# American Institute of Mathematical Sciences

2018, 13: 271-284. doi: 10.3934/jmd.2018021

## Decomposition of infinite-to-one factor codes and uniqueness of relative equilibrium states

 Department of Mathematics, Sungkyunkwan University, Suwon 440-746, Republic of Korea

Received  April 30, 2017 Revised  November 22, 2017 Published  December 2018

Fund Project: Supported by the National Research Foundation of Korea (NRF) grant funded by the MEST 2015R1A3A2031159

We show that an arbitrary factor map $\pi :X \to Y$ on an irreducible subshift of finite type is a composition of a finite-to-one factor code and a class degree one factor code. Using this structure theorem on infinite-to-one factor codes, we then prove that any equilibrium state $\nu$ on $Y$ for a potential function of sufficient regularity lifts to a unique measure of maximal relative entropy on $X$. This answers a question raised by Boyle and Petersen (for lifts of Markov measures) and generalizes the earlier known special case of finite-to-one factor codes.

Citation: Jisang Yoo. Decomposition of infinite-to-one factor codes and uniqueness of relative equilibrium states. Journal of Modern Dynamics, 2018, 13: 271-284. doi: 10.3934/jmd.2018021
##### References:
 [1] M. Allahbakhshi, J. Antonioli and J. Yoo, Relative equilibrium states and class degree, Ergodic Theory Dynam. Systems, accepted, (2017). doi: 10.1017/etds.2017.50. [2] M. Allahbakhshi and A. Quas, Class degree and relative maximal entropy, Trans. Amer. Math. Soc., 365 (2013), 1347-1368. doi: 10.1090/S0002-9947-2012-05637-6. [3] M. Allahbakhshi, S. Hong and U. Jung, Structure of transition classes for factor codes on shifts of finite type, Ergodic Theory and Dynamical Systems, 35 (2015), 2353-2370. doi: 10.1017/etds.2014.39. [4] J. Antonioli, Compensation functions for factors of shifts of finite type, Ergodic Theory and Dynamical Systems, 36 (2016), 375-389. doi: 10.1017/etds.2014.63. [5] R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Mathematics, 470, Springer, Berlin, 1975. [6] M. Boyle and K. Petersen, Hidden Markov processes in the context of symbolic dynamics, in Entropy of Hidden Markov Processes and Connections to Dynamical Systems, London Mathematical Society Lecture Note Series, 385, Cambridge, 2011, 5-71. [7] M. Boyle and S. Tuncel, Infinite-to-one codes and Markov measures, Trans. Amer. Math. Soc., 285 (1984), 657-684. doi: 10.1090/S0002-9947-1984-0752497-0. [8] D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511626302. [9] K. Petersen, A. Quas and S. Shin, Measures of maximal relative entropy, Ergodic Theory Dynam. Systems, 23 (2003), 207-223. doi: 10.1017/S0143385702001153. [10] S. Tuncel, Conditional pressure and coding, Israel Journal of Mathematics, 39 (1981), 101-112. doi: 10.1007/BF02762856. [11] J. Yoo, Measures of maximal relative entropy with full support, Ergodic Theory Dynam. Systems, 31 (2010), 1889-1899. doi: 10.1017/S0143385710000581.

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##### References:
 [1] M. Allahbakhshi, J. Antonioli and J. Yoo, Relative equilibrium states and class degree, Ergodic Theory Dynam. Systems, accepted, (2017). doi: 10.1017/etds.2017.50. [2] M. Allahbakhshi and A. Quas, Class degree and relative maximal entropy, Trans. Amer. Math. Soc., 365 (2013), 1347-1368. doi: 10.1090/S0002-9947-2012-05637-6. [3] M. Allahbakhshi, S. Hong and U. Jung, Structure of transition classes for factor codes on shifts of finite type, Ergodic Theory and Dynamical Systems, 35 (2015), 2353-2370. doi: 10.1017/etds.2014.39. [4] J. Antonioli, Compensation functions for factors of shifts of finite type, Ergodic Theory and Dynamical Systems, 36 (2016), 375-389. doi: 10.1017/etds.2014.63. [5] R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Mathematics, 470, Springer, Berlin, 1975. [6] M. Boyle and K. Petersen, Hidden Markov processes in the context of symbolic dynamics, in Entropy of Hidden Markov Processes and Connections to Dynamical Systems, London Mathematical Society Lecture Note Series, 385, Cambridge, 2011, 5-71. [7] M. Boyle and S. Tuncel, Infinite-to-one codes and Markov measures, Trans. Amer. Math. Soc., 285 (1984), 657-684. doi: 10.1090/S0002-9947-1984-0752497-0. [8] D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511626302. [9] K. Petersen, A. Quas and S. Shin, Measures of maximal relative entropy, Ergodic Theory Dynam. Systems, 23 (2003), 207-223. doi: 10.1017/S0143385702001153. [10] S. Tuncel, Conditional pressure and coding, Israel Journal of Mathematics, 39 (1981), 101-112. doi: 10.1007/BF02762856. [11] J. Yoo, Measures of maximal relative entropy with full support, Ergodic Theory Dynam. Systems, 31 (2010), 1889-1899. doi: 10.1017/S0143385710000581.
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