Rigidity of center Lyapunov exponents for Anosov diffeomorphisms on 3-torus Daohua Yu Ruihao Gu

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Rigidity of center Lyapunov exponents for Anosov

diffeomorphisms on 3-torus

Daohua Yu, Ruihao Gu

June 16, 2023

Abstract

Let fand gbe two Anosov diffeomorphisms on T3with three-subbundles partially

hyperbolic splittings where the weak stable subbundles are considered as center subbun-

dles. Assume that fis conjugate to gand the conjugacy preserves the strong stable foli-

ation, then their center Lyapunov exponents of corresponding periodic points coincide.

This is the converse of the main result of Gogolev and Guysinsky in [9]. Moreover, we get

the same result for partially hyperbolic diffeomorphisms derived from Anosov on T3.

1 Introduction

Let Mbe a smooth closed Riemannian manifold. We say a diffeomrophism f:M→Mis Anosov,

if there exists a D f -invariant continuous splitting T M =Es⊕Eusuch that D f is uniformly con-

tracting on the stable bundle Esand uniformly expanding on the unstable bundle Eu. It is known

that an Anosov diffeomorphism with one-dimensional stable or unstable bundle exists only on d-

torus Td(d≥2) [5, 18], in particular T3is the only 3-manifolds admitting Anosov diffeomorphisms.

Moreover an Anosov diffeomorphism fon Tdis always topologically conjugate to its linearization

f∗:π1(Td)→π1(Td) which induces a toral Anosov automorphism [4, 17], i.e., there exists a homeo-

morphism h:Td→Tdhomotopic to IdTdsuch that h◦f=f∗◦h. However the conjugacy is usually

only H¨

older continuous. Indeed, if two Anosov diffeomorphisms are smoothly conjugate, then the

derivatives of their return maps on corresponding periodic points are conjugate via the derivative of

the conjugacy and more weakly their corresponding periodic Lyapunov exponents coincide. There

are plenty of enlightening works researching the regularity of the conjugacy under the assumption of

the same Lyapunov exponents, usually called rigidity, e.g. [8,9, 16, 24] in which the rigidity for Anosov

diffeomorphisms with partially hyperbolic splittings has been studied extensively.

Deﬁnition 1.1. A diffeomorphism f:M→Mis called partially hyperbolic, if there is an invariant

splitting for D f ,T M =Es

f⊕Ec

f⊕Eu

f, which satisﬁes

||D f vs|| < τ(x)< ||D f vc|| < µ(x)< ||D f vu||,

for all vσ∈Eσ

f(x) with ||vσ|| = 1, where σ=s,c,uand 0 <τ<1<µare two continuous functions on M.

We call Es

f,Ec

fand Eu

f, the stable bundle, the center bundle and the unstable bundle, repectively.

It is well known that the stable/unstable bundle of a partially hyperbolic diffeomorphism is always

uniquely integrable and the integral manifolds form a foliation on M[20], called stable/unstable fo-

liation and denoted by Fs/u

f. However the center bundle in general is not integrable, see [15] for a

counterexample on T3. A partially hyperbolic diffeomorphism fis called dynamically coherent, if

arXiv:2210.12664v2 [math.DS] 15 Jun 2023

there are two f−invariant foliations denoted by Fcu

fand Fcs

ftangent respectively to Ecu

f:=Ec

f⊕Eu

and Ecs

f:=Es

f⊕Ec

f. It is clear that if fis dynamically coherent, then Fc

f:=Fcu

fTFcs

fis a f−invariant

foliation tangent to Ec

f. In particular, for the case of ||D f vc|| < 1 for all unit vc∈Ec

f, we also call Ec

fthe

weak stable bundle,Es

fthe strong stable bundle and the corresponding foliation Fs

fis called strong

stable foliation.

In [9], Gogolev and Guysinsky considered the local rigidity of an Anosov automorphism Lon T3

with partially hyperbolic splitting. For any two partially hyperbolic Anosov diffeomorphisms fand

gon T3which are conjugate via h, we say fand ghave the same center periodic data, if their cor-

responding, by h, periodic points have the same Lyapunov exponents on center bundles. The main

technical lemma in [9] is that for Lwith two-dimensional stable bundle, if its C1-perturbations fand

ghave the same center periodic data, then their conjugacy h preserves the strong stable foliation, i.e.

h¡Fs

f(x)¢=Fs

g¡h(x)¢for all x∈T3. In this paper, we show the converse of this property.

Theorem 1.2. Let f ,g:T3→T3be two C1+α-smooth partially hyperbolic Anosov diffeomorphisms

whose center bundles are weak stable. Assume that f is conjugate to g via a homeomorphism h. If h

preserves the strong stable foliations, then f and g have the same center periodic data.

An interesting corollary of Theorem 1.2 is that the topological conjugacy preserving the strong

stable foliation implies it is smooth along the center (weak stable) foliation. Here the conjugacy being

C1-smooth along the center foliation is deﬁned as the derivative along each center leaf being contin-

uous with respect to the topology of the whole manifold. Note that in our case, the center bundle is

integrable and the conjugacy preserves the center foliation [21], also see Remark 3.2.

Corollary 1.3. Let f ,g:T3→T3be two C 1+α-smooth partially hyperbolic Anosov diffeomorphisms

whose center bundles are weak stable. Assume that f is conjugate to g via a homeomorphism h. Then

h preserves the strong stable foliation, if and only if, h is C 1along the center foliation.

