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Doctoral Thesis
DOI
10.11606/T.45.2017.tde-25112016-214355
Document
Author
Full name
Victor Andres Vargas Cubides
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Paulo, 2015
Supervisor
Committee
Freire Junior, Ricardo dos Santos (President)
Garibaldi, Eduardo
Leplaideur, Renaud Daniel Jacques
Lopes, Artur Oscar
Proença, Rodrigo Bissacot
Title in Portuguese
Sobre existência de estados de equilíbrio e limite em temperatura zero para shifts de Markov topologicamente mixing
Keywords in Portuguese
Estados de equilíbrio
Estados de Gibbs
Limite em temperatura zero
Medidas maximizantes
Potenciais de Markov
Potenciais somáveis
Subshifts de Markov
Abstract in Portuguese
O objetivo desta tese é demonstrar que para um subshift de Markov topologicamente transitivo com alfabeto enumerável e um potencial ƒ com pressão de Gurevic finita e variação limitada (ƒ) < ∞, existe um único estado de equilíbrio µtƒ para cada t > 1, e a família (µtƒ)t>1 tem um ponto de acumulação quando t > ∞. Além disso se também supomos que o ƒ é um potencial de Markov, demonstramos que a família de estados de equilíbrio (µtƒ)t>1 converge quando t > ∞. Finalmente demonstramos a continuidade em ∞ da entropia com respeito ao parâmetro t. Estes resultados não dependem da hipótese de existência de medidas de Gibbs.
Title in English
On equilibrium states existence and zero temperature limit for topologically mixing Markov shifts.
Keywords in English
Equilibrium states
Gibbs states
Markov potentials
Markov subshifts
Maximizing measures
Summable potentials
Zero temperature limit
Abstract in English
The aim of this thesis is to prove that for a topologically transitive Markov subshift with countable alphabet and a summable potential ƒ with finite topological pressure Gurevic and bounded variation (ƒ) < ∞, there exists an equilibrium state µtƒ tf for each t > 1 and the family of equilibrium states (µtƒ)t>1 associated to each potential tf has an accumulation point at t > ∞. Moreover if we also assume that ƒ is a Markov potential we prove that the equilibrium states family (µtƒ)t>1 converges when t > ∞. Finally we prove the continuity at ∞ of the entropy with respect to the parameter t. These results do not depend on assuming the existence of Gibbs measures.
 
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Tese.pdf (849.26 Kbytes)
Publishing Date
2017-04-03
 
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