I recently started reading about hyperbolic dynamics in the notes of L. Wen, http://www6.cityu.edu.hk/rcms/publications/ln5.pdf and in this (page 8) there is the following s …

I have a very stupid question: I often see that the volume entropy of a compact Riemmannian manifold with negative curvature coincide with the Hausdorff dim of the limit set or Pat …

Theorem: Let be $f$ a homeomorphism of a compact metric space $X$, then $$ h_{top}(f)=\sup_{\mu\in \mathcal{M}_{f}}~ h _\mu (f) $$ Question: The above theorem is the famous v …

Notations: $\mathcal{R}(f)$ denotes the chain recurrent set of $f$ $NW(f)$ denotes the non wandering set of $f$ $R(f)$ denotes the recurrent set of $f$ ($x: x\in \omega(x)$) …

The Sz.-Nagy dilation theorem says that for a Hilbert space $H$ with nonexpansive operator $T$, there is a larger space $H'$ containing $H$ and a unitary operator $U$ on $H'$ such …

If we have a Markov coding or another symbolic description of a dynamical system it is usually easy to prove that the system is chaotic (in the sense of of Li-York, Devaney, positi …

Let every point of $\mathbb{Z}^2$ be surrounded by a mirrored disk of radius $r < \frac{1}{2}$, except leave the origin $(0,0)$ unoccupied by a disk. Q. Is it the case that …

Hello I'm looking for some applications of Morse theory to second order differential system,( or boundary value problems ) Someone can help me with a pdf or a book which has thes …

I am wondering if there are any rigorous results telling that some dynamical system hits infinitely many primes (except for the case when orbits are just arithmetic progressions). …

Suppose $Y$ is a Banach space and $X$ is a finite-dimensional subspace of $Y$. Further assume $T:X \rightarrow X$ is a linear operator which is power bounded from above and below, …

Background Let $a,b,c,d$ be nonnegative constants and consider the map $T\colon [0,1]\times[0,1] \rightarrow [0,1]\times[0,1]$ defined by $$ T(x,y) := \left( \frac{1}{1 + ax + by …

Background: A polygonal billiards table $P$ with rational angles gives rise to a flat structure $S(P)$ in a standard way, described here. Curves of constant argument on $S(P)$ whic …

Which smooth, closed surfaces $S \subset \mathbb{R}^3$ have no single geodesic $\gamma$ that fills $S$ densely? Say a geodesic $\gamma$ "fills $S$ densely" if the closure of …

I'm looking for theorems that can be used to show that a topological partition for a given expanding map is Markov. Here are the relevant definitions: Let $\phi\colon\mathbb{R}^ …

Let me begin by giving the relevant definitions. A set $A \subset \mathbb{N}$ is said to be central if and only if there exists a topological system $(X,T)$ (with $X$ a compact met …