Stretching Generic Pesin’s Entropy Formula

Lyapunov exponents and entropy for divergence-free Lipschitz vector fields

“…In order to obtain equality (1.1) we first consider Ruelle's inequality for Lipschitz vector fields which is a fairly straightforward generalization of [8,Proposition 1]. In [8], it was proved for Lipschitz homeomorphisms but the same conclusion can be drawn for time-1 maps of Lipschitz flows.…”

“…The dominated splitting was obtained from an outstanding result having its roots also in the ingenious ideas of Mañé: the Mañé-Bochi-Viana dichotomy [11]. Our main result was motivated by [8] and arguably the main reason for studying maps first was that the proofs of the results are somewhat easier than for vector fields. The step forward is then the study of the continuous-time case.…”

“…2 of class C 1+α , Tahzibi obtained in [38] a simple yet interesting result: C 1 -generic area-preserving diffeomorphisms satisfy Pesin's entropy formula. Later, this result was generalized to any dimension by Sun and Tian [36], to volume-preserving flows and low-dimension Hamiltonians by the author and Varandas [11] and to volume-preserving lipeomorphisms displaying some C 1 -regularity of measure theoretic type, by the author, Silva and Vilarinho [8]. Sun and Tian (see [36]) follow Mañé's clever arguments in [20,21] on the proof of Pesin's entropy formula using merely C 1 -regularity of diffeomorphisms, but they appended an additional hypothesis of a dominated splitting for almost every point.…”

“…The proof follows the same method as in [8] but all the constructions are rearranged to work in a differential equations environment. The proof will be divided into three main steps:…”

“…This is obtained by considering Theorem C and will be proved in Sect. 4 by adapting and generalizing the work made in [8,36].…”