text: 9 simulations performed with a radius similar to those In conclusion, this work provides the means to seek op- experimentally fabricated to date return linewidths in timized experimental parameters for a given application. the same order of magnitude when both non-local and To wit, we ﬁnd that an environment with a large NC current-induced Oersted ﬁelds are included. The rele- radius, low ﬁeld, modest current, and large anisotropy vance of such ﬁelds in the generation linewidth moti- are less susceptible to drift and thus lead to a much vates their inclusion in the analytical theory. For thin narrower generation linewidth. Our results motivate a ﬁlms, the eﬀect of non-local dipole ﬁelds on determinis- more detailed experimental study on the current and tic droplet dynamics has been shown to be a frequency temperature-dependent generation linewidth and ejec- downshift when v = 08. It remains to incorporate these tion statistics of droplets. (cid:54) eﬀects into the stochastic theory when v = 0. Because ACKNOWLEDGMENTS the Oersted ﬁeld is not a singular perturbation22, its inclusion in this collective theory would necessitate the E. I. acknowledges support from the Swedish Research incorporation of droplet coupling to spin waves. Such Council, Reg. No. 637-2014-6863. M. A. H. was partially coupling is in principle possible, see, e.g., Ref. 34. supported by NSF CAREER DMS-1255422. ∗ fxundeﬁned [1]ifx1 fnum [1]1ﬁrstoftwo fx [1]secondoftwo noop P. Bruno and J. P. Renard, noop Appl. Phys. A 49, 499 [0]secondoftworef[1]@startlink1@hrefhref[1]1@endlinkanitize@url(1989). [0]startlink[1]endlink[0]rl [1]href 1bib@innerbibempty A. Kosevich, B. Ivanov, and A. Kovalev, Phys. Rep. N. Nagaosa and Y. Tokura, noop Nature Nano. 8, 898 194, 117 (1990). (2013). L. Berger, Phys. Rev. B 54, 9353 (1996). N. Romming, C. Hanneken, M. Menzel, J. E. Bickel, J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 B. Wolter, K. von Bergmann, A. Kubetzka, and R. Wiesen- (1996). danger, Science 341, 636 (2013). R. Dumas, S. Sani, S. Mohseni, E. Iacocca, Y. Pogo- J. Sampaio, V. Cros, S. Rohart, A. Thiaville, and A. Fert, noop Nature Nano. 8, 839 (2013). ryelov, P. Muduli, S. Chung, P. Du¨rrenfeld, and J. ˚Akerman, noop IEEE Trans. Mag. 50, 1 (2014). Y. Zhou, E. Iacocca, A. Awad, R. Dumas, H. Zhang, H. B. Braun, and J. ˚Akerman, Nature Comm. 6 (2015). S. Petit-Watelot, R. M. Otxoa, and M. Manfrini, Appl. Phys. Lett. 100, 083507 (2012). M. A. Hoefer, T. J. Silva, and M. W. Keller, Phys. Rev. B 82, 054432 (2010). M. A. Hoefer, M. Sommacal, and T. J. Silva, noop Phys. Rev. B 85, 214433 (2012). M. Hoefer and M. Sommacal, Physica D 241, 890 (2012). L. D. Bookman and M. A. Hoefer, noop P. Roy. Soc. M. D. Maiden, L. D. Bookman, and M. A. Hoefer, Phys. Lond. A 471 (2015). Rev. B 89, 180409 (2014). R. K. Dumas, E. Iacocca, S. Bonetti, S. R. Sani, S. M. L. D. Bookman and M. A. Hoefer, Phys. Rev. B 88, Mohseni, A. Eklund, J. Persson, O. Heinonen, and 184401 (2013). J. ˚Akerman, Phys. Rev. Lett. 110, 257202 (2013). E. Iacocca, R. K. Dumas, L. Bookman, M. Mohseni, S. Chung, M. A. Hoefer, and J. ˚Akerman, Phys. Rev. M. Madami, E. Iacocca, S. Sani, G. Gubbiotti, S. Tacchi, R. K. Dumas, J. ˚Akerman, and G. Carlotti, Phys. Rev. Lett. 112, 047201 (2014). B 92, 024403 (2015). F. Maci`a, D. Backes, and A. Kent, noop Nature Nano. 10, 1038 (2014). T. Silva and M. Keller, noop IEEE Trans. Mag. 46, 3555 (2010). S. M. Mohseni, S. R. Sani, J. Persson, T. N. A. Nguyen, S. Chung, Y. Pogoryelov, P. K. Muduli, E. Iacocca, W. F. Brown, Phys. Rev. 130, 1677 (1963). A. Eklund, R. K. Dumas, S. Bonetti, A. Deac, M. A. M. d’Aquino, C. Serpico, G. Coppola, I. D. Mayergoyz, Hoefer, and J. ˚Akerman, Science 339, 1295 (2013). and G. Bertotti, noop J. Appl. Phys. 99 (2006). D. Backes, F. Maci`a, S. Bonetti, R. Kukreja, H. Ohldag, A. Eklund, S. Bonetti, S. R. Sani, S. Majid Mohseni, and A. D. Kent, Phys. Rev. Lett. 115, 127205 (2015). J. Persson, S. Chung, S. Amir Hossein Banuazizi, E. Ia- S. Lend´ınez, N. Statuto, D. Backes, A. Kent, and cocca, M. O¨stling, J. ˚Akerman, and B. Gunnar Malm, F. Maci`a, noop Phys. Rev. B 92, 174426 (2015). noop Appl. Phys. Lett. 104, 092405 (2014). S. Chung, A. Eklund, E. Iacocca, S. Mohseni, S. Sani, S. Mohseni, S. Sani, R. Dumas, J. Persson, T. A. Nguyen, L. Bookman, M. A. Hoefer, R. Dumas, and J. ˚Akerman, S. Chung, Y. Pogoryelov, P. Muduli, E. Iacocca, A. Ek- noop submitted (2015). lund, and J. ˚Akerman, noop Physica B 435, 84 (2014). ##LTLine## ##LTLine## ##LTLine##

text: simulations

text: performed

text: with

text: a

text: radius

text: similar

text: to

text: those

text: In

text: conclusion,

text: this

text: work

text: provides

text: the

text: means

text: to

text: seek

text: op-

text: experimentally

text: fabricated

text: to

text: date

text: return

text: linewidths

text: in

text: timized

text: experimental

text: parameters

text: for

text: a

text: given

text: application.

text: the

text: same

text: order

text: of

text: magnitude

text: when

text: both

text: non-local

text: and

text: To

text: wit,

text: we

text: ﬁnd

text: that

text: an

text: environment

text: with

text: a

text: large

text: NC

text: current-induced

text: Oersted

text: ﬁelds

text: are

text: included.

text: The

text: rele-

text: radius,

text: low

text: ﬁeld,

text: modest

text: current,

text: and

text: large

text: anisotropy

text: vance

text: of

text: such

text: ﬁelds

text: in

text: the

text: generation

text: linewidth

text: moti-

text: are

text: less

text: susceptible

text: to

text: drift

text: and

text: thus

text: lead

text: to

text: a

text: much

text: vates

text: their

text: inclusion

text: in

text: the

text: analytical

text: theory.

text: For

text: thin

text: narrower

text: generation

text: linewidth.

text: Our

text: results

text: motivate

text: a

text: ﬁlms,

text: the

text: eﬀect

text: of

text: non-local

text: dipole

text: ﬁelds

text: on

text: determinis-

text: more

text: detailed

text: experimental

text: study

text: on

text: the

text: current

text: and

text: tic

text: droplet

text: dynamics

text: has

text: been

text: shown

text: to

text: be

text: a

text: frequency

text: temperature-dependent

text: generation

text: linewidth

text: and

text: ejec-

text: downshift

text: when

text: v

text: It

text: remains

text: to

text: incorporate

text: these

text: tion

text: statistics

text: of

text: droplets.

text: eﬀects

text: into

text: the

text: stochastic

text: theory

text: when

text: v

text: Because

text: ACKNOWLEDGMENTS

text: the

text: Oersted

text: ﬁeld

text: is

text: not

text: a

text: singular

text: its

text: inclusion

text: in

text: this

text: collective

text: theory

text: would

text: necessitate

text: the

text: E.

text: I.

text: acknowledges

text: support

text: from

text: the

text: Swedish

text: Research

text: incorporation

text: of

text: droplet

text: coupling

text: to

text: spin

text: waves.

text: Such

text: Council,

text: Reg.

text: No.

text: M.

text: A.

text: H.

text: was

text: partially

text: coupling

text: is

text: in

text: principle

text: possible,

text: see,

text: e.g.,

text: Ref.

text: supported

text: by

text: NSF

text: CAREER

text: fxundeﬁned

text: fnum

text: fx

text: noop

text: P.

text: Bruno

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text: J.

text: P.

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text: A.

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text: L.

text: Bookman,

text: M.

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text: Hoefer,

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text: and

text: J.

text: ˚Akerman,

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text: Y.

text: Pogoryelov,

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text: E.

text: Iacocca,

text: A.

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text: ˚Akerman,

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text: B

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: where r is the ﬁlter radius chosen based on numerical experiments [37]; ρ˜ is the ﬁltered design variable. And projecting is implemented by the threshold projection to remove the gray area in the derived geometrical conﬁguration [38] − tanh (βξ) + tanh (β (ρ˜ ξ)) ρ¯ = − (5) tanh (βξ) + tanh (β (1 ξ)) ∈ where ξ [0, 1] and β are the threshold and projection parameters for the threshold projection, respectively. On the choice of the values of ξ and β, one can refer to [40,41]. EOT is featured by its high transmission of the photonic energy through periodic subwavelength metallic apertures. The input and output transmission power for one period of the metallic slits can be measured by the integration of the average Poynting vector on the t(cid:90)he inlet and outlet sides of th(cid:90)e comp(cid:18)utational domain (cid:19) −1 × ∗ · 1 ∗ ∇ · P = Re (E H ) n dΓ = Re H H n dΓ (6) i 2 i i 2jω(cid:15) (cid:15) zi zi Γi Γi r 0 (cid:90) (cid:90) (cid:18) (cid:19) − 1 × ∗ · 1 ∗ ∇ · P = Re (E H ) n dΓ = Re (H + H ) (H + H ) n dΓ o 2 2jω(cid:15) (cid:15) zi zs zi zs Γo Γo r 0 (7) where P and P are the input and output transmission power, respectively; E is i o i the electric ﬁelds corresponding to the incident magnetic wave H = (0, 0, H ); E is i zi the total electric ﬁelds; H is the total magnetic ﬁelds; Re operator is used to extract ∗ the real part of an expression; represents the conjugate of the complex variable. The objective of the inverse design procedure can be chosen to maximize the transmissivity deﬁned as the normalized transmission power T = P /P (8) r o i Then the inverse design problem for the bilateral nanostructures of periodic metallic slits is to maximize the transmissivity deﬁned in Equ. 8, which is constrained by Equ. 1 and Equ. 4 with design variable valued in [0, 1]. The gradient-based iterative procedure, method of moving asymptotes [35], is applied to update the design variable and maximize the transmissivity. An(cid:90)d the gradient information is obtained using the continuous adjoint method [42] − ∗ δT = Re (ρ˜ ) δρ dΩ (9) r a Ω where ρ˜ is the adjoint of the ﬁltered design variable ρ˜. ρ˜ is derived by solving the a a ad(cid:90)joint equations of the wave equation i(cid:90)n E(cid:18)qu. 1 an(cid:113)d density ﬁlter in Equ.(cid:19)4 (cid:90)− − ∇ ∗ · ∇ ∗ − − ∗ 1 ∂B (cid:15) 1 H˜ φ + k2µ H˜ φ dΩ + jk (cid:15) 1µ H˜ + φ dΓ r zs 0 r zs 0 r r zs P ∂H Ω Γo i zs (10) 1 ∂B · ∇ ∀ ∈ H + ∇ φ dΓ = 0, φ 1 (Ω) P ∂ H (cid:90) Γo i zs (cid:20) (cid:21) − ∇ ∗ · ∇ ∗ − ∂(cid:15) 1 ∂ρ¯∇ · ∇ ∗ ∀ ∈ H r2 ρ˜ ψ + ρ˜ r (H + H ) H˜ ψ dΩ = 0, ψ 1 (Ω ) a a ∂ρ¯ ∂ρ˜ zs zi zs d Ωd (11) where H˜ and B represent the adjoint of H and the integral function of Equ. 7, zsH H zs respectively; 1 (Ω) and 1 (Ω ) are the ﬁrst order Hilbert functional spaces deﬁned d on Ω and Ω . d 4 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: where

label: paragraph

text: r

text: is

text: the

text: ﬁlter

text: radius

text: chosen

text: based

text: on

text: numerical

text: experiments

text: ρ˜

text: is

text: the

text: ﬁltered

text: design

text: variable.

