ON THERMODYNAMICS OF RATIONAL MAPS. II: NON-RECURRENT MAPS

Abstract

The pressure function p(t )o fa non-recurrent map is real analytic on some interval (0 ,t ∗ )w it ht∗ strictly greater than the dimension of the Julia set. The proof is an adaptation of the well known tower techniques to the complex dynamics situation. In general, p(t )n eed not be analytic on the whole positive axis. In this paper we study analyticity properties of the pressure function of non-recurrent maps. Our approach is based on the well known tower techniques adapted to the complex dynamics situation. The pressure function p(t), which is defined in terms of the Poincar´ es erie s( see (1.4)), carries essential information about ergodic and dimensional properties of the maximal measure. In particular, it characterizes the dimension spectrum of harmonic measure on the Julia set in the case of a polynomial dynamics. According to the classical theory of Sinai, Ruelle and Bowen, p(t )i s real analytic if the dynamics is hyperbolic ,t hat is, expanding on the Julia set. This fact is closely related to the so called ‘spectral gap’ phenomenon, which also implies other important features of hyperbolic dynamics such as the existence of equilibrium states, exponential decay of correlations, etc. The problem of extending (some parts of) the classical theory to the non-hyperbolic case has become one of the central themes in the ergodic theory of conformal dynamics. In the first part of this work [8], we provided a detailed analysis of the negative part t 0 is substantially more complicated (and more important). The main difficulty arises from the presence of singularities (critical points) on the Julia set. To circumvent this difficulty, we propose to use a tower construction which forces the dynamics to be expanding on some auxiliary space. The tower method has been widely used in the general theory of dynamical systems with some degree of hyperbolicity (see especially [12]), and in particular in 1-dimensional real dynamics, where the construction is known as Hof bauer’s tower .T oa pply this method in the complex case, it is natural to use some basic elements of the Yoccoz jigsaw puzzle structure (see [9]). We will discuss only the simplest type of non-hyperbolic behavior: the case of nonrecurrent dynamics (every critical point in the Julia set is non-recurrent, that is, is away from its iterates) without parabolic cycles (see [2 ]f orvarious characterizations