Questions tagged [hyperbolic-dynamics]
The hyperbolic-dynamics tag has no usage guidance.
66
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Sufficient conditions for chain recurrent set equal to set of non wandering points
Given a generic diffeomorphism, I know that the set of nonwandering points is contained in the chain recurrent set, but the converse is not always true. Is there some sufficient conditions under which ...
- 101
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1
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98
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Non-absolutely continuous foliation
What is a simple example of a (continuous) foliation of a manifold that is not absolutely continuous?
(A foliation is said to be absolutely continuous if holonomy maps between smooth transversals send ...
- 63
2
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1
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174
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Invariant measure of geodesic flow on unit tangent bundle of a modular surface
This is a paper written by Series "THE MODULAR SURFACE AND CONTINUED FRACTIONS".
I want to know about above construction natural invariant measure $\mu$ for the geodesic flow on $T_{1}M$ ...
- 73
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68
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Understanding logarithmic law for geodesics
I was reading this seminal paper
https://projecteuclid.org/journals/acta-mathematica/volume-149/issue-none/Disjoint-spheres-approximation-by-imaginary-quadratic-numbers-and-the-logarithm/10.1007/...
- 189
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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 $...
- 233
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69
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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 ...
- 270
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71
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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 ...
- 1,880
2
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174
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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 ...
- 45
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2
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225
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Canonically representing the monodromy of a hyperbolic manifold fibered over $S^1$
Let $Y$ be a hyperbolic manifold that fibers over $S^1$, with fibration $\pi:Y \to S^1$ with fiber $\Sigma$. Thurston states that the monodromy $\phi:\Sigma \to
\Sigma$ of this projection is then ...
- 1,161
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112
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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 ...
- 501
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69
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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 ...
- 31
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79
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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 ...
- 45
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96
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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{...
- 291
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309
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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 ...
- 1,578
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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 $Q$, ...
- 824
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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 ...
- 581
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82
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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". ...
- 11
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89
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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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162
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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 \...
- 151
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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}...
- 381
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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 ...
- 419
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67
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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 "...
- 21
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144
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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 ...
- 970
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49
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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
$$ h: W_{loc}^{s} (x) \cap\Lambda \rightarrow W_{loc}^{s} (y) \...
- 970
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117
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On invariant cones of the Katok map
I am studying the Katok map and similarly constructed examples of nonuniformly hyperbolic surface diffeomorphisms. An important part of the analysis of these diffeomorphisms is the invariance of a ...
- 151
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1
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190
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When entropy SRB measure is zero
It is well known that many strongly chaotic dynamical systems have the property that periodic measures are (weak-star) dense in the space of all invariant probability measures.
Let $f:M \rightarrow ...
- 970
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337
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How can I prove that the suspension of an Anosov diffeomorphism is an Anosov flow?
Suppose we have an Anosov diffeo $f$ on $M$. Define $g_t : M\times\mathbb R\to M\times\mathbb R$ by $(x,s)\mapsto (x,s+t)$. Take a quotient of $M\times\mathbb R$ under the relation $(x,s)\sim (f(x),s-...
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Stabilization of non-autonomuous 1-d wavs equation
I want to ask two questions about the stabilization of the equation $$\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 ) ...
- 525
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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 $(...
- 151
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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 ...
- 55.2k
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225
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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 ...
- 189
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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 ...
- 14.1k
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51
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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 ...
- 970
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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 ...
user127549
4
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1
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491
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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 $...
- 41
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112
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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 ...
- 41
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119
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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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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 ...
- 53
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801
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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 ...
- 303
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120
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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+\...
- 327
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127
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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
$$
TM = E^{s} \oplus E^{c} \oplus ...
user60504
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49
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About stable manifold of a point [closed]
Let $(X, d)$ be a compact metric space and $f:X\rightarrow X$ be a homeomorphism and
$$W^{s}(x)=\{y| d(f^{n}(x), f^{n}(y))\rightarrow as \ n\rightarrow \infty\}.$$
Question: What condition on $(X, ...
- 121
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726
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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 ...
- 14.1k
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3
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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 ...
7
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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 ...
- 173
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243
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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 ...
- 21
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107
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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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239
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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 ...
- 1,646
4
votes
2
answers
688
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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 ...
- 103
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600
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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 ...
- 1,010