Questions tagged [hyperbolic-dynamics]
The hyperbolic-dynamics tag has no usage guidance.
73 questions
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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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1
answer
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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 ...
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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 ...
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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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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 ...
- 1,706
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2
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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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1
answer
652
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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 ...
- 1,101
2
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1
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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^...
- 279
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1
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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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1
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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 ...
- 279
4
votes
1
answer
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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 ...
user11178
1
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1
answer
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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 ...
- 1,706
3
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1
answer
586
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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 ...
- 2,244
5
votes
1
answer
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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 : ...
- 437
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1
answer
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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)...
user11178
8
votes
1
answer
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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,...
- 1,143
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2
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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 ...
- 2,244
1
vote
2
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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 ...
- 2,244
6
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4
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763
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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 ...
- 3,101
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votes
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answers
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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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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 ...
- 87
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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. ...
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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 ...
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