Questions tagged [hyperbolic-dynamics]

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Periodic orbits of generalized cat map near the origin

Let $M\in SL(2,\mathbb{Z})$ have eigenvalues $\lambda, \lambda^{-1}$ with $\lambda>1$., and suppose $M$ is diagonalized as $Q\Lambda Q^{-1}$ with $\det Q=1$. (Note that his doesn't determine $Q$, ...
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1 vote
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50 views

Random matrix heuristics for Koopman operators

Consider a nice hyperbolic dynamical system $(X, T)$, for instance a $\mathcal{C}^\infty$ Anosov map. The action of the Koopman operator $$\mathcal{K} : \ f \mapsto f \circ T$$ has a nice spectrum ...
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1 vote
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53 views

Does the presence of a single hyperbolic fixed point partition the space into invariant quadrants?

Consider a $C^1$ flow on $\mathbb R^2$ with a single fixed point $x_0$, that is hyperbolic with one expanding and one contracting direction, and with the stable and unstable manifolds intersecting ...
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  • 1,519
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77 views

Homoclinically related hyperbolic periodic points gives the same pesin homoclinic class up to null sets

In MINIMALITY AND STABLE BERNOULLINESS IN DIMENSION 3 by Nunez and Hertz, the first paragraph in the proof of Corollary 2.4 says the above statement follows by using a "$\lambda$-lemma". ...
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1 vote
0 answers
72 views

Persistence of homoclinic points in the non-compact case

It is well known that a transverse homoclinic point of a hyperbolic periodic point of a $C^1$-diffeomorphism of a compact manifold $M$ persists under small $C^1$ perturbations. This follows easily ...
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124 views

Example of topologically transitive dynamical system with invariant non-ergodic Borel measure

Let $U \subset M$ be an open subset of a Riemannian manifold. I’m trying to find or construct an example of a topologically transitive dynamical system $f : U \to U$ for which $f : \Lambda \to \...
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41 views

Whether or not two distinct points in Teichmuller space induce absolutely continuous volume forms on the unit tangent bundle of a surface?

Let $S$ be a closed orientable surface of genus greater than two. Let $g$ and $g'$ be metrics two of constant curvature. I guess we an think of these as two points in the Teichmüller space $\mathcal{T}...
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2 votes
0 answers
37 views

Some questions about the contruction of center stable manifolds for cubic NLKG by Lyapunov-Perron method

In Nakanish & Schlag: Invariant manifolds and Dispersive Hamiltonian Evolution Equations,, on theorem 3.22, they use Lyapunov-Perron methods to conctruct center stable manifolds for focusing cubic ...
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  • 399
2 votes
0 answers
51 views

Center-stable manifold theorem on manifold with boundary

I would like to see if there is a Center-stable manifold theorem on the phase space that is a manifold with boundary. Suppose $f:M\rightarrow M$ is a diffeomorphism, according to Theorem III.7 in "...
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1 vote
1 answer
131 views

Example of zero Lyapunov exponentes

Assume that $(T, A)$ is a linear cocycle such that $T:X\rightarrow X$ is a homemorphism on compact metric space $X$ and $A:X\rightarrow SL(2, \mathbb{R})$ is a continuous function. We say that an ...
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Unique poine in holonomies

Let $\Lambda$ be Axiom A for $C^{1+\gamma}$ $f$. I am reading this paper. I have a problem to undestand holonomies. The holonomy mapping $$ h: W_{loc}^{s} (x) \cap\Lambda \rightarrow W_{loc}^{s} (y) \...
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2 votes
0 answers
112 views

On invariant cones of the Katok map

I am studying the Katok map and similarly constructed examples of nonuniformly hyperbolic surface diffeomorphisms. An important part of the analysis of these diffeomorphisms is the invariance of a ...
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  • 141
4 votes
1 answer
157 views

When entropy SRB measure is zero

It is well known that many strongly chaotic dynamical systems have the property that periodic measures are (weak-star) dense in the space of all invariant probability measures. Let $f:M \rightarrow ...
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5 votes
1 answer
253 views

How can I prove that the suspension of an Anosov diffeomorphism is an Anosov flow?

