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Reference request on dynamics and hyperbolic dynamics (hyperbolicity in absence of periodic orbits)

I would appreciate if you introduce me a reference (paper or book) who address the concept of hyperbolic dynamics but with emphasis on absence of periodic orbits. a possible ...
Ali Taghavi's user avatar
0 votes
0 answers
36 views

Some equivalent conditions for hyperbolicity of flow

Let $M$ be a manifold and $\phi_t$ be a smooth flow associated to a smooth vector field on $M$. Are the following 3 conditions equivalent? 1)For every fixed $t$ the diffeomorphism $\...
Ali Taghavi's user avatar
4 votes
1 answer
180 views

Fixed points of maps defined on Teichmüller space

Let $\mathcal{T}_A$ be a Teichmüller space of the sphere $S^2$ with a finite set $A$ of marked points, and suppose that $f \colon \mathcal{T}_A \to \mathcal{T}_A$ is a holomorphic map that has a ...
A B's user avatar
  • 41
8 votes
0 answers
156 views

Square root of an Anosov diffeomorphism

Let $T\colon \mathbb T^d\to\mathbb T^d$ be an Anosov diffeomorphism (that is, the tangent bundle splits into invariant stable and unstable bundles; the restriction of $DT$ to the unstable bundle is ...
Anthony Quas's user avatar
  • 23.2k
2 votes
1 answer
254 views

Chaotic dynamics of maps on unit square that are NOT Triangular

We will denote the compact interval $[0,1]$ by $I$ and the unit square $[0,1]\times[0,1]$ by $I^2$. Triangular map on $I^2$ is a continuous map $F:I^2\to I^2$ of the form $F(x,y)=(f(x),g(x,y))$ where $...
confused's user avatar
  • 271
1 vote
0 answers
67 views

Equidistribution on subvarieties of $\mathrm{SL}_n(\mathbb{Z})$

$\DeclareMathOperator\SL{SL}$Let $\Gamma_{\infty}$ be a maximal parabolic subgroup of $\SL_n(\mathbb{Z})$. I believe there are various equidistribution theorems, which say that Iwasawa $x$ coordinates ...
kindasorta's user avatar
  • 2,907
2 votes
0 answers
319 views

A (possible) generic spectral property in one dimensional dynamics

Context and Definitions Consider the interval $I=[0,1]$. We say that $T:I\to I$ satisfies the axiom A (I am following [1]) if: $T$ has a finite number of hyperbolic periodic attractors; and defining $...
Matheus Manzatto's user avatar
3 votes
1 answer
81 views

Can the Reeb foliation of $S^3$ be realized as stable manifold foliation of a smooth hyperbolic discrete dynamic on $S^3$?

Can the Reeb foliation of $S^3$ be realized as foliation associated to stable(or unstable) manifolds of a hyperbolic discrete dynamic on $S^3$?If yes what is a precise formulation for that ...
Ali Taghavi's user avatar
3 votes
0 answers
71 views

Trapped vs. nonwandering points

For a continuous flow on a topological space the concept of a nonwandering point (in forward/backward time) is well-known. Another useful concept (for example from the point of view of scattering ...
B K's user avatar
  • 1,942
2 votes
1 answer
187 views

How to find the hyperbolic dimension of map $f(z) = z^2$ of $\overline{\mathbb{C}}$ onto itself?

I am reading the research article "The Hausdorff dimension of the boundary of Mandelbrot set and Julia sets" by Shishikura. In his article, he defined hyperbolic sets and hyperbolic ...
Nirmal Rawat's user avatar
1 vote
1 answer
163 views

Existence of center-stable manifold when the Jacobian is singular?