text: And

text: projecting

text: is

text: implemented

text: by

text: the

text: threshold

text: projection

text: to

text: remove

text: the

text: gray

text: area

text: in

text: the

text: derived

text: geometrical

text: conﬁguration

label: equation

text: tanh

text: (βξ)

text: tanh

text: (β

text: (ρ˜

text: ξ))

text: ρ¯

text: tanh

text: (βξ)

text: tanh

text: (β

text: ξ))

text: where

text: ξ

text: and

text: β

text: are

text: the

text: threshold

text: and

text: projection

text: parameters

text: for

text: the

text: threshold

text: projection,

text: respectively.

text: On

text: the

text: choice

text: of

text: the

text: values

text: of

text: ξ

text: and

text: β,

text: one

text: can

text: refer

text: to

text: EOT

text: is

text: featured

text: by

text: its

text: high

text: transmission

text: of

text: the

text: photonic

text: energy

text: through

text: periodic

text: subwavelength

text: metallic

text: apertures.

text: The

text: input

text: and

text: output

text: transmission

text: power

text: for

text: one

text: period

text: of

text: the

text: metallic

text: slits

text: can

text: be

text: measured

text: by

text: the

text: integration

text: of

text: the

text: average

text: Poynting

text: vector

text: on

text: the

text: inlet

text: and

text: outlet

text: sides

text: of

text: domain

text: −1

text: P

text: Re

text: (E

text: H

text: n

text: dΓ

text: Re

text: H

text: H

text: n

text: dΓ

text: i

text: i

text: i

text: zi

text: zi

text: Γi

text: Γi

text: r

text: P

text: Re

text: (E

text: H

text: n

text: dΓ

text: Re

text: (H

text: H

text: (H

text: H

text: n

text: dΓ

text: o

text: zi

text: zs

text: zi

text: zs

text: Γo

text: Γo

text: r

text: where

text: P

text: and

text: P

text: are

text: the

text: input

text: and

text: output

text: transmission

text: power,

text: respectively;

text: E

text: is

text: i

text: o

text: i

text: the

text: electric

text: ﬁelds

text: corresponding

text: to

text: the

text: incident

text: magnetic

text: wave

text: H

text: H

text: E

text: is

text: i

text: zi

text: the

text: total

text: electric

text: ﬁelds;

text: H

text: is

text: the

text: total

text: magnetic

text: ﬁelds;

text: Re

text: operator

text: is

text: used

text: to

text: extract

text: the

text: real

text: part

text: of

text: an

text: expression;

text: represents

text: the

text: conjugate

text: of

text: the

text: complex

text: variable.

text: The

text: objective

text: of

text: the

text: inverse

text: design

text: procedure

text: can

text: be

text: chosen

text: to

text: maximize

text: the

text: transmissivity

text: deﬁned

text: as

text: the

text: normalized

text: transmission

text: power

text: T

text: P

text: /P

text: r

text: o

text: i

text: Then

text: the

text: inverse

text: design

text: problem

text: for

text: the

text: bilateral

text: nanostructures

text: of

text: periodic

text: metallic

text: slits

text: is

text: to

text: maximize

text: the

text: transmissivity

text: deﬁned

text: in

text: Equ.

text: which

text: is

text: constrained

text: by

text: Equ.

text: and

text: Equ.

text: with

text: design

text: variable

text: valued

text: in

text: The

text: gradient-based

text: iterative

text: procedure,

text: method

text: of

text: moving

text: asymptotes

text: is

text: applied

text: to

text: update

text: the

text: design

text: variable

text: and

text: maximize

text: the

text: transmissivity.

text: the

text: gradient

text: information

text: is

text: obtained

text: using

text: the

text: continuous

text: adjoint

text: method

text: δT

text: Re

text: (ρ˜

text: δρ

text: dΩ

text: r

text: a

text: Ω

text: where

text: ρ˜

text: is

text: the

text: adjoint

text: of

text: the

text: ﬁltered

text: design

text: variable

text: ρ˜.

text: ρ˜

text: is

text: derived

text: by

text: solving

text: the

text: a

text: a

text: equations

text: of

text: the

text: wave

text: equation

text: density

text: ﬁlter

text: in

text: ∂B

text: H˜

text: φ

text: H˜

text: φ

text: dΩ

text: jk

text: 1µ

text: H˜

text: φ

text: dΓ

text: r

text: zs

text: r

text: zs

text: r

text: r

text: zs

text: P

text: ∂H

text: Ω

text: Γo

text: i

text: zs

text: ∂B

text: H

text: φ

text: dΓ

text: φ

text: (Ω)

text: P

text: H

text: Γo

text: i

text: zs

text: ∂ρ¯∇

text: H

text: ρ˜

text: ψ

text: ρ˜

text: r

text: (H

text: H

text: H˜

text: ψ

text: dΩ

text: ψ

text: (Ω

text: a

text: a

text: ∂ρ¯

text: ∂ρ˜

text: zs

text: zi

text: zs

text: d

text: Ωd

text: where

text: H˜

text: and

text: B

text: represent

text: the

text: adjoint

text: of

text: H

text: and

text: the

text: integral

text: function

text: of

text: Equ.

text: zsH

text: H

text: zs

text: respectively;

text: (Ω)

text: and

text: (Ω

text: are

text: the

text: ﬁrst

text: order

text: Hilbert

text: functional

text: spaces

text: deﬁned

text: d

text: on

text: Ω

text: and

text: Ω

text: d

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: [8] T. Sødergaard, S. I. Bozhevolnyi, S. M. Novikov, J. Beermann, E. Devaux, T. W. Ebbesen, “Extraordinary optical transmission enhanced by nanofocusing,” Nano Lett. 10, 3123–3128 (2010). [9] H. Liu, T. Li, Q. J. Wang, Z. H. Zhu, S. M. Wang, J. Q. Li, S. N. Zhu, Y. Y. Zhu, X. Zhang, “Extraordinary optical transmission induced by excitation of a magnetic plasmon propagation mode in a diatomic chain of slit-hole resonators,” Physical Review B 79, 024304 (2009). [10] F. J. Garc´ıa-Vidal, H. J. Lezec, T.W. Ebbesen, L. Mart´ın-Moreno, “Multiple paths to enhance optical transmission through a single subwavelength slit,” Physical Review Letters 90, 213901 (2003). [11] L. L. Wang, X. F. Ren, R. Yang, G. C. Guo, and G. P. Guo, “Transmission of doughnut light through a bulls eye structure,” Appl. Phys. Lett 95, 111111–111113 (2009). [12] Y. X. Cui, S. He, Y. Okuno, “Giant optical transmission through a metallic nanoslit achieved by the optimization of the groove periodicity and other parameters,” Inter- national Workshop on Metamaterials, 236–239 (2008). [13] E. Popov, M. Nevi`ere, A. L. Fehrembach, N. Bonod, “Optimization of plasmon excitation at structured apertures,” Applied Optics 44, 6141–6154 (2005). [14] M. Bendsøe, “Optimal shape design as a material distribution problem,” Struct. Optim. 1, 193–202 (1989). [15] T. Borrvall, J. Petersson, “Topology optimization of ﬂuids in Stokes ﬂow,” Int. J. Numer. Methods Fluids 41, 77–107 (2003). [16] J. S. Jensen, O. Sigmund, “Topology optimization of photonic crystal structures: a high-bandwidth low-loss T-junction waveguide,” J. Opt. Soc. Am. B 22, 1191–1198 (2005). [17] M. Bendsøe, O. Sigmund, “Topology optimization-theory, methods and applica- tions,” Springer, Berlin (2003). [18] A. Y. Piggott, J. Lu, K. G. Lagoudakis, J. Petykiewicz, T. M. Babinec, J. Vuˇcovi´c, “Inverse design and demonstration of a compact and broadband on-chip wavelength demultiplexer,” Nature Photonics 9, 374–377 (2015). [19] B. Shen, P. Wang, R. Polson, R. Menon, “An integrated-nanophotonics polariza- × tion beamsplitter with 2.4 2.4 µm2 footprint,” Nature Photonics 9, 378–382 (2015). [20] L. H. Frandsen, Y. Elesin, L. F. Frellsen, M. Mitrovic, Y. Ding, O. Sigmund, K. Yvind, “Topology optimized mode conversion in a photonic crystal waveguide fabricated in silicon-on-insulator material,” Optics Express 22, 8525–8532 (2014). [21] O. Sigmund, K. G. Hougaard, “Geometric properties of optimal photonic crystals,” Phys. Rev. Lett. 100, 153904 (2008). [22] H. Men, K. Y. K. Lee, R. M. Freund, J. Peraire, S. G. Johnson, “Robust topol- ogy optimization of three-dimensional photonic-crystal band-gap structures” Optics Express 22(19), 22632–22648 (2014). [23] J. Andkjær, O. Sigmund, “Topology optimized low-contrast all-dielectric optical cloak,” Appl. Phys. Lett. 98, 021112 (2011). [24] B. Vial, Y. Hao, “Topology optimized all-dielectric cloak: design, performances and modal picture of the invisibility eﬀect,” Optics Express 100, 23551–23560 (2015). [25] G. Fujii, H. Watanabe, T. Yamada, T. Ueta, M. Mizuno, “Level set based topology optimization for optical cloaks,” Appl. Phys. Lett. 102, 251106 (2013). [26] S. Zhou, W. Li, Y. Chen, G. Sun, Q. Li, “Topology optimization for negative permeability metamaterials using level-set algorithm,” Acta Materialia 59, 2624– 2636 (2011). [27] A. R. Diaz, O. Sigmund, “A topology optimization method for design of negative permeability metamaterials,” Struct. Multidisc. Optim. 41, 163–177 (2010). [28] M. Otomori, T. Yamada, K. Izui, S. Nishiwaki, J. Andkjær, “A topology optimiza- tion method based on the level set method for the design of negative permeability dielectric metamaterials,” Comput. Methods Appl. Mech. Engrg. 237-240, 192–211 (2012). 10

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text: 2 M. TSUJII present. In this paper, we investigate a related problem: whether exponential mixing is an open dense property for volume-preserving Anosov ﬂows. A few important progresses on the rate of mixing for Anosov ﬂows were made by Chernov[5] and Dolgopyat[7, 8, 9] in late 1990’s. In [7], Dolgopyat proved that a volume- preserving Anosov ﬂow is exponentially mixing if the stable and unstable foliations are C1 and are not jointly integrable. In particular, it is proved in [7] that the geodesic ﬂows on negatively curved surfaces are exponentially mixing. (Later this result is extended to higher dimensional contact Anosov ﬂows [16, 22, 23] and also to those with singularities such as Sinai billiard ﬂows [2, 3].) In [8] and [9], he also studied exponential and rapid (i.e. super-polynomial) mixing for suspension ﬂows of subshifts of ﬁnite type, which abstracts Axiom A ﬂow, and gave several criteria for such ﬂows to be rapid or exponential mixing. Based on the argument in [8], Field, Melbourne and To¨ro¨k proved more recently in [12] that rapid mixing is an open dense property for Axiom A ﬂows and, in particular, for volume-preserving Anosov ﬂows. However, to the author’s knowledge, the problem on exponential mixing mentioned above remains open. The aim of this paper is to study the problem in the simplest pos- sible setting of dimension 3 and present an aﬃrmative answer in Theorem 1.1. This also provides an example of a non-empty open set of volume-preserving Anosov ﬂows which stably exhibit exponential mixing. (Rather surprisingly, no such example has been known.) In the following sections, we ﬁrst investigate the geometry of the stable and unstable foliations and introduce the notion of s-template which describes how the stable subbundle twists along unstable manifolds. In Deﬁnition 2.11, we formulate the non-integrability condition (NI) for ρ > 0 in terms of s-templates. In Theorem 2.15, we show that the ρ ≥ condition (NI) for suﬃciently small ρ > 0 holds for a dense subset in Fr for any r 3. ρ ∈ A Then we prove Theorem 1.1 by showing, in Theorem 2.16, that, if ft F3 satisﬁes (NI) 0 A ρ for some ρ > 0, there is a C3 open neighborhood of ft in F3 in which all the ﬂows are 0 A exponentially mixing with uniform constants c and C in the decay estimate (1). α α The main novelty in our argument consists in the argument related to s-templates in Section . The idea is quite simple and explained in the following few pages. Also the perturbation argument in the proof of Theorem 2.15 in Section may be of some interest, where we consider deformation families of a ﬂow with huge number of parameters and apply large deviation argument in the parameter spaces. The proof of Theorem 2.16 is obtained by modifying the argument in the author’s previous papers [22, 23]. Unfortunately this part is rather long and occupies the remaining two-thirds of this paper, though the core of the argument is presented in a few pages. This is because some objects we consider are not smooth and require careful treatment. Still our argument is basically elementary and straightforward. If the reader is familiar with estimates on non-linearity of hyperbolic ﬂows and/or Fourier analysis, one will be able to skip good part of the argument and/or ﬁnd better ways to prove the propositions in this part. Remark 1.2. The argument presented in this paper depends crucially on the assumptions that M is three dimensional and that ft preserves a smooth volume. So it will not extend to more general cases directly. But the author would like to emphasize that the following idea behind the argument will be useful in much more general cases of partially hyperbolic dy- namical systems: Twist of the stable subbundle along pieces of unstable manifolds viewed in the unit scale will be “random” and “rough” in generic cases and such “random” twist will not be cancelled completely in the process where the ﬂow ft contracts the piece of → −∞ unstable manifold to microscopic scale as t (if we view things in an appropriate

text: M.

