: that

text: (N

text: (i,j)

text: Σ1

text: Σ1

text: To

text: build

text: a

text: Mandelbrot

text: measure

text: on

text: Σ

text: Σ

text: we

text: assume

text: that

text: T

text: and

text: consider

text: a

text: sequence

text: (W

text: (u,

text: v))

text: of

text: independent

text: copies

text: of

text: W

text: deﬁned

text: on

text: a

text: probability

text: A

text: P

text: (u,v)

text: Σn

text: Σn

text: space

text: (Ω,

text: For

text: each

text: n

text: let

text: µ

text: µ

text: be

text: the

text: measure

text: on

text: Σ

text: Σ

text: whose

text: density

text: with

text: respect

text: to

text: the

text: n

text: W,n

text: measure

text: of

text: maximal

text: entropy

text: is

text: constant

text: over

text: each

text: cylinder

text: [u,

text: v]

text: [u]

text: [v]

text: of

text: generation

text: n

text: and

text: given

text: by

text: v),

text: where

text: n

text: Q(u,

text: v)

text: W

text: (u|

text: v|

text: uj,vj

text: j

text: j

text: Denote

text: the

text: total

text: mass

text: of

text: µ

text: as

text: Y

text: i.e.

text: n

text: n

text: Y

text: Q(u,

text: v).

text: n

text: u

text: v

text: =n

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ∈ N (cid:48) (cid:48) (cid:48) for any n , from which (9.1) follows. Indeed setting p = p . . . p and f = (cid:88) (cid:88) j1...jn j1 jn j1...jn (cid:80) f + . . . + f , then (9.3) can be re-written as j1 jn (cid:48) − ≤ (cid:48) − ≤ ≤ pj1...jnm ki=1 fj1...jn(qi) 1 + ≤ ≤ pj1...jnm fj1...jn(q); 1 j1,...,jn m 1 j1,...,jn m (cid:48) but this is just the application of (9.2) to the probability weight (p ) and the concave (cid:80) (cid:80) j1...jn(cid:80) functions f . This ﬁnishes the proof of (9.1). j1...jn (cid:48) − (cid:48) − ≤ In the end, assume that mj=1(cid:88)pjm fj(q) < 1. By (9.1), mj=1 pjm ki=1 fj(qi) 1. We need to show that the inequality is strict. (cid:80)Suppose on the contrary that m (cid:48) − (cid:80) p m ki=1 fj(qi) = 1. j (cid:80) (cid:80)j=1 Deﬁne g(x) = mj=1 p(cid:48)jm−x ki=1 fj(qi) for(cid:48) x−∈ R. Then g is con∈vex. Notice that on a s∈mall neighborhood U of 1, we have m p m xfj(q) < 1 for x U. For any ﬁxed x U, j=1 j ≤ applying (9.1) to the functions xf , we obtain that g(x) 1. Hence g takes a local i R maximum at x = 1. However g is convex and analytic on , it follows that g is constant (cid:88) R on and therefore k f (q ) = 0 j i (cid:80) i=1 ≤ ≤ ≤ ≤ ≤ for any 1 j m. Then by Lemma 9.1(i), we have f (q) 0 for all 1 j m, j (cid:48) − which contradicts the assumption that m p m fj(q) < 1. This ﬁnishes the proof of the j=1 j (cid:3) lemma. (cid:80) (cid:80) Let u(cid:80)s ﬁrst recall some notations. We are given W = (W ) ≤ ≤ − , a non-negative i,j 0 i,j m 1 random vector with E( W ) = 1(cid:26). Let q > 1 and assume that E( W q ) < ∞. Set i,j i,j i,j i,j E p = ( W ). Set i j i,j (cid:54) W /p , if p = 0, V = i,j i i i,j 1/m, if p = 0. i (cid:88) (cid:88) ∈ For t [0, q], set − E − E T (t) = log ( W t ), T (t) = log ( V t ). m i,j i m i,j i,j j ≤ ≤ − Then T and T (0 i m 1) are well deﬁned continuous concave functions on [0, q], i { − }N with T (1) = T (1) = 0. Set Σ = 0, 1, . . . , m 1 . Let µ be the (random) Mandelbrot i × (cid:107) (cid:107) measure on Σ Σ generated by W . Set Y = µ to be the total mass of µ and assume that T (q) > 0. By Kahane-Peyriere [37] and Durrett-Liggett [11], this is equivalent to the E ∞ ∈ × property that 0 < (Y q) < . For each (u, v) (Σ Σ)∗, let Y (u, v) be deﬁned as in (2.3). We deﬁned in Section 68

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text: Section