S-unimodal Misiurewicz maps with flat critical points

Roland Zweimüller. "S-unimodal Misiurewicz maps with flat critical points." Fundamenta Mathematicae 181.1 (2004): 1-25. <http://eudml.org/doc/282714>.

@article{RolandZweimüller2004,
abstract = {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 σ-finite invariant measure μ, 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.},
author = {Roland Zweimüller},
journal = {Fundamenta Mathematicae},
keywords = {neutral orbit; slow mixing; central limit theorem; stable limit law},
language = {eng},
number = {1},
pages = {1-25},
title = {S-unimodal Misiurewicz maps with flat critical points},
url = {http://eudml.org/doc/282714},
volume = {181},
year = {2004},
}

TY - JOUR
AU - Roland Zweimüller
TI - S-unimodal Misiurewicz maps with flat critical points
JO - Fundamenta Mathematicae
PY - 2004
VL - 181
IS - 1
SP - 1
EP - 25
AB - 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 σ-finite invariant measure μ, 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.
LA - eng
KW - neutral orbit; slow mixing; central limit theorem; stable limit law
UR - http://eudml.org/doc/282714
ER -