Questions tagged [hyperbolic-dynamics]
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79 questions
5 votes
1 answer
312 views
Exact boundary of the blobs in Fig 9.5 of Indra's pearls
This question is about Figure 9.5 in Indra's Pearls. In the figure, four blobs $a, A, b, B$ are drawn, along with their boundary curves. My question is about the boundary curves of these blobs. ...
- 673
0 votes
0 answers
76 views
Minimal finite-edit shadowing distance in the full two-shift
Let $\Sigma_2 = \{0,1\}^{\mathbb{Z}}$ be the full two-shift with left-shift map $\sigma$ and the standard product metric $$d(x,y) = 2^{-\inf\{|n| : x_n \neq y_n\}}.$$ Fix $\varepsilon = 2^{-m}$ for ...
5 votes
3 answers
370 views
Where is the application of Anosov's shadowing lemma other than structural stability?
A few weeks ago, I learned Anosov's shadowing lemma and the application to structural stability of hyperbolic map. Despite the statement is amazing, there seems few applications of it. Are there any ...
- 53
2 votes
0 answers
80 views
Distribution of lengths of closed geodesics
(I am trying to get a sense of what the state of the art is regarding the distribution of the length spectrum of a closed surface of negative curvature, I am curious about any good reference/open ...
- 3,056
3 votes
1 answer
157 views
Horocycle foliation and lengths of closed geodesics
This question actually makes sense for an arbitrary Anosov flow on a $3$-dimensional manifold but it easier to state in the particular case of geodesic flows. Let $(\Sigma,h)$ be a closed surface of ...
- 3,056
1 vote
0 answers
87 views
Ergodic stable hamiltonian flows, Anosov flows, and contactness
I have a "lemma" that seems to contradict things I know about Anosov flows in dimension 3. I'll present the lemma and then discuss why it contradicts things I believe to be true. Would love ...
- 259
2 votes
0 answers
93 views
Invariant measures with uniform projection in hyperbolic toral automorphisms
Let $\mu$ be an invariant probability measure of a hyperbolic toral automorphism (i.e., cat map) $M$ on the two-dimensional torus $\Bbb T^2$. Suppose that the projection of $\mu$ onto the $x$-axis ...
- 29
1 vote
1 answer
139 views
Looking for a precise statement about hyperbolic points in the interior of the Mandelbrot set
A Numberphile video piqued my interest regarding the hyperbolicity property of points in the Mandelbrot set. But I can't seem to find a concise statement about the conjecture about hyperbolic points ...
- 135
-1 votes
1 answer
264 views
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 ...
- 302
5 votes
1 answer
245 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 ...
- 51
9 votes
0 answers
182 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 ...
- 23.6k
2 votes
1 answer
409 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 $...
- 281
1 vote
0 answers
87 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 ...
- 3,346
1 vote
0 answers
207 views
Mixing of geodesic flow and rate of mixing
I know the following result which is if $X:=G/\Gamma$ where $G$ be a Lie group and $\Gamma$ is a lattice in $G$. Then geodesic floe $F=(g_t)$ is exponentially mixing i.e., there exist $\gamma > 0$ ...
- 347
1 vote
1 answer
209 views
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 ...
- 143
2 votes
1 answer
312 views
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
1 vote
0 answers
121 views
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/...
- 347
2 votes
0 answers
346 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 $...
- 345
3 votes
1 answer
116 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 ...
- 302
3 votes
0 answers
87 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 ...
- 2,174
2 votes
1 answer
240 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 ...
- 45
5 votes
2 answers
421 views
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 ...
- 1,241
1 vote
1 answer
225 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 ...
- 551
3 votes
0 answers
109 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 ...
- 49
2 votes
1 answer
177 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 ...
- 45
0 votes
0 answers
149 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{...
- 291
6 votes
0 answers
405 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 ...
- 1,678
2 votes
0 answers
102 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 $Q$, ...
- 874
1 vote
0 answers
121 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 ...
- 646
0 votes
0 answers
126 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". ...
- 11
2 votes
0 answers
119 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 ...
- 111
1 vote
0 answers
232 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 \...
- 151
2 votes
0 answers
67 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}...
- 411
2 votes
0 answers
64 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 ...
- 429
2 votes
0 answers
107 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 "...
- 21
1 vote
1 answer
208 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 ...
- 1,045
0 votes
0 answers
85 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 $$ h: W_{loc}^{s} (x) \cap\Lambda \rightarrow W_{loc}^{s} (y) \...
- 1,045
2 votes
0 answers
138 views
On invariant cones of the Katok map
I am studying the Katok map and similarly constructed examples of nonuniformly hyperbolic surface diffeomorphisms. An important part of the analysis of these diffeomorphisms is the invariance of a ...
- 151
4 votes
1 answer
263 views
When entropy SRB measure is zero
It is well known that many strongly chaotic dynamical systems have the property that periodic measures are (weak-star) dense in the space of all invariant probability measures. Let $f:M \rightarrow ...
- 1,045
5 votes
1 answer
538 views
How can I prove that the suspension of an Anosov diffeomorphism is an Anosov flow?
Suppose we have an Anosov diffeo $f$ on $M$. Define $g_t : M\times\mathbb R\to M\times\mathbb R$ by $(x,s)\mapsto (x,s+t)$. Take a quotient of $M\times\mathbb R$ under the relation $(x,s)\sim (f(x),s-...
1 vote
0 answers
38 views
Stabilization of non-autonomuous 1-d wavs equation
I want to ask two questions about the stabilization of the equation $$\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 ) ...
- 617
1 vote
0 answers
118 views
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
5 votes
0 answers
148 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 ...
- 63.5k
4 votes
1 answer
286 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 ...
- 199
2 votes
1 answer
111 views
$C^1$ 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 ...
- 14.7k
1 vote
0 answers
66 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 ...
- 1,045
3 votes
1 answer
421 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 ...
user127549
4 votes
1 answer
610 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 $...
- 41
4 votes
1 answer
181 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 ...
- 41
3 votes
0 answers
162 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 ...
- 373