ERGODIC PROPERTIES OF GIBBS MEASURES FOR EXPANDING MAPS

Abstract

Gibbs measure which are also called Sinai-Ruelle-Bowen Measure describe asymptotic behavior and statistical properties of typical trajectories in many physical systems. In this work we review several methods of studying Gibbs measures by Ya.G. Sinai, D. Ruelle, R. Bowen, and P. Walters. First, using symbolic dynamics we show for subshifts of finite type that the invariant measure obtained in the Ruelle-Perron-Frobenius (R-P-F)Theorem is an ergodic Gibbs measure. Second, the proof of the R-P-F theorem is given following Walters approach, where he considers maps with infinitely many branches. In both cases, the idea is to find a fixed point of the transfer operator which will allow us to define the measure μ . Ergodic properties of μ are studied. In particular results are valid for expanding maps. These ideas are illustrated in the example of an expanding map with two branches where we show explicitly the existence of an invariant measure as well as we prove ergodicity, exactness, and the Rochlin Entropy formula.