168 research outputs found
Exchangeable measures for subshifts
Let \Om be a Borel subset of where is countable. A measure
is called exchangeable on \Om, if it is supported on \Om and is invariant
under every Borel automorphism of \Om which permutes at most finitely many
coordinates. De-Finetti's theorem characterizes these measures when \Om=S^\Bbb
N. We apply the ergodic theory of equivalence relations to study the case
\Om\neq S^\Bbb N, and obtain versions of this theorem when \Om is a
countable state Markov shift, and when \Om is the collection of beta
expansions of real numbers in (a non-Markovian constraint)
Natural equilibrium states for multimodal maps
This paper is devoted to the study of the thermodynamic formalism for a class
of real multimodal maps. This class contains, but it is larger than,
Collet-Eckmann. For a map in this class, we prove existence and uniqueness of
equilibrium states for the geometric potentials , for the largest
possible interval of parameters . We also study the regularity and convexity
properties of the pressure function, completely characterising the first order
phase transitions. Results concerning the existence of absolutely continuous
invariant measures with respect to the Lebesgue measure are also obtained
Renormalisation-induced phase transitions for unimodal maps
The thermodynamical formalism is studied for renormalisable maps of the
interval and the natural potential . Multiple and indeed
infinitely many phase transitions at positive can occur for some quadratic
maps. All unimodal quadratic maps with positive topological entropy exhibit a
phase transition in the negative spectrum.Comment: 14 pages, 2 figures. Revised following comments of referees. First
page is blan
Phase transitions for suspension flows
This paper is devoted to study thermodynamic formalism for suspension flows
defined over countable alphabets. We are mostly interested in the regularity
properties of the pressure function. We establish conditions for the pressure
function to be real analytic or to exhibit a phase transition. We also
construct an example of a potential for which the pressure has countably many
phase transitions.Comment: Example 5.2 expanded. Typos corrected. Section 6.1 superced the note
"Thermodynamic formalism for the positive geodesic flow on the modular
surface" arXiv:1009.462
Equilibrium states and invariant measures for random dynamical systems
Random dynamical systems with countably many maps which admit countable
Markov partitions on complete metric spaces such that the resulting Markov
systems are uniformly continuous and contractive are considered. A
non-degeneracy and a consistency conditions for such systems, which admit some
proper Markov partitions of connected spaces, are introduced, and further
sufficient conditions for them are provided. It is shown that every uniformly
continuous Markov system associated with a continuous random dynamical system
is consistent if it has a dominating Markov chain. A necessary and sufficient
condition for the existence of an invariant Borel probability measure for such
a non-degenerate system with a dominating Markov chain and a finite (16) is
given. The condition is also sufficient if the non-degeneracy is weakened with
the consistency condition. A further sufficient condition for the existence of
an invariant measure for such a consistent system which involves only the
properties of the dominating Markov chain is provided. In particular, it
implies that every such a consistent system with a finite Markov partition and
a finite (16) has an invariant Borel probability measure. A bijective map
between these measures and equilibrium states associated with such a system is
established in the non-degenerate case. Some properties of the map and the
measures are given.Comment: The article is published in DCDS-A, but without the 3rd paragraph on
page 4 (the complete removal of the paragraph became the condition for the
publication in the DCDS-A after the reviewer ran out of the citation
suggestions collected in the paragraph
Equilibrium states for potentials with \sup\phi - \inf\phi < \htop(f)
In the context of smooth interval maps, we study an inducing scheme approach
to prove existence and uniqueness of equilibrium states for potentials
with he `bounded range' condition \sup \phi - \inf \phi < \htop, first used
by Hofbauer and Keller. We compare our results to Hofbauer and Keller's use of
Perron-Frobenius operators. We demonstrate that this `bounded range' condition
on the potential is important even if the potential is H\"older continuous. We
also prove analyticity of the pressure in this context.Comment: Added Lemma 6 to deal with the disparity between leading eigenvalues
and operator norms. Added extra references and corrected some typo
On conformal measures and harmonic functions for group extensions
We prove a Perron-Frobenius-Ruelle theorem for group extensions of
topological Markov chains based on a construction of -finite conformal
measures and give applications to the construction of harmonic functions.Comment: To appear in Proceedings of "New Trends in Onedimensional Dynamics,
celebrating the 70th birthday of Welington de Melo
The Lyapunov spectrum is not always concave
We characterize one-dimensional compact repellers having nonconcave Lyapunov
spectra. For linear maps with two branches we give an explicit condition that
characterizes non-concave Lyapunov spectra
Statistical stability of equilibrium states for interval maps
We consider families of multimodal interval maps with polynomial growth of
the derivative along the critical orbits. For these maps Bruin and Todd have
shown the existence and uniqueness of equilibrium states for the potential
, for close to 1. We show that these
equilibrium states vary continuously in the weak topology within such
families. Moreover, in the case , when the equilibrium states are
absolutely continuous with respect to Lebesgue, we show that the densities vary
continuously within these families.Comment: More details given and the appendices now incorporated into the rest
of the pape
Almost-additive thermodynamic formalism for countable Markov shifts
We introduce a definition of pressure for almost-additive sequences of
continuous functions defined over (non-compact) countable Markov shifts. The
variational principle is proved. Under certain assumptions we prove the
existence of Gibbs and equilibrium measures. Applications are given to the
study of maximal Lypaunov exponents of product of matrices
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