229 research outputs found
Projection of Markov measures may be Gibbsian
We study the induced measure obtained from a 1-step Markov measure, supported
by a topological Markov chain, after the mapping of the original alphabet onto
another one. We give sufficient conditions for the induced measure to be a
Gibbs measure (in the sense of Bowen) when the factor system is again a
topological Markov chain. This amounts to constructing, when it does exist, the
induced potential and proving its Holder continuity. This is achieved through a
matrix method. We provide examples and counterexamples to illustrate our
results.Comment: 4 latex figure
Birkhoff averages of Poincare cycles for Axiom-A diffeomorphisms
We study the time of th return of orbits to some given (union of)
rectangle(s) of a Markov partition of an Axiom A diffeomorphism. Namely, we
prove the existence of a scaled generating function for these returns with
respect to any Gibbs measure (associated to a Holderian potential). As a
by-product, we derive precise large deviation estimates and a central limit
theorem for Birkhoff averages of Poincare cycles. We emphasize that we look at
the limiting behavior in term of number of visits (the size of the visited set
is kept fixed). Our approach relies on the spectral properties of a
one-parameter family of induced transfer operators on unstable leaves crossing
the visited set.Comment: 17 pages; submitte
Multifractals via recurrence times ?
This letter is a comment on an article by T.C. Halsey and M.H. Jensen in
Nature about using recurrence times as a reliable tool to estimate multifractal
dimensions of strange attractors. Our aim is to emphasize that in the recent
mathematical literature (not cited by these authors), there are positive as
well as negative results about the use of such techniques. Thus one may be
careful in using this tool in practical situations (experimental data).Comment: This is a very short and non-technical note written after an article
published in Nature by T.C. Halsey and M.H. Jense
On almost-sure versions of classical limit theorems for dynamical systems
The purpose of this article is to construct a toolbox, in Dynamical Systems,
to support the idea that ``whenever we can prove a limit theorem in the
classical sense for a dynamical system, we can prove a suitable almost-sure
version based on an empirical measure with log-average''. We follow three
different approaches: martingale methods, spectral methods and induction
arguments. Our results apply among others to Axiom A maps or flows, to systems
inducing a Gibbs-Markov map and to the stadium billiard.Comment: 41 pages; submitted v2: replaced the argument for Gibbs-Markov maps
with a general spectral argumen
Testing the irreversibility of a Gibbsian process via hitting and return times
We introduce estimators for the entropy production of a Gibbsian process
based on the observation of a single or two typical trajectories. These
estimators are built with adequate hitting and return times. We then study
their convergence and fluctuation properties. This provides statisticals test
for the irreversibility of Gibbsian processes.Comment: 16 pages; Corrected version; To appear in Nonlinearit
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