Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userMinimal systems with finitely many ergodic measures
184
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
In this paper it is proved that if a minimal system has the property that its sequence entropy is uniformly bounded for all sequences, then it has only finitely many ergodic measures and is an almost finite to one extension of its maximal equicontinuous factor. This result is obtained as an application of a general criteria which states that if a minimal system is an almost finite to one extension of its maximal equicontinuous factor and has no infinite independent sets of length $k$ for some $kge 2$, then it has only finitely many ergodic measures.
rate research
Read More
We discuss an invertible version of Furstenbergs `Ergodic CP Shift Systems. We show that the explicit regularity of these dynamical systems with respect to magnification of measures, implies certain regularity with respect to translation of measures; We show that the translation action on measures is non-singular, and prove pointwise discrete and continuous ergodic theorems for the translation action.
For self-similar sets on $mathbb{R}$ satisfying the exponential separation condition we show that the natural projections of shift invariant ergodic measures is equal to $min{1,frac{h}{-chi}}$, where $h$ and $chi$ are the entropy and Lyapunov exponent respectively. The proof relies on Shmerkins recent result on the $L^{q}$ dimension of self-similar measures. We also use the same method to give results on convolutions and orthogonal projections of ergodic measures projected onto self-similar sets.
For every $rinmathbb{N}_{geq 2}cup{infty}$, we show that the space of ergodic measures is path connected for $C^r$-generic Lorenz attractors while it is not connected for $C^r$-dense Lorenz attractors. Various properties of the ergodic measure space for Lorenz attractors have been showed. In particular, a $C^r$-connecting lemma ($rgeq2$) for Lorenz attractors also has been proved. In $C^1$-topology, we obtain similar properties for singular hyperbolic attractors in higher dimensions.
We establish connections between several properties of topological dynamical systems, such as: - every point is generic for an ergodic measure, - the map sending points to the measures they generate is continuous, - the system splits into uniquely (alternatively, strictly) ergodic subsystems, - the map sending ergodic measures to their topological supports is continuous, - the Cesaro means of every continuous function converge uniformly.
Let ${T^t}$ be a smooth flow with positive speed and positive topological entropy on a compact smooth three dimensional manifold, and let $mu$ be an ergodic measure of maximal entropy. We show that either ${T^t}$ is Bernoulli, or ${T^t}$ is isomorphic to the product of a Bernoulli flow and a rotational flow. Applications are given to Reeb flows.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
