Questions tagged [hyperbolic-dynamics]
The hyperbolic-dynamics tag has no usage guidance.
50
questions
0
votes
0answers
22 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 ...
0
votes
0answers
103 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 \...
2
votes
0answers
38 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}...
3
votes
0answers
101 views
Anosov structure of Fuchsian representations
I was reading a paper by Labourie https://arxiv.org/pdf/math/0401230.pdf and I'm having trouble understanding a proof. First, here's the setup.
$G = PSL(n,\mathbb{R})$ and $U$ is the subgroup of ...
2
votes
0answers
28 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 ...
2
votes
0answers
36 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 "...
1
vote
1answer
118 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 ...
0
votes
0answers
39 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
$$ h: W_{loc}^{s} (x) \cap\Lambda \rightarrow W_{loc}^{s} (y) \...
2
votes
0answers
94 views
On invariant cones of the Katok map
I am studying the Katok map and similarly constructed examples of nonuniformly hyperbolic surface diffeomorphisms. An important part of the analysis of these diffeomorphisms is the invariance of a ...
3
votes
1answer
131 views
When entropy SRB measure is zero
It is well known that many strongly chaotic dynamical systems have the property that periodic measures are (weak-star) dense in the space of all invariant probability measures.
Let $f:M \rightarrow ...
3
votes
1answer
174 views
How can I prove that the suspension of an Anosov diffeomorphism is an Anosov flow?
Suppose we have an Anosov diffeo $f$ on $M$. Define $g_t : M\times\mathbb R\to M\times\mathbb R$ by $(x,s)\mapsto (x,s+t)$. Take a quotient of $M\times\mathbb R$ under the relation $(x,s)\sim (f(x),s-...
1
vote
0answers
19 views
Stabilization of non-autonomuous 1-d wavs equation
I want to ask two questions about the stabilization of the equation $$\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 ) ...
1
vote
0answers
58 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 $(...
4
votes
0answers
79 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 ...
3
votes
1answer
197 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 ...
1
vote
1answer
31 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 ...
1
vote
0answers
49 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 ...
3
votes
1answer
240 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 ...
4
votes
1answer
349 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 $...
4
votes
1answer
92 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 ...
3
votes
0answers
96 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 ...
2
votes
0answers
86 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 ...
5
votes
1answer
581 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 ...
1
vote
1answer
109 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+\...
3
votes
0answers
118 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
$$
TM = E^{s} \oplus E^{c} \oplus ...
1
vote
0answers
46 views
About stable manifold of a point [closed]
Let $(X, d)$ be a compact metric space and $f:X\rightarrow X$ be a homeomorphism and
$$W^{s}(x)=\{y| d(f^{n}(x), f^{n}(y))\rightarrow as \ n\rightarrow \infty\}.$$
Question: What condition on $(X, ...
13
votes
1answer
604 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 ...
2
votes
3answers
771 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 ...
7
votes
1answer
205 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 ...
2
votes
0answers
174 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 ...
0
votes
0answers
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 $...
5
votes
0answers
221 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 ...
2
votes
2answers
507 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 ...
7
votes
1answer
530 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 ...
2
votes
1answer
549 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^...
2
votes
1answer
240 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^...
6
votes
1answer
189 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 ...
4
votes
1answer
311 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 ...
1
vote
1answer
185 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 ...
2
votes
1answer
489 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 ...
5
votes
1answer
199 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 : ...
9
votes
1answer
470 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)...
8
votes
1answer
606 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,...
7
votes
2answers
507 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 ...
1
vote
2answers
405 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 ...
6
votes
4answers
674 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 ...
25
votes
7answers
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 ...
6
votes
6answers
1k views
The relationship between low dimensional topology and dynamics
I am just curious how dynamics get connected with low dimensional topology. Or it is just that we have now powerful computing machines therefore it is natural to use them on topological problems. What ...
0
votes
0answers
104 views
Hausdorff Dimension of non-locally maximal hyperbolic sets
We're referencing Yakov Pesin's "Dimension Theory in Dynamical Systems" in an effort to compute the Hausdorff dimension of a particular invariant set $\Lambda$ of a hyperbolic toral automorphism. ...
3
votes
0answers
268 views
Do there exist Markov partitions with (nearly) uniform Riemannian measures?
This question complements this one; the difference is in considering Riemannian versus SRB measures.
Let $M$ be a compact, finite-dimensional Riemannian manifold, let $T: M \rightarrow M$ be an ...
