S-unimodal Misiurewicz maps with flat critical points

Roland Zweimüller¹

¹ Mathematics Department Imperial College London 180 Queen's Gate London SW7 2AL, UK

Fundamenta Mathematicae, Tome 181 (2004) no. 1, pp. 1-25

Résumé

We consider S-unimodal Misiurewicz maps $T$ with a flat critical point $c$ and show that they exhibit ergodic properties analogous to those of interval maps with indifferent fixed (or periodic) points. Specifically, there is a conservative ergodic absolutely continuous $\sigma$-finite invariant measure $\mu$, exact up to finite rotations, and in the infinite measure case the system is pointwise dual ergodic with many uniform and Darling–Kac sets. Determining the order of return distributions to suitable reference sets we obtain bounds on the decay of correlations and on wandering rates. Assuming some control of the local behaviour of $T$ at $c$, we show that in most cases, e.g. whenever the postcritical orbit has a Lyapunov exponent, the tail of the return distribution is in fact regularly varying, which implies various distributional limit theorems.

Détail

DOI : 10.4064/fm181-1-1

Keywords: consider s unimodal misiurewicz maps flat critical point exhibit ergodic properties analogous those interval maps indifferent fixed periodic points specifically there conservative ergodic absolutely continuous sigma finite invariant measure exact finite rotations infinite measure system pointwise dual ergodic many uniform darling kac sets determining order return distributions suitable reference sets obtain bounds decay correlations wandering rates assuming control local behaviour cases whenever postcritical orbit has lyapunov exponent tail return distribution regularly varying which implies various distributional limit theorems