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Equilibriums of some non-Hölder potentials


Author: Huyi Hu
Journal: Trans. Amer. Math. Soc. 360 (2008), 2153-2190
MSC (2000): Primary 37C40, 37A60; Secondary 28D05
DOI: https://doi.org/10.1090/S0002-9947-07-04412-1
Published electronically: October 22, 2007
MathSciNet review: 2366978
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Abstract: We consider one-sided subshifts $ \sigma $ with some potential functions $ \varphi $ which satisfy the Hölder condition everywhere except at a fixed point and its preimages. We prove that the systems have conformal measures $ \nu $ and invariant measures $ \mu $ absolutely continuous with respect to $ \nu $, where $ \mu $ may be finite or infinite. We show that the systems $ (\sigma , \mu )$ are exact, and $ \mu $ are weak Gibbs measures and equilibriums for $ \varphi $. We also discuss uniqueness of equilibriums and phase transition.

These results can be applied to some expanding dynamical systems with an indifferent fixed point.


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Additional Information

Huyi Hu
Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824
Email: hu@math.msu.edu

DOI: https://doi.org/10.1090/S0002-9947-07-04412-1
Keywords: Potential, equilibrium, invariant measure, exactness, ergodicity, weak Gibbs measure
Received by editor(s): January 9, 2006
Received by editor(s) in revised form: June 10, 2006
Published electronically: October 22, 2007
Additional Notes: Part of this work was done when the author was at Penn State University and the University of Southern California. This work was supported by NSF under grants DMS-9970646 and DMS-0240097.
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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