The Dolgopyat inequality in bounded variation for non-Markov maps

This is the author accepted manuscript. The final version is available from World Scientific via the DOI in this record.Let F be a (non-Markov) countably piecewise expanding interval map satisfying certain regularity conditions, and Ł˜Ł̃ the corresponding transfer operator. We prove the Dolgopyat inequality for the twisted operator Ł˜s(v)=Ł˜s(esφv)Ł̃s(v)=Ł̃s(esφv) acting on the space BV of functions of bounded variation, where φφ is a piecewise C1C1 roof function.We are also grateful for the support the Erwin Schrodinger Institute in Vienna, where this paper was completed

Ulam method for the Chirikov standard map

We introduce a generalized Ulam method and apply it to symplectic dynamical maps with a divided phase space. Our extensive numerical studies based on the Arnoldi method show that the Ulam approximant of the Perron-Frobenius operator on a chaotic component converges to a continuous limit. Typically, in this regime the spectrum of relaxation modes is characterized by a power law decay for small relaxation rates. Our numerical data show that the exponent of this decay is approximately equal to the exponent of Poincar\'e recurrences in such systems. The eigenmodes show links with trajectories sticking around stability islands.Comment: 13 pages, 13 figures, high resolution figures available at: http://www.quantware.ups-tlse.fr/QWLIB/ulammethod/ minor corrections in text and fig. 12 and revised discussio

Operator renewal theory and mixing rates for dynamical systems with infinite measure

We develop a theory of operator renewal sequences in the context of infinite ergodic theory. For large classes of dynamical systems preserving an infinite measure, we determine the asymptotic behaviour of iterates $L^n$ of the transfer operator. This was previously an intractable problem. Examples of systems covered by our results include (i) parabolic rational maps of the complex plane and (ii) (not necessarily Markovian) nonuniformly expanding interval maps with indifferent fixed points. In addition, we give a particularly simple proof of pointwise dual ergodicity (asymptotic behaviour of $\sum_{j=1}^nL^j$) for the class of systems under consideration. In certain situations, including Pomeau-Manneville intermittency maps, we obtain higher order expansions for $L^n$ and rates of mixing. Also, we obtain error estimates in the associated Dynkin-Lamperti arcsine laws.Comment: Preprint, August 2010. Revised August 2011. After publication, a minor error was pointed out by Kautzsch et al, arXiv:1404.5857. The updated version includes minor corrections in Sections 10 and 11, and corresponding modifications of certain statements in Section 1. All main results are unaffected. In particular, Sections 2-9 are unchanged from the published versio

Darling–Kac Theorem for Renewal Shifts in the Absence of Regular Variation

Regular variation is an essential condition for the existence of a Darling–Kac law. We weaken this condition assuming that the renewal distribution belongs to the domain of geometric partial attraction of a semistable law. In the simple setting of one-sided null recurrent renewal chains, we derive a Darling–Kac limit theorem along subsequences. Also in this context, we determine the asymptotic behaviour of the renewal function and obtain a Karamata theorem for positive operators. We provide several examples of dynamical systems to which these results apply

Operator renewal theory and mixing rates for dynamical systems with infinite measure

We develop a theory of operator renewal sequences in the context of infinite ergodic theory. For large classes of dynamical systems preserving an infinite measure, we determine the asymptotic behaviour of iterates $L^n$ of the transfer operator. This was previously an intractable problem. Examples of systems covered by our results include (i) parabolic rational maps of the complex plane and (ii) (not necessarily Markovian) nonuniformly expanding interval maps with indifferent fixed points. In addition, we give a particularly simple proof of pointwise dual ergodicity (asymptotic behaviour of ____sum_{j=1}^nL^j) for the class of systems under consideration. In certain situations, including Pomeau-Manneville intermittency maps, we obtain higher order expansions for $L^n$ and rates of mixing. Also, we obtain error estimates in the associated Dynkin-Lamperti arcsine laws

Limit theorems for wobbly interval intermittent maps

This is the author accepted manuscript. The final version is available from the Polskiej Akademii Nauk via the DOI in this recordWe consider perturbations of interval maps with indifferent fixed points, which we refer to as wobbly interval intermittent maps, for which stable laws for general H¨older observables fail. We obtain limit laws for such maps and H¨older observables. These limit laws are similar to the classical semistable laws previously established for random processes. One of the considered examples is an interval map with a countable number of discontinuities, and to analyse it we need to construct a Markov/Young tower.Engineering and Physical Sciences Research Council (EPSRC