The following is a result from Shub's monograph "Global Stability of Dynamical Systems". I dabble in the proof, and it appears to me that the existence of $W^{\rm cu}_{\rm loc}$ does not ...
Thomas's user avatar
  • 511
2 votes
0 answers
83 views

Question about stable manifold theorem and Frobenius integrability theorem

I have a question about Anosov diffeomorphism (Wikipedia: Anosov diffeomorphisms) For hyperbolic fixed point $p$, $W^{s}(p)$ is a smooth manifold and its tangent space has the same dimension as the ...
WaoaoaoTTTT's user avatar
2 votes
1 answer
120 views

Examples of hyperbolic set and J-stable sets

I am reading the research article "The Hausdorff dimension of the boundary of Mandelbrot set and Julia sets" by Shishikura. The following two definitions are given without any examples in ...
Nirmal Rawat's user avatar
0 votes
0 answers
103 views

Polynomial / quadratic autonomous system of ODEs – proving monotonicity / convexity

Problem: Consider the autonomous ODE system \begin{align*} \dot{x} &= (1-x) (z-xy)\\ \dot{y} &= \tfrac 1 2 y^2 - (a+xy)(1-y) \\ \dot{z} &= \tfrac 1 2 z^2 - \tfrac 1 2 y^2 + (a+xy)z \end{...
Pavel Kocourek's user avatar
6 votes
0 answers
348 views

Examples of expansive homeomorphisms with the specification property that are neither symbolic nor factors of mixing SFT nor product of thereof

I am looking for nontrivial examples of expansive homeomorphisms with the specification property on compact metric spaces. Here, by a ``trivial'' example I understand a subshift with the specification ...
Dominik Kwietniak's user avatar
2 votes
0 answers
76 views

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 $Q$, ...
Yonah Borns-Weil's user avatar
1 vote
0 answers
91 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 ...
D. Thomine's user avatar
0 votes
0 answers
92 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". ...
monell20's user avatar
2 votes
0 answers
95 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 ...
Leon Staresinic's user avatar
1 vote
0 answers
175 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 \...
D. Ford's user avatar
  • 151
2 votes
0 answers
48 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}...
user135520's user avatar
2 votes
0 answers
83 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 "...
Xiiao's user avatar
  • 21
1 vote
1 answer
164 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 ...
Adam's user avatar
  • 1,043
0 votes
0 answers
61 views

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) \...
Adam's user avatar
  • 1,043
2 votes
0 answers
124 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 ...
D. Ford's user avatar
  • 151
4 votes
1 answer
212 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 ...
Adam's user avatar
  • 1,043
4 votes
1 answer
246 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 ...
Michal's user avatar
  • 199
1 vote
0 answers
53 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 ...
Adam's user avatar
  • 1,043
3 votes
1 answer
360 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 ...
user avatar
4 votes
1 answer
536 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 $...
Alex's user avatar
  • 41
4 votes
1 answer
132 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 ...
Julian's user avatar
  • 41
3 votes
0 answers
134 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 ...
asldjk's user avatar
  • 318
2 votes
0 answers
94 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 ...
Skid Row's user avatar
5 votes
1 answer
988 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 ...
Markiff's user avatar
  • 333
1 vote
1 answer
128 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+\...
JustSomeGuy's user avatar
3 votes
0 answers
133 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 ...
user avatar
1 vote
0 answers
51 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, ...
Ali  Barzanouni's user avatar
13 votes
1 answer
775 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 ...
Benoît Kloeckner's user avatar
3 votes
3 answers
1k 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 ...
debabrata chakraborty's user avatar
7 votes
1 answer
281 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 ...
Dmitry Todorov's user avatar
2 votes
0 answers
280 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 ...
aristote's user avatar
0 votes
0 answers
107 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 $...
Ali Barzanouni's user avatar
5 votes
0 answers
243 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 ...
yaoxiao's user avatar
  • 1,706
4 votes
2 answers
729 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 ...
Kmra's user avatar
  • 103
8 votes
1 answer
652 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 ...
shurtados's user avatar
  • 1,101
2 votes
1 answer
752 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^...
mac's user avatar
  • 279
2 votes
1 answer
283 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^...
mac's user avatar
  • 279
7 votes
1 answer
209 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 ...
mac's user avatar
  • 279
4 votes
1 answer
384 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 ...
user avatar
1 vote
1 answer
227 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 ...
yaoxiao's user avatar
  • 1,706