Suppose we have an Anosov diffeo $f$ on $M$. Define $g_t : M\times\mathbb R\to M\times\mathbb R$ by $(x,s)\mapsto (x,s+t)$. Take a quotient of $M\times\mathbb R$ under the relation $(x,s)\sim (f(x),s-...
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1 vote
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Stabilization of non-autonomuous 1-d wavs equation

I want to ask two questions about the stabilization of the equation $$\eqalign{ & {y_{tt}} = k(t,x){y_{xx}}+a(t,x){y_t}+ b(t,x){y_x}+ c(t,x){y_x} +d(t,x)y \ \ (t,x) \in {\text{ }}(0,\infty ) ...
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Transverse measures in pseudo-Anosov diffeomorphisms

I've recently begun doing research involving pseudo-Anosov diffeomorphisms, which are diffeomorphisms on surfaces $f : M \to M$ admitting two singular measured foliations $(\mathcal F^s, \nu^s)$ and $(...
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  • 141
4 votes
0 answers
80 views

Terminology for a foliation that is only tangentially smooth

I'd like to get some information and references, starting with a name, for the following quite common situation, for a smooth (i.e. $C^\infty$) $n$-manifold $M$. A partition $\mathcal{L}$ of $M$ is ...
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4 votes
1 answer
212 views

Lipschitz property of holonomies fails when stable leaves $W^s(x)$ inside the leaves $W^{ss}(x)$

Let $M$ be compact manifold. suppose $f:M\rightarrow M$ is $C^{2}$. There is a continuous splitting of the tangent bundle $TM=E^{ss}+E^{s}+E^{u}$ invariant under the derivative $Df$ of the ...
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1 answer
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$C^1$ partially hyperbolic diffeomorphism have Hölder stable holonomies (reference request)

I have spent an insane amount of time searching for a preprint I have printed a few months ago but misplaced. I cannot find it anymore and this drives me crazy. It might not have been meant for ...
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1 vote
0 answers
50 views

Limit contration rates and expansion rate solenoid map

Let M:=$S^{1}\times \mathcal{D}^1$ where $\mathcal{D}=\{v\in \mathcal{R}^2 | |v|<1\}$ carries the product distance and suppose $f:M\rightarrow M$,$(x,y,z)\rightarrow (\gamma x, \lambda y+v(x), \mu ...
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  • 783
3 votes
1 answer
295 views

Question on a proof of density of periodic orbits

In page 215 and 216 of the book "Introduction to the Modern Theory of Dynamical Systems" by Anatole Katok, Boris Hasselblatt, there is a theorem stated as following: Theorem: Let $\Gamma$ be a ...
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4 votes
1 answer
432 views

Teichmueller disk and the $\mathrm{SL}_2\mathbb{R}$ action

Let $(X,\omega)$ be a Riemann surface of genus $g$ with holomorphic 1-form $\omega$ (or equivalently a translation structure). Let $\Omega\mathcal{T}_g$ be the space of holomorphic 1-forms over genus $...
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  • 41
4 votes
1 answer
101 views

The continuity of the the stable and unstable in definition of hyperbolic sets for flows

I would like to know whether the continuity of the stable and unstable subbundles $E^{s}$ and $E^{u}$ follows from the growth conditions as in the discrete case, or must be hypothesized, in the ...
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  • 41
3 votes
0 answers
108 views

Why is a hyperbolic basic set of dimension 2 either an attractor or a repeller?

