Dominated splitting and Pesin's entropy formula

Approximation of Bernoulli measures for non-uniformly hyperbolic systems

“…The classical non-uniformly hyperbolic theory always assumes that the derivative of a diffeomorphism is Hölder continuous, which could not be removed in general [4,15]. However, if the Hölder condition is replaced by the domination property between Oseledets decompositions E s and E u , some properties such as stable manifold theory and the Pesin entropy formula could still be preserved [1,19]. Liao, Sun and Wang [9] proved that upper semi-continuity of the entropy map holds for a C 1 non-uniformly hyperbolic system with domination but may fail for a C 1+α non-uniformly hyperbolic system without domination.…”

Formula of entropy along unstable foliations for $C^1$ diffeomorphisms with dominated splitting

“…The following lemma about graph transform on dominated bundles is a generalization of Lemma 3 in [8] by Sun and Tian [16].…”

“…Except for Ruelle's inequality, all other results above require that f is C 1+α or C 2 , so it is interesting to investigate Pesin's formula under C 1 differentiability hypothesis plus some additional conditions, for example, dominated splitting. Recently, Sun and Tian [16] applied Mañé's method to prove that Pesin's entropy formula holds if f is a C 1 diffeomorphism with dominated splitting. In [3], Catsigeras, Cerminara, and Enrich considered a nonempty set of invariant measures which describe the asymptotic statistics of Lebesgue almost all orbits, and they proved that the measure-theoretic entropy of each of these measures is bounded from below by the sum of the Lyapunov exponents on the dominating subbundle.…”

“…For more details about the dynamics of a system with dominated splitting, one can refer to [12] and [15]. Instead of the condition in [16] that f admits a dominated splitting, Tian [17] gave the concept of nonuniformly-Hölder-continuity for an f -invariant measure, and proved that Pesin's entropy formula holds under that assumption.…”

“…Before going into the proof of Theorem B, we need a technical lemma from [16]. In the statement of the lemma, we will use the following definition from [8].…”