{"id":{"repo_id":"unt","oai_identifier":"info:ark/67531/metadc4631"},"canonical_url":"https://search.dev.ndltd.org/etd/unt/info:ark/67531/metadc4631","repository":{"repo_id":"unt","name":"University of North Texas","base_url":"https://digital.library.unt.edu/oai/"},"display":{"title":"Thermodynamical Formalism","abstract":"Thermodynamical formalism is a relatively recent area of pure mathematics owing a lot to some classical notions of thermodynamics. On this thesis we state and prove some of the main results in the area of thermodynamical formalism. The first chapter is an introduction to ergodic theory. Some of the main theorems are proved and there is also a quite thorough study of the topology that arises in Borel probability measure spaces. In the second chapter we introduce the notions of topological pressure and measure theoretic entropy and we state and prove two very important theorems, Shannon-McMillan-Breiman theorem and the Variational Principle. Distance expanding maps and their connection with the calculation of topological pressure cover the third chapter. The fourth chapter introduces Gibbs states and the very important Perron-Frobenius Operator. The fifth chapter establishes the connection between pressure and geometry. Topological pressure is used in the calculation of Hausdorff dimensions. Finally the sixth chapter introduces the notion of conformal measures.","abstract_html":"Thermodynamical formalism is a relatively recent area of pure mathematics owing a lot to some classical notions of thermodynamics. On this thesis we state and prove some of the main results in the area of thermodynamical formalism. The first chapter is an introduction to ergodic theory. Some of the main theorems are proved and there is also a quite thorough study of the topology that arises in Borel probability measure spaces. In the second chapter we introduce the notions of topological pressure and measure theoretic entropy and we state and prove two very important theorems, Shannon-McMillan-Breiman theorem and the Variational Principle. Distance expanding maps and their connection with the calculation of topological pressure cover the third chapter. The fourth chapter introduces Gibbs states and the very important Perron-Frobenius Operator. The fifth chapter establishes the connection between pressure and geometry. Topological pressure is used in the calculation of Hausdorff dimensions. Finally the sixth chapter introduces the notion of conformal measures.","abstract_has_math":false,"creators":["Chousionis, Vasileios"],"institution":"University of North Texas","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":["Urbański, Mariusz","Alikakos, Nicholas","Douglass, Matthew"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2004,"date_issued":"2004-08","date_published":"2004-08","updated_at":"2026-07-24T05:35:09Z","subjects":["Measure theory.","Mathematical physics.","Thermodynamics.","dynamical systems","ergodic theory","chaos","conformal measures","topological pressure"],"languages":["English"],"rights":["Public","Copyright","Chousionis, Vasileios","Copyright is held by the author, unless otherwise noted. 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In the second chapter we introduce the notions of topological pressure and measure theoretic entropy and we state and prove two very important theorems, Shannon-McMillan-Breiman theorem and the Variational Principle. Distance expanding maps and their connection with the calculation of topological pressure cover the third chapter. The fourth chapter introduces Gibbs states and the very important Perron-Frobenius Operator. The fifth chapter establishes the connection between pressure and geometry. Topological pressure is used in the calculation of Hausdorff dimensions. 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In the second chapter we introduce the notions of topological pressure and measure theoretic entropy and we state and prove two very important theorems, Shannon-McMillan-Breiman theorem and the Variational Principle. Distance expanding maps and their connection with the calculation of topological pressure cover the third chapter. The fourth chapter introduces Gibbs states and the very important Perron-Frobenius Operator. The fifth chapter establishes the connection between pressure and geometry. Topological pressure is used in the calculation of Hausdorff dimensions. 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