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Factor map, diamond and density of pressure functions

Authors: Jung-Chao Ban and Chih-Hung Chang
Journal: Proc. Amer. Math. Soc. 139 (2011), 3985-3997
MSC (2010): Primary 37D35; Secondary 37B10, 37A35, 28A78
Published electronically: March 17, 2011
MathSciNet review: 2823044
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Abstract: Letting $ \pi:X\rightarrow Y$ be a one-block factor map and $ \Phi$ be an almost-additive potential function on $ X,$ we prove that if $ \pi$ has diamond, then the pressure $ P(X,\Phi)$ is strictly larger than $ P(Y,\pi\Phi)$. Furthermore, if we define the ratio $ \rho(\Phi)=P(X,\Phi)/P(Y,\pi\Phi)$, then $ \rho(\Phi)>1$ and it can be proved that there exists a family of pairs $ \left\{ (\pi_{i},X_{i})\right\} _{i=1}^{k}$ such that $ \pi_{i}:X_{i} \rightarrow Y$ is a factor map between $ X_{i}$ and $ Y$, $ X_{i}\subseteq X$ is a subshift of finite type such that $ \rho(\pi_{i},\Phi\vert _{X_{i}})$ (the ratio of the pressure function for $ P(X_{i},\Phi \vert _{X_{i}})$ and $ P(Y,\pi\Phi)$) is dense in $ [1,\rho(\Phi)]$. This extends the result of Quas and Trow for the entropy case.

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Additional Information

Jung-Chao Ban
Affiliation: Department of Mathematics, National Dong Hwa University, Hualien 970003, Taiwan

Chih-Hung Chang
Affiliation: Department of Mathematics, National Central University, Taoyuan 32001, Taiwan

Keywords: Factor map, diamond, $\mathbf{a}$-weighted thermodynamic formalism, density of pressure
Received by editor(s): May 3, 2010
Received by editor(s) in revised form: September 19, 2010
Published electronically: March 17, 2011
Additional Notes: The first author is partially supported by the National Science Council, ROC (Contract No. NSC 98-2628-M-259-001), National Center for Theoretical Sciences (NCTS) and CMPT (Center for Mathematics and Theoretical Physics) in National Central University.
The second author wishes to express his gratitude to Professor Cheng-Hsiung Hsu for his valuable comments and thanks the National Central University for financial support.
Communicated by: Yingfei Yi
Article copyright: © Copyright 2011 American Mathematical Society

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