66 research outputs found
Induced topological pressure for countable state Markov shifts
We introduce the notion of induced topological pressure for countable state
Markov shifts with respect to a non-negative scaling function and an arbitrary
subset of finite words. Firstly, the scaling function allows a direct access to
important thermodynamical quantities, which are usually given only implicitly
by certain identities involving the classically defined pressure. In this
context we generalise Savchenko's definition of entropy for special flows to a
corresponding notion of topological pressure and show that this new notion
coincides with the induced pressure for a large class of H\"older continuous
height functions not necessarily bounded away from zero. Secondly, the
dependence on the subset of words gives rise to interesting new results
connecting the Gurevi{\vc} and the classical pressure with exhausting
principles for a large class of Markov shifts. In this context we consider
dynamical group extentions to demonstrate that our new approach provides a
useful tool to characterise amenability of the underlying group structure.Comment: 28 page
Recurrence and pressure for group extensions
We investigate the thermodynamic formalism for recurrent potentials on group
extensions of countable Markov shifts. Our main result characterises recurrent
potentials depending only on the base space, in terms of the existence of a
conservative product measure and a homomorphism from the group into the
multiplicative group of real numbers. We deduce that, for a recurrent potential
depending only on the base space, the group is necessarily amenable. Moreover,
we give equivalent conditions for the base pressure and the skew product
pressure to coincide. Finally, we apply our results to analyse the Poincar\'e
series of Kleinian groups and the cogrowth of group presentations
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
Taxonomies of Model-theoretically Defined Topological Properties
A topological classification scheme consists of two ingredients: (1) an abstract class K of topological spaces; and (2) a taxonomy , i.e. a list of first order sentences, together with a way of assigning an abstract class of spaces to each sentence of the list so that logically equivalent sentences are assigned the same class.K, is then endowed with an equivalence relation, two spaces belonging to the same equivalence class if and only if they lie in the same classes prescribed by the taxonomy. A space X in K is characterized within the classification scheme if whenever Y E K, and Y is equivalent to X, then Y is homeomorphic to X. As prime example, the closed set taxonomy assigns to each sentence in the first order language of bounded lattices the class of topological spaces whose lattices of closed sets satisfy that sentence. It turns out that every compact two-complex is characterized via this taxonomy in the class of metrizable spaces, but that no infinite discrete space is so characterized. We investigate various natural classification schemes, compare them, and look into the question of which spaces can and cannot be characterized within them
Entropy sensitivity of languages defined by infinite automata, via Markov chains with forbidden transitions
A language L over a finite alphabet is growth-sensitive (or entropy
sensitive) if forbidding any set of subwords F yields a sub-language L^F whose
exponential growth rate (entropy) is smaller than that of L. Let (X, E, l) be
an infinite, oriented, labelled graph. Considering the graph as an (infinite)
automaton, we associate with any pair of vertices x,y in X the language
consisting of all words that can be read as the labels along some path from x
to y. Under suitable, general assumptions we prove that these languages are
growth-sensitive. This is based on using Markov chains with forbidden
transitions.Comment: to appear in Theoretical Computer Science, 201
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
Lelek's problem is not a metric problem
We show that Lelek's problem on the chainability of continua with span zero
is not a metric problem: from a non-metric counterexample one can construct a
metric one.Comment: Final version as sent to edito
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
- …