DISTORTION ESTIMATES FOR PLANAR DIFFEOMORPHISMS

Dedicated to Yasha Pesin on the occasion of his sixtieth birthday. ABSTRACT. We define new distortion quantities for diffeomorphisms of the Euclidean plane and study their properties. In particular, we obtain composition rules for these quantities analogous to standard rules for maps of an interval. Our results apply to maps with unbounded derivatives and have important applications in the theory of SRB measures for surface diffeomorphisms

On some results of Hofbauer on maps of the interval

We present a new treatment of results of F. Hofbauer on piecewise monotone mappings of the interval with positive topological entropy. Countable state symbolic systems are obtained as models for the dynamics of the natural extensions of the interval mappings. It follows that the set of measures of maximal entropy is a finite dimensional simplex and that each ergodic measure of maximal entropy is a Markov measure. 1 Introduction Symbolic dynamics plays a fundamental role in smooth Dynamical Systems. Frequently, deep geometric results can be obtained by passing to a symbolic model in which special combinatorial methods become available. The earliest use of symbolic dynamics goes back to Hadamard, Morse, Hedlund, and Birkhoff. In the 1960 0 s Sinai established the existence of symbolic dynamics for Anosov diffeomorphisms [8]. Later this was done by Bowen for general hyperbolic basic sets [1]. In a series of papers, F. Hofbauer obtained remarkable results on the structure of piecewise m..