I'm currently trying to understand the Birman-Williams Template Theorem, proved in the paper "Knotted periodic orbits II: Fibered knots, Low Dimensional Topology". Unfortunately, there doesn't seem to ...
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  • 308
2 votes
0 answers
89 views

Algebraic approach to prove the mixing property of Lorenz flow on hyperbolic surface

We knew that the noncompact subgroups of SL(2,$\mathbb{R}$) are mixing by Howe-Moore ergodicity theorem. I am curious about Lorenz flow, if we have a algebraic approach to prove the mixing property of ...
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5 votes
1 answer
644 views

Ergodicity and mixing of geodesic and horocyclic flows

I am reading a theorem about the ergodicity and mixing of geodesic and horocyclic flows on unit tangent bundle of a compact hyperbolic surface. I find that there are two ways (be listed below) to ...
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  • 303
1 vote
1 answer
111 views

Continuity of Lyapunov spaces

The following appears in the paper "Continuity properties of entropy" by Newhouse from 1989: Let $M$ be some smooth Riemannian compact manifold (you may assume boundary-less), and let $f\in Diff^{1+\...
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  • 285
3 votes
0 answers
122 views

If $f$ is dynamically coherent, is there a unique invariant foliation tangent to $E^{c}$?

Let $f$ be a diffeomorphism of a closed manifold $M$ such that $f$ is partially hyperbolic if the tanget bundle of $M$, $TM$ splits into three invariant sub-bunbles $$ TM = E^{s} \oplus E^{c} \oplus ...
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1 vote
0 answers
47 views

About stable manifold of a point [closed]

Let $(X, d)$ be a compact metric space and $f:X\rightarrow X$ be a homeomorphism and $$W^{s}(x)=\{y| d(f^{n}(x), f^{n}(y))\rightarrow as \ n\rightarrow \infty\}.$$ Question: What condition on $(X, ...
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12 votes
1 answer
657 views

Applications of the Central Limit Theorem in dynamical systems

There are very many papers in the area of (possibly non-uniformly) hyperbolic dynamical systems whose aim is to prove the Central Limit Theorem. In a dynamical context, this means that one: has a ...
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3 votes
3 answers
980 views

reference on complex dynamics

Please someone suggest me some reference on the topic "Complex Dynamics". I want a brief geometric treatment from the root level. I have graduate level background on complex analysis, riemannian ...
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7 votes
1 answer
224 views

Quantitative approximation of invariant measures by periodic ones

It is well known that many strongly chaotic dynamical systems have the property that periodic measures are (weak-star) dense in the space of all invariant probability measures. Is there some knowledge ...
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2 votes
0 answers
192 views

Uniqueness of analytic center manifold

In a book, i have read a remark which says that the center manifold of an equilibrium point of a differential equation is not unique in general but is unique in the class of analytic manifold. The ...
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0 votes
0 answers
106 views

Asymptotic pseudo orbit of an action

Let $G$ be finitely generated group (i.e $G= <S>$ $S=\{ s_1, ..., s_n\}$) and $\varphi:G\times M\longrightarrow M$ is an action then $f:G\longrightarrow M$ is called $\delta$- pseudo orbit if $...
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5 votes
0 answers
227 views

introduction books for Dynamic systems of discrete Schrodinger operator for beginner

In this semester, I study in a class of dynamic system. recently the French professor turn to the dynamic system of discrete operator. I find it is difficult to find a book in English. (I have found ...
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  • 1,492
2 votes
2 answers
605 views

Centre manifold theory for a curve of equilibrium points

I am looking for advice concerning a specific situation related to centre manifold theory (compare Perko 2001). The part which is known Let's consider a differential equation in higher-dimensional ...
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  • 83
8 votes
1 answer
577 views

Lebesgue entropy zero and positive topological entropy

I am looking for examples of volume preserving $C^{\infty}$ diffeomorphisms $f$ of a surface, which have positive topological entropy ($h(f) > 0$), but that the Lebesgue measure entropy (metric ...
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  • 1,010
2 votes
1 answer
602 views