text: TSUJII

text: present.

text: In

text: this

text: paper,

text: we

text: investigate

text: a

text: related

text: problem:

text: whether

text: exponential

text: mixing

text: is

text: an

text: open

text: dense

text: property

text: for

text: volume-preserving

text: Anosov

text: ﬂows.

text: A

text: few

text: important

text: progresses

text: on

text: the

text: rate

text: of

text: mixing

text: for

text: Anosov

text: ﬂows

text: were

text: made

text: by

text: and

text: in

text: late

text: In

text: Dolgopyat

text: proved

text: that

text: a

text: volume-

text: preserving

text: Anosov

text: ﬂow

text: is

text: exponentially

text: mixing

text: if

text: the

text: stable

text: and

text: unstable

text: foliations

text: are

text: and

text: are

text: not

text: jointly

text: integrable.

text: In

text: particular,

text: it

text: is

text: proved

text: in

text: that

text: the

text: geodesic

text: ﬂows

text: on

text: negatively

text: curved

text: surfaces

text: are

text: exponentially

text: mixing.

text: (Later

text: this

text: result

text: is

text: extended

text: to

text: higher

text: dimensional

text: contact

text: Anosov

text: ﬂows

text: and

text: also

text: to

text: those

text: with

text: singularities

text: such

text: as

text: Sinai

text: billiard

text: ﬂows

text: In

text: and

text: he

text: also

text: studied

text: exponential

text: and

text: rapid

text: (i.e.

text: super-polynomial)

text: mixing

text: for

text: suspension

text: ﬂows

text: of

text: subshifts

text: of

text: ﬁnite

text: type,

text: which

text: abstracts

text: Axiom

text: A

text: ﬂow,

text: and

text: gave

text: several

text: criteria

text: for

text: such

text: ﬂows

text: to

text: be

text: rapid

text: or

text: exponential

text: mixing.

text: Based

text: on

text: the

text: argument

text: in

text: Field,

text: Melbourne

text: and

text: To¨ro¨k

text: proved

text: more

text: recently

text: in

text: that

text: rapid

text: mixing

text: is

text: an

text: open

text: dense

text: property

text: for

text: Axiom

text: A

text: ﬂows

text: and,

text: in

text: particular,

text: for

text: volume-preserving

text: Anosov

text: ﬂows.

text: However,

text: to

text: the

text: author’s

text: knowledge,

text: the

text: problem

text: on

text: exponential

text: mixing

text: mentioned

text: above

text: remains

text: open.

text: The

text: aim

text: of

text: this

text: paper

text: is

text: to

text: study

text: the

text: problem

text: in

text: the

text: simplest

text: pos-

text: sible

text: setting

text: of

text: dimension

text: and

text: present

text: an

text: aﬃrmative

text: answer

text: in

text: Theorem

text: This

text: also

text: provides

text: an

text: example

text: of

text: a

text: non-empty

text: open

text: set

text: of

text: volume-preserving

text: Anosov

text: ﬂows

text: which

text: stably

text: exhibit

text: exponential

text: mixing.

text: (Rather

text: surprisingly,

text: no

text: such

text: example

text: has

text: been

text: known.)

text: In

text: the

text: following

text: sections,

text: we

text: ﬁrst

text: investigate

text: the

text: geometry

text: of

text: the

text: stable

text: and

text: unstable

text: foliations

text: and

text: introduce

text: the

text: notion

text: of

text: s-template

text: which

text: describes

text: how

text: the

text: stable

text: subbundle

text: twists

text: along

text: unstable

text: manifolds.

text: In

text: Deﬁnition

text: we

text: formulate

text: the

text: non-integrability

text: condition

text: (NI)

text: for

text: ρ

text: in

text: terms

text: of

text: s-templates.

text: In

text: Theorem

text: we

text: show

text: that

text: the

text: ρ

text: condition

text: (NI)

text: for

text: suﬃciently

text: small

text: ρ

text: holds

text: for

text: a

text: dense

text: subset

text: in

text: Fr

text: for

text: any

text: r

text: ρ

text: A

text: Then

text: we

text: prove

text: Theorem

text: by

text: showing,

text: in

text: Theorem

text: that,

text: if

text: ft

text: satisﬁes

text: (NI)

text: A

text: ρ

text: for

text: some

text: ρ

text: there

text: is

text: a

text: open

text: neighborhood

text: of

text: ft

text: in

text: in

text: which

text: all

text: the

text: ﬂows

text: are

text: A

text: exponentially

text: mixing

text: with

text: uniform

text: constants

text: c

text: and

text: C

text: in

text: the

text: decay

text: estimate

text: α

text: α

text: The

text: main

text: novelty

text: in

text: our

text: argument

text: consists

text: in

text: the

text: argument

text: related

text: to

text: s-templates

text: in

text: Section

text: The

text: idea

text: is

text: quite

text: simple

text: and

text: explained

text: in

text: the

text: following

text: few

text: pages.

text: Also

text: the

text: perturbation

text: argument

text: in

text: the

text: proof

text: of

text: Theorem

text: in

text: Section

text: may

text: be

text: of

text: some

text: interest,

text: where

text: we

text: consider

text: deformation

text: families

text: of

text: a

text: ﬂow

text: with

text: huge

text: number

text: of

text: parameters

text: and

text: apply

text: large

text: deviation

text: argument

text: in

text: the

text: parameter

text: spaces.

text: The

text: proof

text: of

text: Theorem

text: is

text: obtained

text: by

text: modifying

text: the

text: argument

text: in

text: the

text: author’s

text: previous

text: papers

text: Unfortunately

text: this

text: part

text: is

text: rather

text: long

text: and

text: occupies

text: the

text: remaining

text: two-thirds

text: of

text: this

text: paper,

text: though

text: the

text: core

text: of

text: the

text: argument

text: is

text: presented

text: in

text: a

text: few

text: pages.

text: This

text: is

text: because

text: some

text: objects

text: we

text: consider

text: are

text: not

text: smooth

text: and

text: require

text: careful

text: treatment.

text: Still

text: our

text: argument

text: is

text: basically

text: elementary

text: and

text: straightforward.

text: If

text: the

text: reader

text: is

text: familiar

text: with

text: estimates

text: on

text: non-linearity

text: of

text: hyperbolic

text: ﬂows

text: and/or

text: Fourier

text: analysis,

text: one

text: will

text: be

text: able

text: to

text: skip

text: good

text: part

text: of

text: the

text: argument

text: and/or

text: ﬁnd

text: better

text: ways

text: to

text: prove

text: the

text: propositions

text: in

text: this

text: part.

text: Remark

text: The

text: argument

text: presented

text: in

text: this

text: paper

text: depends

text: crucially

text: on

text: the

text: assumptions

text: that

text: M

text: is

text: three

text: dimensional

text: and

text: that

text: ft

text: preserves

text: a

text: smooth

text: volume.

text: So

text: it

text: will

text: not

text: extend

text: to

text: more

text: general

text: cases

text: directly.

text: But

text: the

text: author

text: would

text: like

text: to

text: emphasize

text: that

text: the

text: following

text: idea

text: behind

text: the

text: argument

text: will

text: be

text: useful

text: in

text: much

text: more

text: general

text: cases

text: of

text: partially

text: hyperbolic

text: dy-

text: namical

text: systems:

text: Twist

text: of

text: the

text: stable

text: subbundle

text: along

text: pieces

text: of

text: unstable

text: manifolds

text: viewed

text: in

text: the

text: unit

text: scale

text: will

text: be

text: “random”

text: and

text: “rough”

text: in

text: generic

text: cases

text: and

text: such

text: “random”

text: twist

text: will

text: not

text: be

text: cancelled

text: completely

text: in

text: the

text: process

text: where

text: the

text: ﬂow

text: ft

text: contracts

text: the

text: piece

text: of

text: unstable

text: manifold

text: to

text: microscopic

text: scale

text: as

text: t

text: (if

text: we

text: view

text: things

text: in

text: an

text: appropriate


text: 12 M. TSUJII { | } (32) 10τ∗ < min per(w) w is a periodic point of ft. Then we set 2 λ∗ := e10χ∗τ∗ > 1 and c∗ = − − . 1 λ∗1 (cid:107) | (cid:107) ∈ so that the modulus of hyperbolicity Dfper(w) of a periodic point w M is always w Eu greater than λ∗. Let R > 0 be an integer constant that we will take soon in the next ∈ subsection. (One may suppose R = r + 11 if one like.) For b > 1 and p M, we set { ≥ | (cid:107) − | (cid:107) ≤ − } (33) T(p, b) = inf t 0 Dfp t Eu b 1/(4R) . ⊂ Deﬁnition 3.1 (The exceptional set E(b)). The exceptional set E(b) M for b > 1 is the ∈ open set of points p M such that there exits a periodic orbit γ whose prime period is less − than T(p, b) and whose (minimum) distance from Wu− (p) is less than b 2/(3R). [ c∗,c∗] The next lemma tells that the periodic orbit γ in the deﬁnition above is unique for each ∈ p E(b), provided that b is suﬃciently large. ≥ ∈ Lemma 3.2. For any C > 1, there exists b0 > 0 such that, for b b0 and p M, there exists at most one periodic orbit γ satisfying both of the following conditions: (1) the prime period of γ is bounded by T(p, b) + C, and − (2) the (minimum) distance from γ to Wu− (p) is less than b 3/(5R). [ c∗,c∗] − Further, if such periodic orbit γ exists, it passes through the b 3/(5R)-neighborhood of W−[u−c∗,c∗](p) only once. More precisely, if there are two po| in| ts p−1, p2 on γ that belongs to the b 3/(5R)-neighborhood of W[u−c∗,c∗](p), there exists t with τ < b 1/(3R) such that ft(p1) = p2. Proof. Suppose that a periodic orbit γ satisﬁes the conditions (1) and (2) in the lemma for ≥ ∈ so−me b b0 and p M. Let p1 be a point on the periodic orbit γ that belongs to the b 3/(5R)-neighborhood of Wu− (p). By a crude estimate we see that the backward orbit − [ c∗,c∗] f t(p) of the point p traces that of p as far as 1 (cid:107) − (cid:107) ∼ (cid:107) − | (cid:107)− (cid:28) Dfp t Dfp t Eu 1 b3/(5R). H−ence, for an−y ε >∈0, we can take t(ε) > 0 a∈nd let b0 large so that the distance between f−t(p) and f∈t(p1) γ is b|ou|nded by ε for t [t(ε), t(ε)| +| T(p, b) + C]. That is, the orbit f t(p) for t [t(ε), t(ε) + γ ] traces the periodic orbit γ within distance ε. By uniform hyperbolicity of the ﬂow ft, this implies uniqueness of the periodic orbit γ and also the last − (cid:29) − claim of the lemma because b 1/(3R) b 3/(5R). (cid:3) ##LTLine##

text: M.

