Exchangeable measures for subshifts

Let \Om be a Borel subset of $S^\Bbb N$ where $S$ 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 $[0,1]$ (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 $-t \log|Df|$, for the largest possible interval of parameters $t$. 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 $-t \log|Df|$. Multiple and indeed infinitely many phase transitions at positive $t$ 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

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 $\sigma$-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

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 $\phi$ 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

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 $\phi_t:x\mapsto-t\log|Df(x)|$, for $t$ close to 1. We show that these equilibrium states vary continuously in the weak$^*$ topology within such families. Moreover, in the case $t=1$, 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