Thermodynamic formalism for symbolic dynamical systems
Kempton, Thomas, 1985- (2011) Thermodynamic formalism for symbolic dynamical systems. PhD thesis, University of Warwick.
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Official URL: http://webcat.warwick.ac.uk/record=b2521727~S15
We derive results in the ergodic theory of symbolic dynamical systems. Our first result concerns β-expansions of real numbers. We show that for a fixed non-integer β > 1 and a fixed real number x ∈ [0, |β|/β-1], the number of words (x1, ..., xn) that can be extended to β-expansions of x grows at least exponentially in n. Our second result concerns definitions of topological pressure for suspension ows over countable Markov shifts. Previously, topological pressure had been considered for a restricted class of suspension ows upon which the thermodynamic formalism can be well understood using the base transformation. We consider a more general class of suspension ows and show the equivalence of several natural definitions of topological pressure, including a definition analogous to that of Gurevich pressure for a Markov shift. Our third result concerns zero temperature limit laws for countable Markov shifts. We show that for a uniformly locally constant potential f on a topologically mixing countable Markov shift satisfying the big images and preimages property, the equilibrium states μtf associated to the potential tf converge as t tends to infinity. Finally we consider the image under a one-block factor map Π of a Gibbs measure μ supported on a finite alphabet Markov shift. We give sufficient conditions on Π for the image measure Π*(μ) to be a Gibbs measure and discuss regularity properties of the potential associated to Π*(μ) in terms of the regularity of the potential associated to μ.
|Item Type:||Thesis or Dissertation (PhD)|
|Subjects:||Q Science > QA Mathematics|
|Library of Congress Subject Headings (LCSH):||Ergodic theory, Dynamics, Markov processes|
|Institution:||University of Warwick|
|Theses Department:||Mathematics Institute|
|Sponsors:||Engineering and Physical Sciences Research Council (EPSRC)|
|Extent:||vi, 130 leaves|
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