text: TSUJII

font: PFBMDO+txsy

text: 10τ∗

font: QUKERW+rtxmi

text: min

text: per(w)

text: w

text: is

text: a

text: periodic

text: point

text: of

text: ft.

text: Then

text: we

text: set

text: λ∗

text: e10χ∗τ∗

text: and

text: c∗

font: JZGPLZ+rtxr

text: λ∗1

text: so

text: that

text: the

text: modulus

text: of

text: hyperbolicity

text: Dfper(w)

text: of

text: a

text: periodic

text: point

text: w

text: M

text: is

text: always

text: w

text: Eu

text: greater

text: than

text: λ∗.

text: Let

text: R

text: be

text: an

text: integer

text: constant

text: that

text: we

text: will

text: take

text: soon

text: in

text: the

text: next

text: subsection.

text: (One

text: may

text: suppose

text: R

text: r

text: if

text: one

text: like.)

text: For

text: b

text: and

text: p

text: M,

text: we

text: set

text: T(p,

text: b)

text: inf

text: t

text: Dfp

text: t

text: Eu

text: b

text: Deﬁnition

text: (The

text: exceptional

text: set

text: E(b)).

text: The

text: exceptional

text: set

text: E(b)

text: M

text: for

text: b

text: is

text: the

text: open

text: set

text: of

text: points

text: p

text: M

text: such

text: that

text: there

text: exits

text: a

text: periodic

text: orbit

text: γ

text: whose

text: prime

text: period

text: is

text: less

text: than

text: T(p,

text: b)

text: and

text: whose

text: (minimum)

text: distance

text: from

text: Wu−

text: (p)

text: is

text: less

text: than

text: b

text: c∗,c∗]

text: The

text: next

text: lemma

text: tells

text: that

text: the

text: periodic

text: orbit

text: γ

text: in

text: the

text: deﬁnition

text: above

text: is

text: unique

text: for

text: each

text: p

text: E(b),

text: provided

text: that

text: b

text: is

text: suﬃciently

text: large.

text: Lemma

text: For

text: any

text: C

text: there

text: exists

text: such

text: that,

text: for

text: b

text: and

text: p

text: M,

text: there

text: exists

text: at

text: most

text: one

text: periodic

text: orbit

text: γ

text: satisfying

text: both

text: of

text: the

text: following

text: conditions:

text: the

text: prime

text: period

text: of

text: γ

text: is

text: bounded

text: by

text: T(p,

text: b)

text: C,

text: and

text: the

text: (minimum)

text: distance

text: from

text: γ

text: to

text: Wu−

text: (p)

text: is

text: less

text: than

text: b

text: c∗,c∗]

text: Further,

text: if

text: such

text: periodic

text: orbit

text: γ

text: exists,

text: it

text: passes

text: through

text: the

text: b

text: of

text: W−[u−c∗,c∗](p)

text: only

text: once.

text: More

text: precisely,

text: if

text: there

text: are

text: two

text: po|

text: in|

text: ts

text: on

text: γ

text: that

text: belongs

text: to

text: the

text: b

text: of

text: W[u−c∗,c∗](p),

text: there

text: exists

text: t

text: with

text: τ

text: b

text: such

text: that

text: Proof.

text: Suppose

text: that

text: a

text: periodic

text: orbit

text: γ

text: satisﬁes

text: the

text: conditions

text: and

text: in

text: the

text: lemma

text: for

text: so−me

text: b

text: and

text: p

text: M.

text: Let

text: be

text: a

text: point

text: on

text: the

text: periodic

text: orbit

text: γ

text: that

text: belongs

text: to

text: the

text: b

text: of

text: Wu−

text: (p).

text: By

text: a

text: crude

text: estimate

text: we

text: see

text: that

text: the

text: backward

text: orbit

text: c∗,c∗]

text: f

text: t(p)

text: of

text: the

text: point

text: p

text: traces

text: that

text: of

text: p

text: as

text: far

text: as

text: Dfp

text: t

text: Dfp

text: t

text: Eu

text: H−ence,

text: for

text: an−y

text: ε

text: we

text: can

text: take

text: t(ε)

text: a∈nd

text: let

text: large

text: so

text: that

text: the

text: distance

text: between

text: f−t(p)

text: and

text: γ

text: is

text: b|ou|nded

text: by

text: ε

text: for

text: t

text: [t(ε),

text: t(ε)|

text: T(p,

text: b)

text: C].

text: That

text: is,

text: the

text: orbit

text: f

text: t(p)

text: for

text: t

text: [t(ε),

text: t(ε)

text: γ

text: traces

text: the

text: periodic

text: orbit

text: γ

text: within

text: distance

text: ε.

text: By

text: uniform

text: hyperbolicity

text: of

text: the

text: ﬂow

text: ft,

text: this

text: implies

text: uniqueness

text: of

text: the

text: periodic

text: orbit

text: γ

text: and

text: also

text: the

text: last

text: claim

text: of

text: the

text: lemma

text: because

text: b

text: b

text: ##LTLine##


text: 20 M. TSUJII perturbation, mentioned in the beginning of this proof, and also from the estimate above at the end points, we see that (cid:107) · − (cid:107) − − ∈ − (46) γ0p,t(wup(h(t) τ; ϕt)) γ0p,0(wup(τ; ϕ0)) < C∗b 1 (1/(4R)) for τ ( 1, 1). We can now conclude the lemma by comparing (41) and (42) and using (44), (45), (46). (cid:3) (cid:12)(cid:90) (cid:90) (cid:12) (cid:12) (cid:16) (cid:16) (cid:17)(cid:17) (cid:16) (cid:16) (cid:17)(cid:17) (cid:12) (cid:12) (cid:12) (cid:12)(cid:12)From Lemma 3.8, we have (cid:12)(cid:12) − − exp ib ψ˜sp,(−1,1)(τ; t) + ατ dτ exp ib ψsp,(−1,1)(τ; ϕt) + ατ dτ < C∗b 1/(4R) I I ⊂ − for any Borel subset I [ 1, 1]. Therefore, in proving (21), we may consider ψ˜s − (τ; t) p,( 1,1) in the place of ψsp,(−1,1)(τ; ϕt), provided that 0 < ρ < 1/(4R). To proceed, we introduce the mapping − → Ψ(cid:18) τ = Ψτ,p,α,b : [ 4, 4]J RJ (cid:19) ∈ − − − deﬁned for each τ ( b 1/R, b 1/R) by (cid:82) · − Ψ (t) = b ψ˜s − (s(j) + τ; a 1t) + α(s(j) + τ) τ p,( 1,1) 0 ∈ j J − where a0 = χ(z/τ∗)dz. (∈The factor a01 before t above is thrown-in just for normalization. See Remark 3.6.) For ϕ0 W in (3(cid:90)7), let A(ϕ0) be the diagonal matrix of size (cid:93)J with the diagonal elements − · − · aj(ϕ0) = a01 (1 + ϕ0(ft(q(j)))) 2 χ(t/τ∗) dt · w−here χ( ) is the functi∈on that appeared in the deﬁnition (38) of ϕj. Clearly we have C∗1 < aj(ϕ) < C∗ for ϕ W and | − | (cid:107) − (cid:107) ∈ (47) aj(ϕ˜) aj(ϕ) < C∗ ϕ˜ ϕ ∞ for ϕ, ϕ˜ W. The next lemma realize the idea explained in Remark 3.6. (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) Proposition 3.9. If b > 0 is suﬃciently large, we have − → ≤ − DΨτ(t) A(ϕ0) : RJ RJ max C∗b 1/(4R) ∈ − ∈ \ ∈ ∈ − − − for any t [ 4, 4]J, p M E(b), α R and τ ( b 1/R, b 1/R), where we consider (cid:107) (cid:107) | | the operator norm with respect to the norm (s ) = max s on RJ. This implies in − j max j j particular that the mapping Ψ restricted to [ 4, 4]J is a diﬀeomorphism onto its image. τ In addition, for the Jacobian determinant, we have | − | ≤ − · (48) log det DΨτ(t) log det A(ϕ0) C∗b 1/(4R) (cid:93)J.

text: M.

text: TSUJII

text: perturbation,

text: mentioned

text: in

text: the

text: beginning

text: of

text: this

text: proof,

text: and

text: also

text: from

text: the

text: estimate

text: above

text: at

text: the

text: end

text: points,

text: we

text: see

text: that

text: τ;

text: ϕt))

text: C∗b

text: for

text: τ

text: We

text: can

text: now

text: conclude

text: the

text: lemma

text: by

text: comparing

text: and

text: and

text: using

font: GVSQHL+txsya

font: FQHAYK+txex

text: Lemma

text: we

text: have

text: exp

text: ib

text: t)

text: ατ

text: dτ

text: exp

text: ib

text: ϕt)

text: ατ

text: dτ

text: C∗b

text: I

text: I

text: for

text: any

text: Borel

text: subset

text: I

text: Therefore,

text: in

text: proving

text: we

text: may

text: consider

text: ψ˜s

text: (τ;

text: t)

text: p,(

text: in

text: the

text: place

text: of

text: ϕt),

text: provided

text: that

text: ρ

text: To

text: proceed,

text: we

text: introduce

text: the

text: mapping

text: τ

text: Ψτ,p,α,b

text: RJ

text: deﬁned

text: for

text: each

text: τ

text: b

text: b

text: by

text: Ψ

text: (t)

text: b

text: ψ˜s

text: (s(j)

text: τ;

text: a

text: α(s(j)

text: τ)

text: τ

text: p,(

text: j

text: J

text: where

text: χ(z/τ∗)dz.

text: (∈The

text: factor

text: before

text: t

text: above

text: is

text: thrown-in

text: just

text: for

text: normalization.

text: See

text: Remark

text: For

text: ϕ0

text: W

text: in

text: let

text: be

text: the

text: diagonal

text: matrix

text: of

text: size

text: with

text: the

text: diagonal

text: elements

text: χ(t/τ∗)

text: dt

text: w−here

text: χ(

text: is

text: the

text: functi∈on

text: that

text: appeared

text: in

text: the

text: deﬁnition

text: of

text: ϕj.

text: Clearly

text: we

text: have

text: aj(ϕ)

text: C∗

text: for

text: ϕ

text: W

text: and

text: aj(ϕ˜)

text: aj(ϕ)

text: C∗

text: ϕ˜

text: ϕ

text: for

text: ϕ,

text: ϕ˜

text: W.

text: The

text: next

text: lemma

text: realize

text: the

text: idea

text: explained

text: in

text: Remark

text: Proposition

text: If

text: b

text: is

text: suﬃciently

text: large,

text: we

text: have

text: DΨτ(t)

text: RJ

text: RJ

text: max

text: C∗b

text: for

text: any

text: t

text: p

text: M

text: E(b),

text: α

text: R

text: and

text: τ

text: b

text: b

text: where

text: we

text: consider

text: the

text: operator

text: norm

text: with

text: respect

text: to

text: the

text: norm

text: (s

text: max

text: s

text: on

text: RJ.

text: This

text: implies

text: in

text: j

text: max

text: j

text: j

text: particular

text: that

text: the

text: mapping

text: Ψ

text: restricted

text: to

text: is

text: a

text: diﬀeomorphism

text: onto

text: its

text: image.

