A note on two approaches to the thermodynamic formalism
Vaughn Climenhaga
Discrete & Continuous Dynamical Systems - A 2010, 27(3): 995-1005 doi: 10.3934/dcds.2010.27.995
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
Multifractal formalism derived from thermodynamics for general dynamical systems
Vaughn Climenhaga
Electronic Research Announcements 2010, 17(0): 1-11 doi: 10.3934/era.2010.17.1
We show that under quite general conditions, various multifractal spectra may be obtained as Legendre transforms of functions $T$: $ \RR\to \RR$ arising in the thermodynamic formalism. We impose minimal requirements on the maps we consider, and obtain partial results for any continuous map $f$ on a compact metric space. In order to obtain complete results, the primary hypothesis we require is that the functions $T$ be continuously differentiable. This makes rigorous the general paradigm of reducing questions regarding the multifractal formalism to questions regarding the thermodynamic formalism. These results hold for a broad class of measurable potentials, which includes (but is not limited to) continuous functions. Applications include most previously known results, as well as some new ones.
keywords: Birkhoff spectrum dimension spectrum. thermodynamic formalism topological pressure Multifractal analysis
Open problems in the theory of non-uniform hyperbolicity
Yakov Pesin Vaughn Climenhaga
Discrete & Continuous Dynamical Systems - A 2010, 27(2): 589-607 doi: 10.3934/dcds.2010.27.589
This is a survey-type article whose goal is to review some recent developments in studying the genericity problem for non-uniformly hyperbolic dynamical systems with discrete time on compact smooth manifolds. We discuss both cases of systems which are conservative (preserve the Riemannian volume) and dissipative (possess hyperbolic attractors). We also consider the problem of coexistence of hyperbolic and regular behaviour.
keywords: SRB measures $u$-measures. Non-uniform hyperbolicity generic properties mixed hyperbolicity dissipative systems conservative systems

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