Angle between two subspaces

Let $f:M\to M$ be a diffeomorphism on a compact riemannian manifold $M$.In the definition of a hyperbolic set we know that for all $x\in M$ there is a splitting of tangent space $T_xM=E^s(x)\oplus E^...
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  • 269
2 votes
1 answer
248 views

the union of local stable manifolds along local unstable manifolds

Let $f:M\rightarrow M$ be a $C^2$ hyperbolic diffeomorphism on compact connected riemannian manifold $M$. then there are local stable and unstable manifolds at each point denoted by $W^s_\delta(x), W^...
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  • 269
6 votes
1 answer
192 views

Is there a similar theorem in the partially hyperbolic case?

Theorem 5.10.3 from Introduction to dynamical systems, by Brin & Stuck: Let $f:M\rightarrow M$ be an Anosov diffeomorphism. Then the following are equivalent: $NW(f)=M$, every unstable manifold ...
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  • 269
4 votes
1 answer
342 views

Extending the hyperbolic splitting on $\Lambda$ to a neighborhood of $\Lambda$

Let $M$ be a compact Riemannian manifold and let $f:M→M$ a diffeomorphism. Let $\Lambda\subset M$ be a compact invariant subset of $M$. We say that $\Lambda $ is a hyperbolic set for $f$ when there ...
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1 vote
1 answer
201 views

whether there are some books and original papers ergodic theory approach to ODE

Recently I become more and more interested in the field of ergodic theory, especially in the dimension theory and thermal formalism and its applications. People always said that most of the ideas in ...
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  • 1,492
2 votes
1 answer
530 views

Center-stable manifolds

Let $f:M\to M$ be a partially hyperbolic diffeomorphism. That is, there exists a continuous splitting $TM=E^u\oplus E^c\oplus E^s$ into unstable, center and stable bundles. It is well known that there ...
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  • 2,062
5 votes
1 answer
210 views

Uniform hyperbolicity decay estimate

I have been trying to understand the proof of the following result, which is considered well-known. Theorem: Fix a compact metric space $X$, a homeomorphism $T:X \to X$, and a continuous map $ A : ...
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9 votes
1 answer
477 views

Relationship between basic sets and attractors

Definition: Let be $f:M\to M$ a diffeomorphism of a compact manifold. We say that $A\subset M$ is an attractor when there exists a neighborhood $U\supset A$ such that $f( \overline{U})\subset int (U)...
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8 votes
1 answer
629 views

A concept of dynamical coherence

I'm trying to make an overview of the study of partial hyperbolicity and there is an interesting concept of dynamical coherence which appears there. Some call it mild (see the Thesis of Pablo Carrasco,...
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  • 1,093
7 votes
2 answers
528 views

Curvatures of stable and unstable manifolds

Let $(M,g)$ be a closed Riemannian manifold and $f:M\to M$ be a $C^r$ ($r\ge2$) Anosov diffeomorphism, that is, there is a continuous hyperbolic splitting $TM=E^s\oplus E^u$ with respect to the ...
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  • 2,062
1 vote
2 answers
409 views

Extension of integrable distribution over a subset

Let $M$ be a smooth manifold and $G_k(M)$ be the $k$-dimensional Grassmian bundle of $M$. Let $K\subset M$ be a compact subset and $E:K\to G_k(M)$ be a continuous distribution on $K$. We say $E$ is ...
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  • 2,062
6 votes
4 answers
692 views

A follow up question related to entropy

For a self-map $\varphi:X\longrightarrow X$ of a space $X$, many important notions of entropy are defined through a limit of the form $$\lim_{n\rightarrow\infty}\frac{1}{n}\log a_n,$$ where in each ...
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26 votes
7 answers
2k views

If you were to axiomatize the notion of entropy

What are the axioms that a good notion of entropy must satisfy? Please note that I am not asking for the definitions of various types of entropy such as topological entropy or measure-theoretic ...
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