text: τ

text: In

text: addition,

text: for

text: the

text: Jacobian

text: determinant,

text: we

text: have

text: log

text: det

text: DΨτ(t)

text: log

text: det

text: C∗b


text: EXPONENTIAL MIXING FOR ANOSOV FLOWS 31 so that we can approximate the sum of them by an aﬃne function, say ατ+β, with an error (cid:104) (cid:105)− term bounded by C∗κ2 ω 1. For the remaining term on the last line, we have (cid:93) | − | ≤ (cid:104) (cid:105)− (cid:104) (cid:105) ψs − (τ) ψs − (0) C∗κ ω 1/2 log ω q,( h,h) q,( h,h) (cid:93) from (14) and also (cid:104) ⊥ (cid:105) (cid:104) (cid:105)− γq,(−h,h)(wuq(τ)), ey(wuq(τ)) = ca,ω,n + O∗(κ(cid:93) ω 1/2) (cid:48) (cid:48) ∈ correspondingly to (77). Hence, with some α , β R, it holds | ◦ − · − (cid:48) − (cid:48)| (cid:104) (cid:105)− (cid:104) (cid:105) θas,ω,n(κa,ω,n wup(τ)) ca,ω,n ψqs,(−h,h)(τ) α τ β < C∗κ2(cid:93) ω 1 log ω − ≤ ≤ for h < τ < h. Since we assume b0 b κ−(cid:93) −in the cl−ai−m(cid:104)(6(cid:105)7−), we may suppose that the right-hand side above is much smaller than b 1 ρh > κ 1 ρ ω 1+θ∗ by letting ω be larger (cid:93) (cid:93) · if necessary. Therefore we obtain the claim (67) from (80) with b replaced by c b, − | | a,ω,n noting that C∗1 < ca,ω,n < C∗ from Remark 4.3. (cid:3) Remark 4.8. (cid:107) − (cid:48) − (cid:107) ≤ · (cid:104) (cid:105)− · (cid:104)(cid:104) (cid:105) (cid:107) (cid:48) − (cid:107)(cid:105) (81) ∆a,1ω,n(ea,ω,n(w ) ea,ω,n(w)) C∗t(cid:93)2 ω 1/2 ω 1/2 w w × (cid:32) (cid:33) where ∆ is the 2 2 matrix deﬁned by a,ω,n ∆ 0 (cid:104) (cid:104) (cid:105)− (cid:105) (82) ∆a,ω,n = a0,ω,n 1 , ∆a,ω,n := ∆(pa,ω,n, κ(cid:93) ω 1/2) . (Recall that ∆(q, δ) is the approximate non-integrablity deﬁned in Deﬁnition 2.17.) For a technical reason to be explained in Remark 5.5, we will need to extend the functions · e ( ) to R2. Though the choice of the extension is rather arbitrary, we will suppose that a,ω,n (cid:48) ∈ the extension is continuous and satisﬁes (81) for all w, w R2. In this subsection, we(cid:48)con(cid:48)sid(cid:48)er how the(cid:48)ﬂ∈ow ft loo(cid:48)k∈s in the loca∈l charts κa,ω(cid:48),n.∈Below(cid:48) w(cid:48)e consider (a, ω, n) and (a , ω , n ) with a, a A, ω, ω Z and n N(a, ω), n N(a , ω ) and suppose ≤ ≤ ∩ − ∅ (83) t(cid:93) t 2t(cid:93) and U = Ua,ω,n f t(Ua(cid:48),ω(cid:48),n(cid:48)) (cid:44) . Then the ﬂow ft viewed in the local charts κa,ω,n and κa(cid:48),ω(cid:48),n(cid:48) is ◦ ◦ − → f := κa(cid:48),ω(cid:48),n(cid:48) ft κa,1ω,n : κa,ω,n(U) κa(cid:48),ω(cid:48),n(cid:48)(f(U)). Since κa,ω,n are ﬂow box local charts, this diﬀeomorphism f is written in the form (84) f(x, y, z) = (fˆ(x, y), z + fˇ(x, y)) and therefore we may extend it naturally to × → (cid:48) × ◦ (cid:48) ◦ f : V R V R, with setting V := π κa,ω,n(U), V := π κa(cid:48),ω,n(f(U))

text: EXPONENTIAL

text: MIXING

text: FOR

text: ANOSOV

text: FLOWS

text: so

text: that

text: we

text: can

text: approximate

text: the

text: sum

text: of

text: them

text: by

text: an

text: aﬃne

text: function,

text: say

text: ατ+β,

text: with

text: an

text: error

text: term

text: bounded

text: by

text: C∗κ2

text: ω

text: For

text: the

text: remaining

text: term

text: on

text: the

text: last

text: line,

text: we

text: have

text: ψs

text: (τ)

text: ψs

text: C∗κ

text: ω

text: log

text: ω

text: q,(

text: h,h)

text: q,(

text: h,h)

text: from

text: and

text: also

text: γq,(−h,h)(wuq(τ)),

text: ey(wuq(τ))

text: ca,ω,n

text: ω

text: correspondingly

text: to

text: Hence,

text: with

text: some

text: α

text: β

text: R,

text: it

text: holds

text: θas,ω,n(κa,ω,n

text: wup(τ))

text: ca,ω,n

text: ψqs,(−h,h)(τ)

text: α

text: τ

text: β

text: ω

text: log

text: ω

text: for

text: h

text: τ

text: h.

text: Since

text: we

text: assume

text: b

text: −in

text: the

text: we

text: may

text: suppose

text: that

text: the

text: right-hand

text: side

text: above

text: is

text: much

text: smaller

text: than

text: b

text: ρh

text: κ

text: ρ

text: ω

text: 1+θ∗

text: by

text: letting

text: ω

text: be

text: larger

text: if

text: necessary.

text: Therefore

text: we

text: obtain

text: the

text: claim

text: from

text: with

text: b

text: replaced

text: by

text: c

text: b,

text: a,ω,n

text: noting

text: that

text: ca,ω,n

text: C∗

text: from

text: Remark

text: Remark

text: ea,ω,n(w))

text: ω

text: ω

text: w

text: w

text: where

text: is

text: the

text: matrix

text: deﬁned

text: by

text: a,ω,n

text: ∆a,ω,n

text: ∆a,ω,n

text: ∆(pa,ω,n,

text: ω

text: (Recall

text: that

text: ∆(q,

text: δ)

text: is

text: the

text: approximate

text: non-integrablity

text: deﬁned

text: in

text: Deﬁnition

text: For

text: a

text: technical

text: reason

text: to

text: be

text: explained

text: in

text: Remark

text: we

text: will

text: need

text: to

text: extend

text: the

text: functions

text: e

text: to

text: Though

text: the

text: choice

text: of

text: the

text: extension

text: is

text: rather

text: arbitrary,

text: we

text: will

text: suppose

text: that

text: a,ω,n

text: the

text: extension

text: is

text: continuous

text: and

text: satisﬁes

text: for

text: all

text: w,

text: w

text: In

text: this

text: subsection,

text: how

text: ft

text: in

text: the

text: loca∈l

text: charts

text: consider

text: (a,

text: ω,

text: n)

text: and

text: (a

text: ω

text: n

text: with

text: a,

text: a

text: A,

text: ω,

text: ω

text: Z

text: and

text: n

text: N(a,

text: ω),

text: n

text: N(a

text: ω

text: and

text: suppose

text: t

text: and

text: U

text: Ua,ω,n

text: f

text: Then

text: the

text: ﬂow

text: ft

text: viewed

text: in

text: the

text: local

text: charts

text: κa,ω,n

text: and

text: is

text: f

text: ft

text: κa,ω,n(U)

text: Since

text: κa,ω,n

text: are

text: ﬂow

text: box

text: local

text: charts,

text: this

text: diﬀeomorphism

text: f

text: is

text: written

text: in

text: the

text: form

text: f(x,

text: y,

text: z)

text: (fˆ(x,

text: y),

text: z

text: fˇ(x,

text: y))

text: and

text: therefore

text: we

text: may

text: extend

text: it

text: naturally

text: to

text: f

text: V

text: R

text: V

text: R,

text: with

text: setting

text: V

text: π

text: κa,ω,n(U),

text: V

text: π


text: 38 M. TSUJII ⊂ ⊂ ⊂ − ⊂ (cid:48) (104) Cα0(M) Hα0(M) H H α0(M) (Cα0(M)) where Hr(M) denotes the Sobolev space of order r. By deﬁnition, the operator I extends → to an isometric injection I : H H. ◦ ◦ ∗ We deﬁne the operator Lt formally by Lt = I Lt I , so that the following diagram commutes: (cid:120) (cid:120) H −−−−L−t−→ H I I −−−−L−t−→ (cid:76) (cid:81) H H Remark 5.8. At this moment, we only know that the operator Lt is deﬁned as an op- ∞ ∞ erator from j C0 (supp ψj) to j C0 (supp ψj). We will see that it extends naturally to a bounded operator on H and consequently that Lt extends to a bounded operator on H ≥ when t t . (cid:93) 6. Proof of Theorem 2.16 ∈ We henceforth assume that ft F3 satisﬁes the non-integrability condition (NI) for ≤ ≤ A ρ some ρ > 0 and suppose t t 2t . Most part of the argument below is devoted to (cid:93) (cid:93) show that ft itself is exponentially mixing. In the last subsection, we complete the proof of Theorem 2.16 by examining dependence of the argument on the ﬂow ft. For this last part of the argument, we emphasize at this moment that, for the proof of exponential mixing for ft, we actually need the estimate (21) in the non-integrability condition (NI) only for ρ b in some bounded range. (See Remark 6.10.) This is crucial when we prove stability of exponential mixing. In the following, we suppose that ω(cid:93) > ω0 is the constant in Lemma 4.9, but will let it be larger if necessary. We will also introduce a large constant m > 0 depending on t and ω . (cid:93) (cid:93) (cid:93) Below we will ignore some absolute constants, such as 2π, that appear in Fourier transform and partial Bargmann transform, since they are not essential at all in our argument. Below we will use the notations prepared in the last section, especially (102). We write ∞ → ∞ Ltj→j(cid:48) : (cid:48)C0 (supp ψj) C0 (supp ψj(cid:48)) for the component of Lt that sends the j-component to the j -component. It is written as ◦ ◦ ◦ ∗ (105) Ltj→j(cid:48)u = M(ψj(cid:48)) B Ltj→j(cid:48) B u where Ltj→j(cid:48) is the transfer operator on the local charts deﬁned by · ◦ − (106) Ltj→j(cid:48)v = (ρtj→j(cid:48) v) (fjt→j(cid:48)) 1 with setting ◦ ◦ − ◦ · fjt→j(cid:48) = κj(cid:48) ft κj 1 and ρtj→j(cid:48) = (ρj(cid:48) fjt→j(cid:48)) ρ˜j. As we noted in Subsection , the diﬀeomorphism fjt→j(cid:48) is written fjt→j(cid:48)(x, y, z) = (fˆjt→j(cid:48)(x, y), z + fˇjt→j(cid:48)(x, y)).

text: M.

text: TSUJII

text: H

text: H

text: where

text: Hr(M)

text: denotes

text: the

text: Sobolev

text: space

text: of

text: order

text: r.

text: By

text: deﬁnition,

text: the

text: operator

text: I

text: extends

text: to

text: an

text: isometric

text: injection

text: I

text: H

text: H.

font: LNBSKR+txsyb

text: We

text: deﬁne

text: the

text: operator

text: Lt

text: formally

text: by

text: Lt

text: I

text: Lt

text: I

text: so

text: that

text: the

text: following

text: diagram

text: commutes:

text: H

text: −−−−L−t−→

text: H

text: I

text: I

text: −−−−L−t−→

text: H

text: H

text: Remark

text: At

text: this

text: moment,

text: we

text: only

text: know

text: that

text: the

text: operator

text: Lt

text: is

text: deﬁned

text: as

text: an

text: op-

text: erator

text: from

text: j

text: (supp

text: ψj)

text: to

text: j

text: (supp

text: ψj).

text: We

text: will

text: see

text: that

text: it

text: extends

text: naturally

text: to

text: a

text: bounded

text: operator

text: on

text: H

text: and

text: consequently

text: that

text: Lt

text: extends

text: to

text: a

text: bounded

text: operator

text: on

text: H

text: when

text: t

text: t

text: Proof

text: of

text: Theorem

text: We

text: henceforth

text: assume

text: that

text: ft

text: satisﬁes

text: the

text: non-integrability

text: condition

text: (NI)

text: for

text: A

text: ρ

text: some

text: ρ

text: and

text: suppose

text: t

text: t

text: Most

text: part

text: of

text: the

text: argument

text: below

text: is

text: devoted

text: to

text: show

text: that

text: ft

text: itself

text: is

text: exponentially

text: mixing.

text: In

text: the

text: last

text: subsection,

text: we

text: complete

text: the

text: proof

text: of

text: Theorem

text: by

text: examining

text: dependence

text: of

text: the

text: argument

text: on

text: the

text: ﬂow

text: ft.

text: For

text: this

text: last

text: part

text: of

text: the

text: argument,

text: we

text: emphasize

text: at

text: this

text: moment

text: that,

text: for

text: the

text: proof

text: of

text: exponential

text: mixing

text: for

text: ft,

text: we

text: actually

text: need

text: the

text: estimate

text: in

text: the

text: non-integrability

text: condition

text: (NI)

text: only

text: for

text: ρ

text: b

text: in

text: some

text: bounded

text: range.

