Results 1 - 10 of 99

Existence of Gibbs measures for countable Markov shifts

by Omri Sarig, Communicated Michael H - Proc. Amer. Math. Soc , 2003
"... Abstract. We prove that a potential with summable variations and finite pressure on a topologically mixing countable Markov shift has a Gibbs measure iff the transition matrix satisfies the big images and preimages property. This strengthens a result of D. Mauldin and M. Urbański (2001) who showed t ..."
Abstract - Cited by 56 (5 self) - Add to MetaCart
Abstract. We prove that a potential with summable variations and finite pressure on a topologically mixing countable Markov shift has a Gibbs measure iff the transition matrix satisfies the big images and preimages property. This strengthens a result of D. Mauldin and M. Urbański (2001) who showed that this condition is sufficient. 1.
(Show Context)

A multifractal analysis for Stern-Brocot intervals, continued fractions and Diophantine growth rates

by Marc Kesseböhmer, Bernd O. Stratmann , 2005
"... In this paper we obtain multifractal generalizations of classical results by Lévy and Khintchin in metrical Diophantine approximations and measure theory of continued fractions. We give a complete multifractal analysis for Stern–Brocot intervals, for continued fractions and for certain Diophantine ..."
Abstract - Cited by 33 (10 self) - Add to MetaCart
In this paper we obtain multifractal generalizations of classical results by Lévy and Khintchin in metrical Diophantine approximations and measure theory of continued fractions. We give a complete multifractal analysis for Stern–Brocot intervals, for continued fractions and for certain Diophantine growth rates. In particular, we give detailed discussions of two multifractal spectra closely related to the Farey map and the Gauss map.

EQUILIBRIUM MEASURES FOR MAPS WITH INDUCING SCHEMES

by Yakov Pesin, Samuel Senti , 2008
"... Abstract. We introduce a class of continuous maps f of a compact topological space I admitting inducing schemes and describe the tower constructions associated with them. We then establish a thermodynamic formalism, i.e., describe a class of real-valued potential functions ϕ on I, which admit a uniq ..."
Abstract - Cited by 24 (5 self) - Add to MetaCart
Abstract. We introduce a class of continuous maps f of a compact topological space I admitting inducing schemes and describe the tower constructions associated with them. We then establish a thermodynamic formalism, i.e., describe a class of real-valued potential functions ϕ on I, which admit a unique equilibrium measure µϕ minimizing the free energy for a certain class of invariant measures. We also describe ergodic properties of equilibrium measures including decay of correlation and the central limit theorem. Our results apply to certain maps of the interval with critical points and/or singularities (including some unimodal and multimodal maps) and to potential functions ϕt = −t log |df | with t ∈ (t0, t1) for some t0 < 1 < t1. In the particular case of S-unimodal maps we show that one can choose t0 < 0 and that the class of measures under consideration consists of all invariant Borel probability measures. 1.
(Show Context)

A multifractal formalism for growth rates and applications to geometrically finite Kleinian groups, Ergodic theory and dynamical systems 24

by Marc H. Kesseböhmer, Bernd, O. Stratmann , 2004
"... ABSTRACT. We elaborate thermodynamic and multifractal formalisms for general classes of potential functions and their average growth rates. We then apply these formalisms to certain geometrically finite Kleinian groups which may have parabolic elements of dif-ferent ranks. We show that for these gro ..."
Abstract - Cited by 22 (12 self) - Add to MetaCart
ABSTRACT. We elaborate thermodynamic and multifractal formalisms for general classes of potential functions and their average growth rates. We then apply these formalisms to certain geometrically finite Kleinian groups which may have parabolic elements of dif-ferent ranks. We show that for these groups our revised formalisms give access to a de-scription of the spectrum of ‘homological growth rates ’ in terms of Hausdorff dimension. Furthermore, we derive necessary and sufficient conditions for the existence of ’strong phase transitions’. 1.
(Show Context)

Existence and convergence properties of physical measures for certain dynamical systems with holes

by Henk Bruin, Mark Demers, Ian Melbourne - ERGODIC THEORY DYNAM. SYS , 2010
"... ..."
Abstract - Cited by 20 (5 self) - Add to MetaCart
Abstract not found

