Weight-balanced measures and free energy for one-dimensional dynamics

Abstract

In this paper we consider Thermodynamic Formalism propcrties o fone dimensional maps. Wc consider the existence ofweight-balanced measures and large dcviation propcrtics o f the Frcc-Encrgy of the Jacobian of measures. We show that a wcight-balanced mcasure exists under the hypotheses that the map is piccewisc-homeomorphic and the weights picccwisc constant. We considcr also a certain class of measures with the property that thc Frce-Encrgy of the Jacobian is difTerentiable by parts. For measures in this class wc show that a certa in measure is the maximal entropy measurc i f and only ifthe Frcc-Energy of thc Jacobian i~ linear. Tite rcsult follows from general propcrtics of Large-Deviation Thcory and does not use the more classical approach of Thcrmodynamic Formalism.