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A renewal scheme for non uniformly hyperbolic flows

Banff International Research Station for Mathematical Innovation and Discovery

In recent work, I. Melbourne and D. Terhesiu, 2014 obtain optimal results for the asymptotic of the correlation function associated with both finite and infinite measure preserving suspen- sion semiflows over Gibbs Markov maps. The involved observables are supported on a thick- ened Poincare section. The involved renewal scheme relies on inducing to such a section. In more recent work with H. Bruin, we investigate a di↵erent renewal scheme for suspension flows over non uniformly hyperbolic maps: we induce to a well chosen region Y of the same dimension as the manifold (on which the flow is defined); we do not require that Y is of bounded length. By forcing expansion on the flow direction, we can ensure that the induced version of the flow is a hyperbolic map F. Although at first counter-intuitive, such a scheme ensures that the ob- tained hyperbolic map F satisfies good spectral properties. Combined with the type of renewal equation established in Melbourne and Terhesiu, 2014 and several abstract assumptions on the hyperbolic map F (and thus on the underlying map of the suspension flow), this scheme allows us to estimate the correlation function of observables supported on the whole region Y.

Mathematics; Dynamical systems and ergodic theory; Probability theory and stochastic processes; Dynamical systems

video/mp4

Author affiliation: University of Vienna

2015-07-23

Attribution-NonCommercial-NoDerivs 2.5 Canada

http://hdl.handle.net/2429/54137

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/2.5/ca/