Equilibrium States and SRB-like measures of C1 Expanding Maps of the Circle

For any C1 expanding map f of the circle we study the equilibrium states for the potential -log |f'|. We formulate a C1 generalization of Pesin's Entropy Formula that holds for all the SRB measures if they exist, and for all the (necessarily existing) SRB-like measures. In the C1-generic case Pesin's Entropy Formula holds for a unique SRB measure which is not absolutely continuous with respect to Lebesgue. The result also stands in the non generic case for which no SRB measure exists.Comment: Un this version we include some addings and corrections that were suggested by the referees. The final version will appear in Portugaliae Mathematica and will be available at http://www.ems-ph.org/journals/journal.php?jrn=p

Pesin Entropy Formula for C1 Diffeomorphisms with Dominated Splitting

For any C1 diffeomorphism with dominated splitting we consider a nonempty set of invariant measures which describes the asymptotic statistics of Lebesgue-almost all orbits. They are the limits of convergent subsequences of averages of the Dirac delta measures supported on those orbits. We prove that the metric entropy of each of these measures is bounded from below by the sum of the Lyapunov exponents on the dominating subbundle. As a consequence, if those exponents are non negative, and if the exponents on the dominated subbundle are non positive, those measures satisfy the Pesin Entropy Formula.Comment: We added the corrections suggested by the referee. Accepted for publication in the journal "Ergodic Theory and Dynamical Systems". Final version will appear in http://journals.cambridge.org/action/displayJournal?jid=et