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A GENERIC C 1 MAP HAS NO ABSOLUTELY CONTINUOUS INVARIANT PROBABILITY MEASURE (2006)
| Citations: | 2 - 0 self |
BibTeX
@MISC{Avila06ageneric,
author = {Artur Avila and Jairo Bochi and A. Avila and J. Bochi},
title = {A GENERIC C 1 MAP HAS NO ABSOLUTELY CONTINUOUS INVARIANT PROBABILITY MEASURE},
year = {2006}
}
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Abstract
Let M be a smooth compact manifold (maybe with boundary, maybe disconnected) of any dimension d ≥ 1. Let m be some (smooth) volume probability measure in M. Let C1 (M, M) be the set of C1 maps M → M, endowed with the C1 topology. Given f ∈ C1 (M, M), we say thatµis an acim for f ifµis an f-invariant probability measure which is absolutely continuous with respect to m. Theorem 1. The setRof C 1 maps f: M→M which have no acim is a residual (dense Gδ) subset of C 1 (M, M). Since the set the set of all expanding maps and the set of all diffeomorphisms are open subsets of C 1 (M, M), we have the following immediate consequences: (i) The C 1-generic expanding map has no acim. (ii) The C 1-generic diffeomorphism has no acim. Result (i) was previously obtained in the case M is the circle by Quas [Q]. Of course, (i) does not hold in the C 1+Hölder topology. It seems possible that result (ii) holds in higher topologies. An old result by Livsic and Sinai implies that the C ∞-generic Anosov map has no acim, see [LS], also [C]. (In fact, the existence of a single periodic point of the Anosov map over which the Jacobian is different from 1 prohibits the existence of an acim.) On the other hand, the existence of acim is certainly not rare (in the probabilistic sense) among smooth enough diffeomorphisms of tori close to translations (by KAM). In the course of the proof, we will need a generalization of the usual Rokhlin tower lemma to non-invariant measures. That result, theorem 2, may be of independent interest. 2. Proof In§§2.1–2.4 we give some results that are added up to prove theorem 1 in§2.5.
Keyphrases
single periodic point lder topology c1 topology usual rokhlin tower lemma smooth enough diffeomorphisms old result c1 map f-invariant probability measure anosov map volume probability measure 1-generic diffeomorphism independent interest following immediate consequence generic anosov map probabilistic sense smooth compact manifold non-invariant measure open subset
