Absolutely continuous invariant measures for non-uniformly expanding maps

Abstract

For a large class of nonuniformly expanding maps of R m , with indifferent fixed points and unbounded distorsion and non necessarily Markovian, we construct an absolutely continuous invariant measure. We extend to our case techniques previously used for expanding maps on quasi-Hölder spaces. We give general conditions and provide examples to which apply our result. IntroductionA challenge problem in smooth ergodic theory is to construct invariant measures for multidimensional maps T with some sort of weak hyperbolicity and then to study their statistical properties (decay of correlations, central limit theorem, distribution of return times, etc.). For nonuniformly expanding endomorphisms of R m , only few results exist at the moment. When the system has a Bernoulli structure, or verifies the so-called "finite range structure", and it enjoys a suitable distortion relation (Renyi's condition), M. Yuri [22,23] was able to construct an invariant, possibly σ-finite, measure absolutely continuous with respect to the Lebesgue measure. Young's tower ([20, 21]), which is mainly for nonuniformly hyperbolic systems, also works for nonuniformly expanding maps, and the invariant measures and other statistical properties can be obtained, if some bounded distortion properties are assumed. In Alves-Bonatti-Viana's work ([2], also see [3,4,5]), nonuniformly expansion are understood as the average value of log ||DT (x) −1 || along the orbits to be less than zero for almost every points. Under some conditions on the set of critical points, they can

Statistical properties of one-dimensional expanding maps with singularities of low regularity

“…In this way, the density function of the new conditional measure µ # W is given by ρ # W = |W |ρ W , and by (26) and (27), ρ # W is uniformly bounded above and away from zero, and d dx ρ # W (x) ≤ C|W | 1/r h −1 for some constant C > 0. Moreover, the new factor measure η # is a finite measure, and further, as shown by Lemma 4.1 in [16], η # is non-atomic, i.e, η # (W ) = 0 for any W ∈ Γ.…”

“…[32,33,12,26,37,30,5,9,41,1,38,19,39,36] for references). The bounded variation condition could be extended to weaker regularity conditions, such as the generalized bounded variation [31,42,35,14], and the quasi-Hölder condition [27,28], etc.…”

Pseudo-orbits, stationary measures and metastability

“…These ergodic components are with respect to absolutely continuous invariant measures whose densities are the fixed points of the Perron-Frobenius operator associated to Φ 0 . The existence of such fixed points follow easily by obtaining a Lasota-Yorke inequality on a suitable function space, such as the space of functions of bounded variation [9] or quasi-Hölder functions, [29,19], which satisfy all the assumptions required in this paper. Notice that a Lasota-Yorke inequality can be obtained as well for the random Perron-Frobenius operator associated to the random system, by using the closeness of the perturbed maps Φ ω , |ω| ≤ for small (this means that the constants η and D in (LY) and (RLY) can be chosen to be the same for the unperturbed and the perturbed systems 10 ).…”

Decay of correlations for non-uniformly expanding systems with general return times

“…However, it turns out that we can obtain decay estimates (often optimal estimates) under a weaker condition (hypothesis (*) below) that seems much more tractable. Verification of hypothesis (*) in situations such as [2] and [9] will be addressed in future work.…”