text: (See

text: Remark

text: This

text: is

text: crucial

text: when

text: we

text: prove

text: stability

text: of

text: exponential

text: mixing.

text: In

text: the

text: following,

text: we

text: suppose

text: that

text: ω0

text: is

text: the

text: constant

text: in

text: Lemma

text: but

text: will

text: let

text: it

text: be

text: larger

text: if

text: necessary.

text: We

text: will

text: also

text: introduce

text: a

text: large

text: constant

text: m

text: depending

text: on

text: t

text: and

text: ω

text: Below

text: we

text: will

text: ignore

text: some

text: absolute

text: constants,

text: such

text: as

text: that

text: appear

text: in

text: Fourier

text: transform

text: and

text: partial

text: Bargmann

text: transform,

text: since

text: they

text: are

text: not

text: essential

text: at

text: all

text: in

text: our

text: argument.

text: Below

text: we

text: will

text: use

text: the

text: notations

text: prepared

text: in

text: the

text: last

text: section,

text: especially

text: We

text: write

text: (supp

text: ψj)

text: (supp

text: for

text: the

text: component

text: of

text: Lt

text: that

text: sends

text: the

text: j-component

text: to

text: the

text: j

text: -component.

text: It

text: is

text: written

text: as

text: B

text: B

text: u

text: where

text: is

text: the

text: transfer

text: operator

text: on

text: the

text: local

text: charts

text: deﬁned

text: by

text: v)

text: with

text: setting

text: ft

text: κj

text: and

text: ρ˜j.

text: As

text: we

text: noted

text: in

text: Subsection

text: the

text: diﬀeomorphism

text: is

text: written

text: y,

text: z)

text: y),

text: z

text: y)).


text: EXPONENTIAL MIXING FOR ANOSOV FLOWS 43 The next proposition is the key to the proof of Proposition 6.1, which gives an estimate {| | | (cid:48) |} on the components Ltj→j(cid:48) for which max m(j) , m(j ) is relatively small. The proof of this proposition is the main ingredient of this section and will be given in the next subsection. Proposition 6.8. There exist constants 0 < δ∗ < ρ∗ such that, if we let the constants t and (cid:48) ∈ (cid:93) ω be large depending on δ∗, then, for all j, j J satisfying (cid:93) {| | | (cid:48) |} ≤ | | ≥ | (cid:48) − | ≤ (118) max m(j) , m(j ) δ∗t , ω(j) ω /2 and ω(j ) ω(j) exp(ρ∗t /10), (cid:93) (cid:93) (cid:93) we have (cid:107) (cid:107) ≤ − ≤ ≤ Lˆtj→j(cid:48) L2 exp( ρ∗t(cid:93)) for t(cid:93) t 2t(cid:93). Now we deduce Proposition 6.1 from the estimates on the norms of the components Ltj→j(cid:48) given Lemma 6.3, Lemma 6.7 and Proposition 6.8. Proof of Proposition 6.1. Let 0 < δ∗ < ρ∗ be those constants in Proposition 6.8. Note that we may and do suppose that δ∗ is much smaller than ρ∗, because the claims of Proposition 6.8 remains valid when we let δ∗ be smaller (and Lemma 6.6 holds for any choice δ∗). ≤ ≤ Below we suppose t t 2t and proceed with the assumption (cid:93) (cid:93) | | ≥ | (cid:48) − | (119) ω ω(cid:93)/2 and ω ω < exp(α0δ∗t(cid:93)/10) (cid:48) ∈ ∈ for ω, ω Z, where α0 (0, 1/6) is that in the deﬁnition of the anisotropic Sobolev space H. The argument in the remaining case is much simpler and will be mentioned at the end. (cid:48) ∈ (cid:48) ∈ Let us take m, m Z and consider the components Ltj→j(cid:48) for j, j J satisfying (cid:48) (cid:48) (cid:48) (cid:48) (120) ω(j) = ω, m(j) = m, ω(j ) = ω , m(j ) = m . (cid:48) ∈ Note that, for each j (resp. j J), the cardinality of the set { (cid:48) ∈ | (cid:48) (cid:48) (cid:48) (cid:48)} { ∈ | } j J ρtj→j(cid:48) (cid:44) 0, ω(j ) = ω , m(j ) = m (resp. j J ρtj→j(cid:48) (cid:44) 0, ω(j) = ω, m(j) = m ) may be large (i.e. grow exponentially with respect to t). This causes a problem when we (cid:48) ∈ sum the estimates for the components Ltj→j(cid:48) for j, j J. Our idea to do with this problem is as follows: (A) if we consider the operator Lˆtj→j(cid:48) deﬁned in (111) instead of Ltj→j(cid:48), we will not have this problem by virtue of the assumption noted in Remark 6.2, and (cid:107) − (cid:107) (B) the norm Lˆtj→j(cid:48) Ltj→j(cid:48) L2 of the diﬀerence is very small and(cid:28)dominates the cardi- nalities of the sets above, which is bounded by C∗ exp(C∗t ) κ . (cid:93) (cid:48)(cid:93) ∈ Below we consider the following three cases for the combination (m, m ) Z2, but note (cid:48) ∈ that we continue to assume (119) for (ω, ω ) Z2: (cid:48) {| | | (cid:48)|} ≤ (i) those (m, m ) satisfying max m , m δ∗t , (cid:48) (cid:54)(cid:93)→ (cid:48) (cid:48) (ii) those (m, m ) not in (i), but satisﬁes (m, ω) (cid:44) t (m , ω ), or (cid:48) (iii) those (m, m ) not either in (i) and (ii).

text: EXPONENTIAL

text: MIXING

text: FOR

text: ANOSOV

text: FLOWS

text: The

text: next

text: proposition

text: is

text: the

text: key

text: to

text: the

text: proof

text: of

text: Proposition

text: which

text: gives

text: an

text: estimate

text: on

text: the

text: components

text: for

text: which

text: max

text: m(j)

text: m(j

text: is

text: relatively

text: small.

text: The

text: proof

text: of

text: this

text: proposition

text: is

text: the

text: main

text: ingredient

text: of

text: this

text: section

text: and

text: will

text: be

text: given

text: in

text: the

text: next

text: subsection.

text: Proposition

text: There

text: exist

text: constants

text: δ∗

text: ρ∗

text: such

text: that,

text: if

text: we

text: let

text: the

text: constants

text: t

text: and

text: ω

text: be

text: large

text: depending

text: on

text: δ∗,

text: then,

text: for

text: all

text: j,

text: j

text: J

text: satisfying

text: max

text: m(j)

text: m(j

text: δ∗t

text: ω(j)

text: ω

text: and

text: ω(j

text: ω(j)

text: exp(ρ∗t

text: we

text: have

text: exp(

text: for

text: t

text: Now

text: we

text: deduce

text: Proposition

text: from

text: the

text: estimates

text: on

text: the

text: norms

text: of

text: the

text: components

text: given

text: Lemma

text: Lemma

text: and

text: Proposition

text: Proof

text: of

text: Proposition

text: Let

text: δ∗

text: ρ∗

text: be

text: those

text: constants

text: in

text: Proposition

text: Note

text: that

text: we

text: may

text: and

text: do

text: suppose

text: that

text: δ∗

text: is

text: much

text: smaller

text: than

text: ρ∗,

text: because

text: the

text: claims

text: of

text: Proposition

text: remains

text: valid

text: when

text: we

text: let

text: δ∗

text: be

text: smaller

text: (and

text: Lemma

text: holds

text: for

text: any

text: choice

text: δ∗).

text: Below

text: we

text: suppose

text: t

text: t

text: and

text: proceed

text: with

text: the

text: assumption

text: ω

text: and

text: ω

text: ω

text: for

text: ω,

text: ω

text: Z,

text: where

text: α0

text: is

text: that

text: in

text: the

text: deﬁnition

text: of

text: the

text: anisotropic

text: Sobolev

text: space

text: H.

text: The

text: argument

text: in

text: the

text: remaining

text: case

text: is

text: much

text: simpler

text: and

text: will

text: be

text: mentioned

text: at

text: the

text: end.

text: Let

text: us

text: take

text: m,

text: m

text: Z

text: and

text: consider

text: the

text: components

text: for

text: j,

text: j

text: J

text: satisfying

text: ω(j)

text: ω,

text: m(j)

text: m,

text: ω(j

text: ω

text: m(j

text: m

text: Note

text: that,

text: for

text: each

text: j

text: (resp.

text: j

text: J),

text: the

text: cardinality

text: of

text: the

text: set

text: j

text: J

text: ω(j

text: ω

text: m(j

text: m

text: (resp.

text: j

text: J

text: ω(j)

text: ω,

text: m(j)

text: m

text: may

text: be

text: large

text: (i.e.

text: grow

text: exponentially

text: with

text: respect

text: to

text: t).

text: This

text: causes

text: a

text: problem

text: when

text: we

text: sum

text: the

text: estimates

text: for

text: the

text: components

text: for

text: j,

text: j

text: J.

text: Our

text: idea

text: to

text: do

text: with

text: this

text: problem

text: is

text: as

text: follows:

text: (A)

text: if

text: we

text: consider

text: the

text: operator

text: deﬁned

text: in

text: instead

text: of

text: we

text: will

text: not

text: have

text: this

text: problem

text: by

text: virtue

text: of

text: the

text: assumption

text: noted

text: in

text: Remark

text: and

text: (B)

text: the

text: norm

text: of

text: the

text: diﬀerence

text: is

text: very

text: small

text: the

text: cardi-

text: nalities

text: of

text: the

text: sets

text: above,

text: which

text: is

text: bounded

text: by

text: C∗

text: exp(C∗t

text: κ

text: Below

text: we

text: consider

text: the

text: following

text: three

text: cases

text: for

text: the

text: combination

text: (m,

text: m

text: but

text: note

text: that

text: we

text: continue

text: to

text: assume

text: for

text: (ω,

text: ω

text: (i)

text: those

text: (m,

text: m

text: satisfying

text: max

text: m

text: m

text: δ∗t

text: (ii)

text: those

text: (m,

text: m

text: not

text: in

text: (i),

text: but

text: satisﬁes

text: (m,

text: ω)

text: t

text: (m

text: ω

text: or

text: (iii)

text: those

text: (m,

text: m

text: not

text: either

text: in

text: (i)

text: and

text: (ii).


