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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
DOI: https://doi.org/10.1090/S0002-9939-2011-10803-7
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 $\rho (\pi _{i},\Phi |_{X_{i}})$ (the ratio of the pressure function for $P(X_{i},\Phi |_{X_{i}})$ 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
MR Author ID: 684625
Email: jcban@mail.ndhu.edu.tw

Chih-Hung Chang
Affiliation: Department of Mathematics, National Central University, Taoyuan 32001, Taiwan
Email: chchang@mx.math.ncu.edu.tw

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