19 research outputs found
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 ĆËĆÌ the corresponding transfer operator. We prove the Dolgopyat inequality for the twisted operator ĆËs(v)=ĆËs(esÏv)ĆÌs(v)=ĆÌs(esÏv) acting on the space BV of functions of bounded variation, where ÏÏ 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 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 ) for the class of systems under
consideration.
In certain situations, including Pomeau-Manneville intermittency maps, we
obtain higher order expansions for 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
DarlingâKac Theorem for Renewal Shifts in the Absence of Regular Variation
Regular variation is an essential condition for the existence of a DarlingâKac 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 DarlingâKac 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 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 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 Hšolder observables fail. We obtain limit laws for such maps and
Hšolder 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