Weak pseudo-physical measures and Pesin’s entropy formula for Anosov C 1 - diffeomorphisms.

Blokh, Alexander M. (ed.) et al., Dynamical systems, ergodic theory, and probability. In memory of Kolya Chernov. Conference dedicated to the memory of Nikolai Chernov, University of Alabama at Birmingham, Birmingham, AL, USA, May 18–20, 2015. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-2773-3/pbk; 978-1-4704-4224-8/ebook). Contemporary Mathematics 698, 69-89 (2017). The authors give a characterization of the invariant measures that satisfy Pesin’s entropy formula by means of their physical-like properties. The authors deal with C 1 Anosov systems to examine a converse of the results given by E. Catsigeras et al. [Ergodic Theory Dyn. Syst. 35, No. 3, 737–761 (2015; Zbl 1356.37047)]. The authors study general features of the weak pseudo physical measures, which do always exist. The authors prove that the weak pseudo-physical condition for the ergodic components of an invariant measure is necessary and sufficient to satisfy Pesin’s entropy formula. They also prove that for C 1 Anosov diffeomorphisms, the set of ergodic weak pseudo-physical measures is nonempty, and the set of invariant probability measures that satisfy Pesin’s entropy formula is its closed convex hull. In order to prove their theorem, the authors consider the exponential rate a(µ) with difference hµ(f) − ∫ ∑ i χ + i dµ, where hµ(f) is the metric entropy and {χ + i } are the positive Lyapunov exponents. For the entire collection see [Zbl 1376.37001]