Decay of correlations in suspension semi-flows of angle-multiplying maps

Abstract

We consider suspension semi-flows of angle-multiplying maps on the circle for Cr ceiling functions with r≥3. Under a Crgeneric condition on the ceiling function, we show that there exists a Hilbert space (anisotropic Sobolev space) contained in the L2 space such that the Perron–Frobenius operator for the time-t-map acts naturally on it and that the essential spectral radius of that action is bounded by the square root of the inverse of the minimum expansion rate. This leads to a precise description of decay of correlations. Furthermore, the Perron–Frobenius operator for the time-t-map is quasi-compact for a Cr open and dense set of ceiling functions.

On the Smooth Dependence of SRB Measures for Partially Hyperbolic Systems

“…In this section, we will collection some basic notions from [25]. Throughout this section, we denote R = (− 1 4 , 1 4 ) 2 and Q = (− 1 3 , 1 3 ) 2 .…”

Entropy and Decay of Correlations for Real Analytic Semi-Flows

“…In a very similar setting, when μ τ g is the Lebesgue measure, Tsujii essentially shows in [24] that for generic ceiling functions, one can construct a Hilbert space H with the above properties such that…”

“…Liverani [13] later extended this result to higher dimensional contact Anosov flows, overcoming the difficult issue of the low regularity of hyperbolic foliations. Using a pseudo-differential approach, Tsujii [24] showed that generic suspensions over uniformly expanding maps of the circle have exponential decay of correlations and went a step further by giving an asymptotic expansion with an explicit upper bound on the error term, involving the expansion rate of the flow. He recently obtained the same result [25], unconditionally, for contact Anosov flows.…”

Local limit theorem for nonuniformly partially hyperbolic skew-products and Farey sequences

“…To the best of our knowledge, the only partially hyperbolic transformations for which a local limit theorem is proved in the literature are the Anosov flows, in [Wad96] (the specific algebraic structure of flows makes it possible to reduce the problem to the study of Axiom A maps, which are uniformly hyperbolic). With the techniques of [Tsu05], it is probably possible to obtain it also for skew-products over uniformly expanding maps, for an absolutely continuous measure. Unfortunately, the main motivating example of our study, described in the next paragraph, is nonuniformly hyperbolic, and its invariant measure is singular.…”

“…However, in our setting, the map T is an isometry in the fibers, and a spectral gap seems therefore difficult to obtain. Note that [Tsu05] manages to construct a space with a spectral gap for such maps, but under strong assumptions: the map T should be uniformly expanding, andμ should be absolutely continuous with respect to Lebesgue measure. These properties are unfortunately not satisfied in our setting, and we will thus have to work without a spectral gap (on the space X × S 1 ).…”