2
votes
0answers
110 views

On the volume entropy of negatively curved manifolds

Let $X$ be the universal cover of a closed negatively curved Riemannian manifold. Let $x_0\in X$ be a base point, $S$ be the unit sphere in $T_{x_0}X$ and $\exp:T_{x_0}X\rightarrow X$ be the ...
2
votes
1answer
111 views

Is there a one-dimensional subshift of positive entropy s, all of whose sub-subshifts also have entropy s?

A subshift is a subset $X$ of $A^\mathbb{N}$ or $A^\mathbb{Z}$ (with $A$ finite), such that $X$ is topologically closed and closed under the shift operation. The shift operation is defined by ...
2
votes
0answers
122 views

Reference for and Properties of the $alpha$-entropy

Let $T : X \to X$ be a continuous map on, say, a compact metric space $X$. Let $\mu$ be an invariant borel measure. Under suitable conditions, a result of Brin and Katok states that $\mu$-almost ...
2
votes
2answers
249 views

probability measures with entropy equal to nonnegative number

Is it true that for a given nonnegative number, there exists a measure-theoretical entropy value (supremum of entropies of all partitions under a measure-preserving transformation) that equals this ...
0
votes
1answer
220 views

Entropy of inverse map for endomorphism case on surfaces

Hi, I know that in the diffeomorphism case the measure entropy of the T:M^{2}-->M^{2} (M smooth Rimannian surface) will be the same as the measure entropy of T^{-1}. But i need to know about the ...
2
votes
1answer
212 views

Compact group extension of a zero entropy system.

Suppose $T: X \to X$ is a continuous map and $\mu$ a $T$-ergodic probability measure over the Borel sets of $X$. Now, suppose $K \subset \mathrm{Hom}(X)$ is a compact group of measure-preserving ...
10
votes
1answer
1k views

System with invariant measure, but no ergodic measure.

Question Examples of continuous transformations $T: X \to X$ such that the family of invariant probability measures $M(T)$ is NOT empty but there is no ergodic measure ($E(T) = \emptyset$). Notice ...
10
votes
3answers
727 views

Supremum amongst Kolmogorov-Sinai entropies: ergodic or just invariant measures.

Cases where $sup_{\mu \in E(T)} h_\mu(T) \neq \sup_{\mu \in M(T)} h_\mu(T)$. Background For a topological space $X$, let $T: X \to X$ be a continuous application. Then, call the set of ...
6
votes
1answer
750 views

Entropy of first return map and suspension flows

There are some well know formulas of Abramov about derived systems. Firstly let $(X,\mu,f)$ be a probability preserving system and $A\subset X$ is measurable such that $\bigcup_{n\ge0}f^nA=X$. Let ...
5
votes
2answers
271 views

Entropy of nested compact invariant sets

Let $f$ be a homeomorphism on a compact metric space $X$. $K_1\supset K_2\supset\cdots \supset K$ are compact subsets of $X$ such that $f(K_n)=K_n$ and $K=\bigcap K_n$. If $h(f, K_1)<\infty$, do we ...
1
vote
0answers
85 views

Median entropy to observe evolution of system?

Hello, I am studying a dynamical system that takes as an initial condition a list. I want to analyze the evolution of Shannon's entropy in this system. I know the maximum entropy (50) and the minimum ...
3
votes
0answers
598 views

Entropy conjecture for flows

The entropy conjecture for diffeomorphisms (see for example this paper) asserts that for diffeomorphisms of manifolds, the log of the spectral radius of the actions of the diffeomorphism on the ...