Remark 1.4.One can get Corollary 1.3 from Theorem 1.2 and [9, Lemma 5 and Lemma 6] which state

that the same center periodic data of fand gimplies that his smooth along the center foliation and

hpreserves the strong stable foliation. We mention that the assumption of local perturbation in [9]

is just for getting that the center bundle of fis integrable and preserved by h. As mentioned above,

we now have these two properties. For the inverse conclusion, for any periodic point pwith period n,

since his smooth along the center foliation, we can take the directional derivative of fn=h−1◦gn◦h

in the direction of Ec, we can get that λc(p,f)=λc(h(p), g). From the conclusion of [9], we can know

that hpreserves the strong stable foliation.

We are also curious about the higher-dimensional case of Theorem 1.2 and Corollary 1.3, since

Gogolev extended the result of local rigidity in [9] to the higher-dimensional case [8]. For convenience,

we state this question in Section 3, see Question 3.5.

In this paper, we also consider partially hyperbolic diffeomorphisms on general closed Rieman-

nian manifolds with integrable one-dimensional center bundles and with accessibility. We call a par-

tially hyperbolic diffeomorphism f:M→Mis accessible, if any two points on Mcan be connected

by curves each of which is tangent to Es

for Eu

f. It is well known that accessibility is Cr-dense and

C1-open among Cr(r≥1) partially hyperbolic diffeomorphisms with one-dimensional center bun-

dles [1–3, 14].

Let f,g:M→Mbe two partially hyperbolic diffeomorphisms and dynamically coherent. We call

fis fully conjugate to g, if there exists a homeomorphsim h:M→Msuch that fis conjugate to gvia

hand hpreserves the stable foliations, unstable foliations and center foliations, respectively, i.e.,

h¡Fσ

f(x)¢=Fσ

g¡h(x)¢,∀x∈Mand σ=s,c,u.

It is clear that if fis accessible and fully conjugate to g, then gis also accessible. Now, we can extend

Theorem 1.2 to the following one in which the dimension of the manifold Mcan be bigger than 3. We

mention that hdoes not need to be homotopic to identity in Theorem 1.5.

Theorem 1.5. Let f ,g:M→M be two C 1+α-smooth partially hyperbolic diffeomorphisms with one-

dimensional center bundles and dynamically coherent. If f is accessible and fully conjugate to g via a

homeomorphism h, then h is C 1along the center foliation.

As a corollary, we apply Theorem 1.5 to partially hyperbolic diffeomorphisms derived from Anosov

on T3. We call f:T3→T3is derived from Anosov, also called a DA diffeomorphism, i.e., its lineariza-

tion f∗:π1(T3)→π1(T3) induces an Anosov automorphism. We mention that the partially hyperbolic

diffeomorphism derived from Anosov is one of the three types of partially hyperbolic diffeomorphisms

on 3-manifolds with solvable fundamental group [11].

Corollary 1.6. Let f ,g:T3→T3be two C1+α-smooth partially hyperbolic diffeomorphisms and con-

jugate via a homeomorphism h. Suppose the linearization of f is an Anosov automorphism with one-

dimensional unstable bundle. If h preserves the stable foliation, then h is C1along the center foliation.

It is clear that Theorem 1.2 is a special case of Corollary 1.6, as we can take the directional deriva-

tive of fn=h−1◦gn◦hin the direction of Ecat periodic point pwith period nto get the same center

periodic data. Another special case of Corollary 1.6 is that g=f∗which has been studied in [7]. We

refer readers to [10] for higher-dimensional case under the assumption of g=f∗.

Acknowledgements

We are grateful for the valuable communication and suggestions from Shaobo Gan and Yi Shi.

Thanks for the comments from the annoymous referees about Remark 2.7 and Remark 3.4. The au-

thors were partially supported by National Key R&D Program of China (2021YFA1001900).

2 Rigidity of center Lyapunov exponents in the accessible case

In this section, we prove Theorem 1.5. First of all, we recall some basic notions and useful proper-

ties of partially hyperbolic diffeomorphisms. Let Mbe a smooth cloesd Riemannian manifold and

f:M→Mbe a diffeomorphism admitting partially hyperbolic splitting T M =Es

f⊕Ec

f⊕Eu

fwith

dimEc

f=1. The Lyapunov exponent on Ec

fat point x, if it exists, is denoted by λc(x,f) and deﬁned

λc(x,f)=lim

n→+∞

nlog∥D f n|Ec

f(x)∥.

Assume further that fis dynamically coherent. Deﬁne the local leaf with size δ>0 by

Fσ

f(x,δ) :=©y∈Fσ

f(x)|dFσ

f(x,y)<δª,

where σ=s,c,u,cs,cu and dFσ

f(·,·) is the metric on Fσ

finduced by the Riemannian metric on the base

space. By coherence, the local stable/unstable foliation Fs

f±Fu

finduces holonomy maps restricted on

Fcs

f±Fcu

fas follow,

Hols/u

f,x,y:Fc

f(x,δ1)−→ Fc

f(y,δ2),

z7−→ Fc

f(y,δ2)∩Fs/u

f(z,R),

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RigidityofcenterLyapunovexponentsforAnosovdiffeomorphismson3-torusDaohuaYu,RuihaoGuJune16,2023AbstractLetfandgbetwoAnosovdiffeomorphismsonT3withthree-subbundlespartiallyhyperbolicsplittingswheretheweakstablesubbundlesareconsideredascentersubbun-dles.Assumethatfisconjugatetogandtheconjugacypreservest...

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Rigidity of center Lyapunov exponents for Anosov diffeomorphisms on 3-torus Daohua Yu Ruihao Gu

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