22 - Conformal Invariant Measures for Compactly Nonrecurrent Regular Elliptic Functions

Summary

This chapter is in a sense a core of our book. Using what has been done in all previous chapters, we prove here the existence and uniqueness, up to a multiplicative constant, of a $\sg$-finite $f$-invariant measure $\mu_h$ equivalent to the $h$-conformal measure $m_h$. Furthermore, still heavily relying on what has been done in all previous chapters, particularly on conformal graph directed Markov systems, nice sets, first return map techniques, and Young towers, we provide here a systematic account of ergodic and refined stochastic properties of the dynamical system $(f,\mu_h)$ generated by such subclasses of compactly nonrecurrent regular elliptic functions as normal subexpanding elliptic functions of finite character and parabolic elliptic functions. By stochastic properties, we mean the exponential decay of correlations, the Central Limit Theorem, and the Law of the Iterated Logarithm for subexpanding functions, the Central Limit Theorem for those parabolic elliptic functions for which the invariant measure $\mu_h$ is finite and an appropriate version of the Darling–Kac Theorem establishing the strong convergence of weighted Birkhoff averages to Mittag–Leffler distributions for those parabolic elliptic functions for which the invariant measure $\mu_h$ is infinite.