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4
votes
1answer
136 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
1answer
85 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 ...
5
votes
2answers
243 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
191 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 ...
2
votes
1answer
110 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), ...
5
votes
1answer
131 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 ...
3
votes
1answer
156 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
96 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
197 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
160 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
387 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 ...
8
votes
1answer
383 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 ...
7
votes
2answers
346 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
249 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
534 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 ...
20
votes
6answers
1k 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
829 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
79 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. ...
2
votes
0answers
212 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 ...