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August  2010, 27(3): 995-1005. doi: 10.3934/dcds.2010.27.995

A note on two approaches to the thermodynamic formalism

Vaughn Climenhaga 1,

1. 

Department of Mathematics, McAllister Building, Pennsylvania State University, University Park, PA 16802

Received  February 2009 Revised  January 2010 Published  March 2010

Inducing schemes provide a means of using symbolic dynamics to study equilibrium states of non-uniformly hyperbolic maps, but necessitate a solution to the liftability problem. One approach, due to Pesin and Senti, places conditions on the induced potential under which a unique equilibrium state exists among liftable measures, and then solves the liftability problem separately. Another approach, due to Bruin and Todd, places conditions on the original potential under which both problems may be solved simultaneously. These conditions include a bounded range condition, first introduced by Hofbauer and Keller. We compare these two sets of conditions and show that for many inducing schemes of interest, the conditions from the second approach are strictly stronger than the conditions from the first. We also show that the bounded range condition can be used to obtain Pesin and Senti's conditions for any inducing scheme with sufficiently slow growth of basic elements.
Keywords: Thermodynamic formalism, equilibrium states, inducing schemes, multimodal maps., unimodal maps.
Mathematics Subject Classification: Primary: 37D25, 37D35; Secondary: 37E05, 37E1.
Citation: Vaughn Climenhaga. A note on two approaches to the thermodynamic formalism. Discrete & Continuous Dynamical Systems, 2010, 27 (3) : 995-1005. doi: 10.3934/dcds.2010.27.995
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