Equilibrium measures for some partially hyperbolic systems
Vaughn Climenhaga, Yakov Pesin, Agnieszka Zelerowicz
Abstract
We study thermodynamic formalism for topologically transitive partially hyperbolic systems in which the center-stable bundle satisfies a bounded expansion property, and show that every potential function satisfying the Bowen property has a unique equilibrium measure. Our method is to use tools from geometric measure theory to construct a suitable family of reference measures on unstable leaves as a dynamical analogue of Hausdorff measure, and then show that the averaged pushforwards of these measures converge to a measure that has the Gibbs property and is the unique equilibrium measure.
Topics & Concepts
MathematicsBounded functionTransitive relationMeasure (data warehouse)Hausdorff measurePure mathematicsGibbs measureIterated function systemProperty (philosophy)Dynamical systems theoryBundleHausdorff dimensionBounded variationEntropy (arrow of time)Thermodynamic equilibriumHausdorff spaceHölder conditionMathematical analysisMeasurable functionUniform boundednessFormalism (music)Open setFunction (biology)Hyperbolic setDynamical system (definition)Carnot cycleMathematical Dynamics and FractalsTheoretical and Computational PhysicsControl and Stability of Dynamical Systems