text: EXPONENTIAL MIXING FOR ANOSOV FLOWS 51 not follow the argument in the last part of the proof in the case (I) using the Schur test. In order to resolve this problem17, we actually need a little more information on the result of (regularized) integration by parts. As the result of (regularized) integration by parts, the (cid:90)(cid:48) (cid:48) (cid:48) integral I(w, ξ; w , ξ ; w˜, w˜ ; η) is written in the form − (cid:48) − (cid:48)(cid:48) · (cid:48)(cid:48) (cid:48) (cid:48) (cid:48) (cid:48)(cid:48) (145) exp(i(ξ ξ )Λ 1w ) I0(w ; w, ξx; w , ξx; w˜, w˜ ; η)dw · where, correspondingly to the two terms on right-hand side of (141), I ( ) is of the form 0 (cid:48)(cid:48) (cid:48) (cid:48) I0(w(cid:48)(cid:48); w, ξx; w(cid:48), ξ(cid:48)x; w˜, w˜(cid:48); η) =I1(wλ;−w1(,ξwx −; w˜ξ,(cid:48)xw)˜ ; η) + I2(w(cid:48)(cid:48); w, w(cid:48); w˜, w˜(cid:48); η). From the argument above, we already have | (cid:48)(cid:48) (cid:48) (cid:48) (cid:48) | ≤ − · (cid:48)(cid:48) I0(w ; w, ξx; w , ξx; w˜, w˜ ; η) C∗(ν)∆j 1/2 hν(w ), | (cid:48)(cid:48) (cid:48) (cid:48) | ≤ − · − · (cid:48)(cid:48) I1(w ; w, w ; w˜, w˜ ; η) C∗(ν)∆j 1/2 ε 1 hν(w ), and | (cid:48)(cid:48) (cid:48) (cid:48) | ≤ − · (cid:48)(cid:48) I2(w ; w, w ; w˜, w˜ ; η) C∗(ν)∆j 1/2 hν(w ). · (cid:48) Note that the dependence of I0( ) on the variables ξ and ξ is rather simple. Indeed, using (140) and the assumption (142), it is easy to see | (cid:48) (cid:48)(cid:48) (cid:48) (cid:48) (cid:48) | ≤ (cid:48) · | | − − (cid:48) (cid:48)(cid:48) (146) ∂kξx∂kξ(cid:48)xI0(w ; w, ξx; w , ξx; w˜, w˜ ; η) C∗(ν, k, k ) (λ1/2∆j ω 1/2) k k hν(w ) (cid:48) ≥ (cid:48) · for any k, k 0 with (k, k ) (cid:44) (0, 0). This together with the estimate on I0( ) above gives (cid:48) · { − − } · (cid:48)(cid:48) (147) |∂kξx∂kξ(cid:48)(cid:48)xI0(w(cid:48)(cid:48); w, ξx; w(cid:48), ξ(cid:48)x; w˜, w˜(cid:48); η)| ≤ C∗(ν, k, k )(emδ∗at(cid:93)x∆jλ|ω1|1/4/2,)∆k+jk1(cid:48)/2 hν(w ) (cid:48) ≥ for any k, k 0. Remark 6.13. Notices that the estimates above are obtained under the assumptions (142) and (143). For the proof of the next lemma we note that, in the case where the condition − (143) fails while (142) remains valid, we have the estimate (144) with ∆ 1/2 replaced by j − λ 1, that is, · − · | |− | (cid:48) (cid:48) (cid:48) | ≤ C∗(ν) λ 1 ω 1 (148) I(w, ξ; w , ξ ; w˜, w˜ ; η) (cid:104)| | | − − (cid:48) − − (cid:48) |(cid:105) . ω 1/2 Λ 1(w˜ w˜ ) (w w ) ν Indeed it is easy to see that simple estimates without using integration by parts yield this − − estimate (148) without the factor λ 1 (or the estimate (144) without the factor ∆ 1/2). But, j if the condition (143) fails while (142) remains valid, we have | − (cid:48)(cid:48) − − − (cid:48) (cid:48)(cid:48) − (cid:48) | ≥ | − − (cid:48) − − (cid:48) | ≥ | − (cid:48)| ≥ | |− (λ˜ 1(y˜ + y ) y) (λ˜ 1(y˜ + y ) y ) λ˜ 1(y˜ y˜ ) (y y ) y y /2 ω 1/2λ1/2 17The precise point of the problem is that the non-integrability condition (NI) does not tell how the quantity ρ on the right hand side of (2| 1|) depends on α. Below we make use of the fact that we can get (21) by (regularized) integration by parts when α is suﬃciently large and then we can see how the result depend on α. This argument is valid only in the case (II). ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: EXPONENTIAL

text: MIXING

text: FOR

text: ANOSOV

text: FLOWS

text: not

text: follow

text: the

text: argument

text: in

text: the

text: last

text: part

text: of

text: the

text: proof

text: in

text: the

text: case

text: (I)

text: using

text: the

text: Schur

text: test.

text: In

text: order

text: to

text: resolve

text: this

text: we

text: actually

text: need

text: a

text: little

text: more

text: information

text: on

text: the

text: result

text: of

text: (regularized)

text: integration

text: by

text: parts.

text: As

text: the

text: result

text: of

text: (regularized)

text: integration

text: by

text: parts,

text: the

text: integral

text: I(w,

text: ξ;

text: w

text: ξ

text: w˜,

text: w˜

text: η)

text: is

text: written

text: in

text: the

text: form

text: exp(i(ξ

text: ξ

text: )Λ

text: w,

text: ξx;

text: w

text: ξx;

text: w˜,

text: w˜

text: η)dw

text: where,

text: correspondingly

text: to

text: the

text: two

text: terms

text: on

text: right-hand

text: side

text: of

text: I

text: is

text: of

text: the

text: form

text: w,

text: ξx;

text: w˜,

text: η)

text: η)

text: w,

text: w˜,

text: η).

text: From

text: the

text: argument

text: above,

text: we

text: already

text: have

text: w,

text: ξx;

text: w

text: ξx;

text: w˜,

text: w˜

text: η)

text: C∗(ν)∆j

text: hν(w

text: w,

text: w

text: w˜,

text: w˜

text: η)

text: C∗(ν)∆j

text: ε

text: hν(w

text: and

text: w,

text: w

text: w˜,

text: w˜

text: η)

text: C∗(ν)∆j

text: hν(w

text: Note

text: that

text: the

text: dependence

text: of

text: on

text: the

text: variables

text: ξ

text: and

text: ξ

text: is

text: rather

text: simple.

text: Indeed,

text: using

text: and

text: the

text: assumption

text: it

text: is

text: easy

text: to

text: see

text: w,

text: ξx;

text: w

text: ξx;

text: w˜,

text: w˜

text: η)

text: C∗(ν,

text: k,

text: k

text: ω

text: k

text: k

text: hν(w

text: for

text: any

text: k,

text: k

text: with

text: (k,

text: k

text: This

text: together

text: with

text: the

text: estimate

text: on

text: above

text: gives

text: w,

text: ξx;

text: w˜,

text: η)|

text: C∗(ν,

text: k,

text: k

text: hν(w

text: for

text: any

text: k,

text: k

text: Remark

text: Notices

text: that

text: the

text: estimates

text: above

text: are

text: obtained

text: under

text: the

text: assumptions

text: and

text: For

text: the

text: proof

text: of

text: the

text: next

text: lemma

text: we

text: note

text: that,

text: in

text: the

text: case

text: where

text: the

text: condition

text: fails

text: while

text: remains

text: valid,

text: we

text: have

text: the

text: estimate

text: with

text: replaced

text: by

text: j

text: λ

text: that

text: is,

text: C∗(ν)

text: λ

text: ω

text: I(w,

text: ξ;

text: w

text: ξ

text: w˜,

text: w˜

text: η)

text: ω

text: Λ

text: w˜

text: (w

text: w

text: ν

text: Indeed

text: it

text: is

text: easy

text: to

text: see

text: that

text: simple

text: estimates

text: without

text: using

text: integration

text: by

text: parts

text: yield

text: this

text: estimate

text: without

text: the

text: factor

text: λ

text: (or

text: the

text: estimate

text: without

text: the

text: factor

text: But,

text: j

text: if

text: the

text: condition

text: fails

text: while

text: remains

text: valid,

text: we

text: have

text: (λ˜

text: y

text: y)

text: (λ˜

text: y

text: y

text: λ˜

text: y˜

text: (y

text: y

text: y

text: y

text: ω

text: precise

text: point

text: of

text: the

text: problem

text: is

text: that

text: the

text: non-integrability

text: condition

text: (NI)

text: does

text: not

text: tell

text: how

text: the

text: quantity

text: ρ

text: on

text: the

text: right

text: hand

text: side

text: of

text: depends

text: on

text: α.

text: Below

text: we

text: make

text: use

text: of

text: the

text: fact

text: that

text: we

text: can

text: get

text: by

text: (regularized)

text: integration

text: by

text: parts

text: when

text: α

text: is

text: suﬃciently

text: large

text: and

text: then

text: we

text: can

text: see

text: how

text: the

text: result

text: depend

text: on

text: α.

text: This

text: argument

text: is

text: valid

text: only

text: in

text: the

text: case

text: (II).

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 52 M. TSUJII and therefore we can get the estimate (148), making use of arbitrariness of ν. Let us note also that, in such argument without using integration by parts, we may express (130) in the · (cid:48) form (145) with the function I0( ) independent of the variables ξ and ξ (cid:101). The next lemma is our remedy to the problem in the case (II) mentioned above. Recall ∞ → the deﬁnition of the function X(cid:104)j→j(cid:48) a(cid:105)n−d consider another C function Xj→j(cid:48) : R2 [0, 1] that takes value 1 on (cid:104)the 4κ(cid:105)−1(cid:93)/2 ω(j)(cid:101)1/2-neighborhood of the subset π˜(supp ρj→j(cid:48)) and is supported in the 8κ1/2 ω(j) (cid:18)1/2-neighborhood of that subset. Cl(cid:19)early the estimate parallel (cid:101) (cid:93) → (cid:101) to (110) holds for this function. Let Ψj : R2+2+1 [0, 1] be the function deﬁned by − − | |− (cid:107) − − · (cid:107) · Ψ0j (w, ξ, η) = χ 8 1e δ∗t(cid:93) ω 1/2 ∆j 1(ξ η ej(w)) Xj→j(cid:48)(w). (cid:101) − Notice that this function diﬀers from Ψj in (123) only in the factor 8 1 and the multiplica- tion by the function Xj→j(cid:48)(w). (cid:101) (cid:101) Lemma 6.14. We have (cid:107) ∗ ◦ ◦ ∗ ◦ · (cid:48) ◦ ◦ ◦ → (cid:107) ≤ { − − } B Ψ0j A M(Ψj(cid:48) Xj→j(cid:48))2 A Ψ0j B : L2(R2+1) L2(R2+1) C∗ max λ 1/8, ∆j 1/4 . Before proving th(cid:101)is lemma, let us see that Lemma 6.9 follows fro·m it. L(cid:101)etting 1j→j(cid:48) be the multiplication by the characteristic function of the support of Ψ0j Xj→j(cid:48) and writing, − ◦ ◦ ∗ ◦ ∗ ◦ − ◦ ∗ ◦ D = (1 Ψ0j B B ) 1j→j(cid:48) and D = 1j→j(cid:48) (1 B B Ψ0j ), we see (cid:20) (cid:21) 1j→j(cid:48) ◦ A∗ ◦ M(Ψj(cid:48) · X(cid:48)j→j(cid:48))2(cid:101)◦ A ◦ 1j→j(cid:48) (cid:101) − ◦ ◦ ∗ ◦ ◦ ∗ ◦ · (cid:48) ◦ ◦ ◦ ◦ ∗ ◦ 1j→j(cid:48) B B Ψ0j A M(Ψj(cid:48) Xj→j(cid:48))2 A Ψ0j B B 1j→j(cid:48) ∗ ◦ ∗ ◦ · (cid:48) ◦(cid:101) ◦ = D A M(Ψj(cid:48) Xj→j(cid:48))2 A 1j→j(cid:48) ◦ ◦ ∗ ◦ ◦ ∗ ◦ · (cid:48) ◦ ◦ + 1j→j(cid:48) B B Ψ0j A M(Ψj(cid:48) Xj→j(cid:48))2 A D. ◦ ∗ From the estimate on the kernel of P = B B in Lemma 5.2, it is clear that the operator ∗ norms of D and D with respect to the L2 norm are bounded by − (cid:28) − C∗ exp( eδ∗t(cid:93)) λ 1/8. ∗ Since the operators A, B and B do not increase the L2 norm, the same estimate holds for the operator norms of the two operators on the right-hand side above. Therefore the required estimate (124) follows from Lemma 6.14. (cid:90) Proof of Lemma 6.14. The proof is easy once we have the estimate (147) and note Remark (cid:101) (cid:101) (cid:101) 6.13. Let us write the operator under consideration as an integral operator ∗ ◦ ◦ ∗ ◦ · ◦ ◦ ◦ (cid:48) (cid:48) (cid:48) (cid:48) (B Ψ0j A M(Ψj(cid:48) Xj→j(cid:48))2 A Ψ0j B) u(w†, z ) = K(w†, z ; w†, z)u(w†, z) dw†dz. (cid:90) ∗ ◦ · ◦ From the d(cid:101)escription (128) of A (Ψj(cid:48) Xj→j(cid:48))2 A, we can write the kernel as (149) K(w(cid:48)†, z(cid:48); w†, z) = d(cid:101)wdξdw(cid:48)dξ(cid:48)d(cid:101)η · e−iξw/2+iξ(cid:48)w(cid:48)/2 · K(w(cid:48), ξ(cid:48); w, ξ; η) · · (cid:48) (cid:48) · · (cid:48) (cid:48) Ψ0j (w, ξ, η) Ψ0j (w , ξ , η) φw,ξ,η(w†, z) φw(cid:48),ξ(cid:48),η(w†, z ) ##LTLine##

text: M.

text: TSUJII

text: and

text: therefore

text: we

text: can

text: get

text: the

text: estimate

text: making

text: use

text: of

text: arbitrariness

text: of

text: ν.