Zero temperature limits of Gibbs-equilibrium states for countable alphabet subshifts of finite type

by O. Jenkinson, R. D. Mauldin, M. Urbański - J. Stat. Phys , 2005
"... Let �A be a finitely primitive subshift of finite type on a countable alphabet. For appropriate functions f: �A → IR, the family of Gibbs-equilibrium states (µtf)t � 1 for the functions tf is shown to be tight. Any weak ∗-accumulation ..."
Abstract - Cited by 19 (3 self) - Add to MetaCart
Let �A be a finitely primitive subshift of finite type on a countable alphabet. For appropriate functions f: �A → IR, the family of Gibbs-equilibrium states (µtf)t � 1 for the functions tf is shown to be tight. Any weak ∗-accumulation
(Show Context)

Statistical properties of onedimensional maps with critical points and singularities

by K. Díaz-ordaz, M. P. Holland, S. Luzzatto - Stoch. Dyn
"... Abstract. We prove that a class of one-dimensional maps with an arbitrary number of non-degenerate critical and singular points admits an induced Markov tower with exponential return time asymptotics. In particular the map has an absolutely continuous invariant probability measure with exponential d ..."
Abstract - Cited by 18 (5 self) - Add to MetaCart
Abstract. We prove that a class of one-dimensional maps with an arbitrary number of non-degenerate critical and singular points admits an induced Markov tower with exponential return time asymptotics. In particular the map has an absolutely continuous invariant probability measure with exponential decay of correlations for Hölder observations.
(Show Context)

Thermodynamical formalism associated with inducing schemes for one-dimensional maps.

by Yakov Pesin , Samuel Senti - Mosc. Math. J., , 2005
"... Abstract. For a smooth map f of a compact interval I admitting an inducing scheme we establish a thermodynamical formalism, i.e., describe a class of real-valued potential functions ϕ on I which admit a unique equilibrium measure µϕ. Our results apply to unimodal maps corresponding to a positive Le ..."
Abstract - Cited by 16 (6 self) - Add to MetaCart
Abstract. For a smooth map f of a compact interval I admitting an inducing scheme we establish a thermodynamical formalism, i.e., describe a class of real-valued potential functions ϕ on I which admit a unique equilibrium measure µϕ. Our results apply to unimodal maps corresponding to a positive Lebesgue measure set of parameters in a one-parameter transverse family.
(Show Context)

Ornstein-Zernike Theory for the Finite Range Ising Models above ...

by Massimo Campanino, Dmitry Ioffe, Yvan Velenik , 2001
"... We derive precise Ornstein-Zernike asymptotic formula for the decay of the two-point function (rr0rr)z in the general context of finite range Ising type models on Z a. ..."
Abstract - Cited by 15 (5 self) - Add to MetaCart
We derive precise Ornstein-Zernike asymptotic formula for the decay of the two-point function (rr0rr)z in the general context of finite range Ising type models on Z a.
(Show Context)

Projective metrics and mixing properties on towers

by Véronique Maume-Deschamps - TRANS. AMER. MATH. SOC , 2001
"... We study the decay of correlations for towers. Using Birkhoff’s projective metrics, we obtain a rate of mixing of the form: cn(f, g) ≤ Ctα(n)‖f‖‖g‖1 where α(n) goes to zero in a way related to the asymptotic mass of upper floors, ‖f ‖ is some Lipschitz norm and ‖g‖1 is some L¹ norm. The fact that ..."
Abstract - Cited by 15 (2 self) - Add to MetaCart
We study the decay of correlations for towers. Using Birkhoff’s projective metrics, we obtain a rate of mixing of the form: cn(f, g) ≤ Ctα(n)‖f‖‖g‖1 where α(n) goes to zero in a way related to the asymptotic mass of upper floors, ‖f ‖ is some Lipschitz norm and ‖g‖1 is some L¹ norm. The fact that the dependence on g is given by an L¹ norm is useful to study asymptotic laws of successive entrance times.