Topological Entropy on Points without Physical-like Behaviour

Eleonora Catsigeras, Xueting Tian, Edson Vargas

Published 2015-12-07Version 1

Let $f:M\rightarrow M$ be a $C^1$ diffeomorphism on a compact Riemannian manifold $M$. Let $\mathcal O_f$ denote the space of all SRB-like measures and for $x\in M$, $pw(x)$ denote the limit set of $ \{\frac1n\sum_{j=0}^{n-1}\delta_{f^j(x)} \}_{n\in\mathbb{N}} $ in weak$^*$ topology where $\delta_y$ is the Dirac probability measure supported at $y\in M.$ We state a sufficient condition to prove that the set of points without physical-like behaviour $$\Gamma_f=\{x: pw(x)\cap \mathcal{O}_{f}=\emptyset\}$$ has full topological entropy, even though in general it always has zero Lebesgue measure. In particular, this phenomena is valid for all $C^1$ transitive Anosov diffeomorphisms and time$-1$ maps of all $C^1$ transitive Anosov flows. We emphasize that the system is just required $C^1.$ The proof ideas are mainly based on Pesin's entropy formula and variational principle of saturated sets.