text: Let

text: us

text: note

text: also

text: that,

text: in

text: such

text: argument

text: without

text: using

text: integration

text: by

text: parts,

text: we

text: may

text: express

text: in

text: the

text: form

text: with

text: the

text: function

text: independent

text: of

text: the

text: variables

text: ξ

text: and

text: ξ

text: The

text: next

text: lemma

text: is

text: our

text: remedy

text: to

text: the

text: problem

text: in

text: the

text: case

text: (II)

text: mentioned

text: above.

text: Recall

text: the

text: deﬁnition

text: of

text: the

text: function

text: consider

text: another

text: C

text: function

text: that

text: takes

text: value

text: on

text: of

text: the

text: subset

text: π˜(supp

text: and

text: is

text: supported

text: in

text: the

text: ω(j)

text: of

text: that

text: subset.

text: the

text: estimate

text: parallel

text: to

text: holds

text: for

text: this

text: function.

text: Let

text: Ψj

text: be

text: the

text: function

text: deﬁned

text: by

text: (w,

text: ξ,

text: η)

text: χ

text: ω

text: ∆j

text: η

text: ej(w))

text: Notice

text: that

text: this

text: function

text: diﬀers

text: from

text: Ψj

text: in

text: only

text: in

text: the

text: factor

text: and

text: the

text: multiplica-

text: tion

text: by

text: the

text: function

text: Lemma

text: We

text: have

text: B

text: A

text: A

text: B

text: C∗

text: max

text: λ

text: ∆j

text: Before

text: proving

text: lemma,

text: let

text: us

text: see

text: that

text: Lemma

text: follows

text: fro·m

text: it.

text: be

text: the

text: multiplication

text: by

text: the

text: characteristic

text: function

text: of

text: the

text: support

text: of

text: and

text: writing,

text: D

text: B

text: B

text: and

text: D

text: B

text: B

text: we

text: see

text: A∗

text: A

text: B

text: B

text: A

text: A

text: B

text: B

text: D

text: A

text: A

text: B

text: B

text: A

text: A

text: D.

text: From

text: the

text: estimate

text: on

text: the

text: kernel

text: of

text: P

text: B

text: B

text: in

text: Lemma

text: it

text: is

text: clear

text: that

text: the

text: operator

text: norms

text: of

text: D

text: and

text: D

text: with

text: respect

text: to

text: the

text: norm

text: are

text: bounded

text: by

text: C∗

text: exp(

text: λ

text: Since

text: the

text: operators

text: A,

text: B

text: and

text: B

text: do

text: not

text: increase

text: the

text: norm,

text: the

text: same

text: estimate

text: holds

text: for

text: the

text: operator

text: norms

text: of

text: the

text: two

text: operators

text: on

text: the

text: right-hand

text: side

text: above.

text: Therefore

text: the

text: required

text: estimate

text: follows

text: from

text: Lemma

text: Proof

text: of

text: Lemma

text: The

text: proof

text: is

text: easy

text: once

text: we

text: have

text: the

text: estimate

text: and

text: note

text: Remark

text: Let

text: us

text: write

text: the

text: operator

text: under

text: consideration

text: as

text: an

text: integral

text: operator

text: (B

text: A

text: A

text: B)

text: u(w†,

text: z

text: K(w†,

text: z

text: w†,

text: z)u(w†,

text: z)

text: dw†dz.

text: From

text: the

text: of

text: A

text: A,

text: we

text: can

text: write

text: the

text: kernel

text: as

text: w†,

text: z)

text: w,

text: ξ;

text: η)

text: (w,

text: ξ,

text: η)

text: (w

text: ξ

text: η)

text: φw,ξ,η(w†,

text: z)

text: z

text: ##LTLine##


text: EXPONENTIAL MIXING FOR ANOSOV FLOWS 55 (cid:90) (cid:101) ∗ ◦ − (cid:48) (cid:48) (cid:48) (cid:48) (153) B (G P)u(w , z ) = K(w , z ; w, ξ, η) u(w, ξ, η) dwdξdη. (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) From Lemma 4.9 and Le(cid:12)mma 4.11 (see also Remark 4.10), the kernel satisﬁes (cid:12) (cid:12) (cid:12) | (cid:48) (cid:48) | ≤ · (cid:16) − (cid:48)(cid:10) (cid:48) − · (cid:11) (cid:17) (cid:48) (cid:48) K(w , z ; w, ξ, η) (ρtj→j(cid:48) φw,ξ,η)((Gtj→j(cid:48)) 1(w , z )) (ρtj→j(cid:48) φw,ξ,η)(w , z ) ≤ · | |− · | | · | | − (cid:48) − · (cid:104) (cid:48)(cid:105)− C∗(ν) ω 1/2+2θ∗ ω 1/2 ω 1/2(w w ) ν z ν (cid:90) (cid:48) (cid:48) ∈ ∈ and vanishes unless (w , z ) supp ρtj→j(cid:48) and (w, ξ, η) supp ψj. Hence we have | (cid:48) (cid:48) | (cid:48) (cid:48) | |− sup K(w , z ; w, ξ, η) dw dz < C∗ ω 1+2θ∗ (cid:90) w,ξ,η and | (cid:48) (cid:48) | | | | | sup K(w , z ; w, ξ, η) dwdξdη < C∗e2m ∆(cid:101)j ω 2θ∗. (cid:48) (cid:48) w ,z suppψj − By Schur test, the operator norm of (152) (and hence that of G P ) is bounded by | | | |− ≤ · − (cid:28) − (154) C∗em ∆1j/2 ω 1/2+2θ∗ C∗eδ∗t(cid:93) ω(cid:93) 1/2+3θ∗ κ(cid:93) 1/2, ≤ (cid:104) (cid:105) where we used the estimate ∆j C(cid:101)∗ log ω , which follows from Lemma 2.19, in the left inequality. (For the r(cid:90)ight inequality, recall that we choose large ω depending on t .) − (cid:48) (cid:48) (cid:48) (cid:93) (cid:48) (cid:93) Next we consider the diﬀerence P P. Its kernel K (w, ξ, η; w , ξ , η ) is written (cid:16) (cid:17) − (cid:48) (cid:48) (cid:104) (cid:105) (cid:48)(cid:48) · − (cid:48) (cid:48)(cid:48)−(cid:104) (cid:105)| (cid:48)− (cid:48)(cid:48)| ei(ξw ξ w )/2 η 1/2 dw ei(ξ ξ )w η w w 2/2 · (cid:104) (cid:48)(cid:105) −(cid:104) (cid:48)(cid:105)(cid:107) − (cid:48)(cid:48)(cid:107) (cid:48) (cid:48) − − (cid:104) (cid:105) −(cid:104) (cid:105)(cid:107) − (cid:48)(cid:48)(cid:107) (cid:48)(cid:48) (cid:48) − η 1/2e η w w /2ρˆtj→j(cid:48)(w , η η) η 1/2e η w w /2ρˆtj→j(cid:48)(w , η η) . By integration by parts using the estimate in Lemma 4.11, we obtain that | (cid:48) (cid:48) (cid:48) (cid:48) | ≤ · · − · (cid:104)| | | − (cid:48)|(cid:105)− · (cid:104)| |− | − (cid:48)|(cid:105)− · (cid:104) (cid:48) − (cid:105)− K (w, ξ, η; w , ξ , η ) C∗(ν) (eC∗t(cid:93) κ 1) ω 1/2 w w ν(cid:101) ω 1/2 ξ ξ ν η η ν (cid:93) − → for arbitrarily large ν >·0.−The(cid:28)refo−re the opera(cid:101)tor norm o(cid:101)f P P : L2(supp ψj) L2(R2+2+1) is bounded by C∗(eC∗t κ 1) κ 1/2. (cid:93) (cid:93) (cid:93) − − From the estimates on the diﬀerences G P and P P above, we obtain the conclusion of the lemma, provided that we take suﬃciently large t and then take suﬃciently large ω (cid:93) (cid:93) according to the choice of t . (cid:3) (cid:93) The proofs of Lemma 6.3 and Lemma 6.7 presented below are based on straightforward estimates on the kernels of the operators using integration by parts. (But recall Remark 6.15.) Below w−e will prove the claims for Ltj→j(cid:48) in the conclusions, and those for the diﬀerence Ltj→j(cid:48) Lˆtj→j(cid:48) will follow immediately from that proof if we note that the kernel of the operator Ltj→j(cid:48) is localized in the space.

text: EXPONENTIAL

text: MIXING

text: FOR

text: ANOSOV

text: FLOWS

text: B

font: LDNQDK+txmia

text: (G

text: P)u(w

text: z

text: K(w

text: z

text: w,

text: ξ,

text: η)

text: u(w,

text: ξ,

text: η)

text: dwdξdη.

text: From

text: Lemma

text: and

text: (see

text: also

text: Remark

text: the

text: kernel

text: satisﬁes

text: K(w

text: z

text: w,

text: ξ,

text: η)

text: z

text: φw,ξ,η)(w

text: z

text: C∗(ν)

text: ω

text: ω

text: ω

text: w

text: ν

text: z

text: ν

text: and

text: vanishes

text: unless

text: (w

text: z

text: supp

text: and

text: (w,

text: ξ,

text: η)

text: supp

text: ψj.

text: Hence

text: we

text: have

text: sup

text: K(w

text: z

text: w,

text: ξ,

text: η)

text: dw

text: dz

text: C∗

text: ω

text: w,ξ,η

text: and

text: sup

text: K(w

text: z

text: w,

text: ξ,

text: η)

text: dwdξdη

text: ω

text: 2θ∗.

text: w

text: ,z

text: suppψj

text: By

text: Schur

text: test,

text: the

text: operator

text: norm

text: of

text: (and

text: hence

text: that

text: of

text: G

text: P

text: is

text: bounded

text: by

text: C∗em

text: ω

text: where

text: we

text: used

text: the

text: estimate

text: ∆j

text: log

text: ω

text: which

text: follows

text: from

text: Lemma

text: in

text: the

text: left

text: inequality.

text: (For

text: the

text: inequality,

text: recall

text: that

text: we

text: choose

text: large

text: ω

text: depending

text: on

text: t

text: Next

text: we

text: consider

text: the

text: diﬀerence

text: P

text: P.

text: Its

text: kernel

text: K

text: (w,

text: ξ,

text: η;

text: w

text: ξ

text: η

text: is

text: written

text: ei(ξw

text: ξ

text: w

text: η

text: dw

text: ei(ξ

text: ξ

text: )w

text: η

text: w

text: w

text: η

text: η

text: w

text: w

text: η

text: η)

text: η

text: η

text: w

text: w

text: η

text: η)

text: By

text: integration

text: by

text: parts

text: using

text: the

text: estimate

text: in

text: Lemma

text: we

text: obtain

text: that

text: K

text: (w,

text: ξ,

text: η;

text: w

text: ξ

text: η

text: C∗(ν)

text: κ

text: ω

text: w

text: w

text: ω

text: ξ

text: ξ

text: ν

text: η

text: η

text: ν

text: for

text: arbitrarily

text: large

text: ν

text: the

text: norm

text: P

text: P

text: ψj)

text: is

text: bounded

text: by

text: C∗(eC∗t

text: κ

text: κ

text: From

text: the

text: estimates

text: on

text: the

text: diﬀerences

text: G

text: P

text: and

text: P

text: P

text: above,

text: we

text: obtain

text: the

text: conclusion

text: of

text: the

text: lemma,

text: provided

text: that

text: we

text: take

text: suﬃciently

text: large

text: t

text: and

text: then

text: take

text: suﬃciently

text: large

text: ω

text: according

text: to

text: the

text: choice

text: of

text: t

text: The

text: proofs

text: of

text: Lemma

text: and

text: Lemma

text: presented

text: below

text: are

text: based

text: on

text: straightforward

text: estimates

text: on

text: the

text: kernels

text: of

text: the

text: operators

text: using

text: integration

text: by

text: parts.

text: (But

text: recall

text: Remark

text: Below

text: w−e

text: will

text: prove

text: the

text: claims

text: for

text: in

text: the

text: conclusions,

text: and

text: those

text: for

text: the

text: diﬀerence

text: will

text: follow

text: immediately

text: from

text: that

text: proof

text: if

text: we

text: note

text: that

text: the

text: kernel

text: of

text: the

text: operator

text: is

text: localized

text: in

text: the

text: space.