Class degree and relative maximal entropy
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- by Mahsa Allahbakhshi and Anthony Quas PDF
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Abstract:
Given a factor code $\pi$ from a one-dimensional shift of finite type $X$ onto an irreducible sofic shift $Y$, if $\pi$ is finite-to-one there is an invariant called the degree of $\pi$ which is defined as the number of preimages of a typical point in $Y$. We generalize the notion of the degree to the class degree which is defined for any factor code on a one-dimensional shift of finite type. Given an ergodic measure $\nu$ on $Y$, we find an invariant upper bound on the number of ergodic measures on $X$ which project to $\nu$ and have maximal entropy among all measures in the fibre $\pi ^{-1}\{\nu \}$. We show that this bound and the class degree of the code agree when $\nu$ is ergodic and fully supported. One of the main ingredients of the proof is a uniform distribution property for ergodic measures of relative maximal entropy.References
- David Blackwell, The entropy of functions of finite-state Markov chains, Transactions of the first Prague conference on information theory, statistical decision functions, random processes held at Liblice near Prague from November 28 to 30, 1956, Publishing House of the Czechoslovak Academy of Sciences, Prague, 1957, pp. 13–20. MR 0100297
- Mike Boyle, Putnam’s resolving maps in dimension zero, Ergodic Theory Dynam. Systems 25 (2005), no. 5, 1485–1502. MR 2173429, DOI 10.1017/S0143385705000106
- Mike Boyle and Selim Tuncel, Infinite-to-one codes and Markov measures, Trans. Amer. Math. Soc. 285 (1984), no. 2, 657–684. MR 752497, DOI 10.1090/S0002-9947-1984-0752497-0
- C. J. Burke and M. Rosenblatt, A Markovian function of a Markov chain, Ann. Math. Statist. 29 (1958), 1112–1122. MR 101557, DOI 10.1214/aoms/1177706444
- Robert Burton and Jeffrey E. Steif, Non-uniqueness of measures of maximal entropy for subshifts of finite type, Ergodic Theory Dynam. Systems 14 (1994), no. 2, 213–235. MR 1279469, DOI 10.1017/S0143385700007859
- Dimitrios Gatzouras and Yuval Peres, The variational principle for Hausdorff dimension: a survey, Ergodic theory of $\textbf {Z}^d$ actions (Warwick, 1993–1994) London Math. Soc. Lecture Note Ser., vol. 228, Cambridge Univ. Press, Cambridge, 1996, pp. 113–125. MR 1411217, DOI 10.1017/CBO9780511662812.004
- Dimitrios Gatzouras and Yuval Peres, Invariant measures of full dimension for some expanding maps, Ergodic Theory Dynam. Systems 17 (1997), no. 1, 147–167. MR 1440772, DOI 10.1017/S0143385797060987
- Robert B. Israel, Convexity in the theory of lattice gases, Princeton Series in Physics, Princeton University Press, Princeton, N.J., 1979. With an introduction by Arthur S. Wightman. MR 517873
- Gerhard Keller, Equilibrium states in ergodic theory, London Mathematical Society Student Texts, vol. 42, Cambridge University Press, Cambridge, 1998. MR 1618769, DOI 10.1017/CBO9781107359987
- O. E. Lanford III and D. Ruelle, Observables at infinity and states with short range correlations in statistical mechanics, Comm. Math. Phys. 13 (1969), 194–215. MR 256687
- François Ledrappier and Peter Walters, A relativised variational principle for continuous transformations, J. London Math. Soc. (2) 16 (1977), no. 3, 568–576. MR 476995, DOI 10.1112/jlms/s2-16.3.568
- F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms. II. Relations between entropy, exponents and dimension, Ann. of Math. (2) 122 (1985), no. 3, 540–574. MR 819557, DOI 10.2307/1971329
- Douglas Lind and Brian Marcus, An introduction to symbolic dynamics and coding, Cambridge University Press, Cambridge, 1995. MR 1369092, DOI 10.1017/CBO9780511626302
- Brian Marcus, Karl Petersen, and Susan Williams, Transmission rates and factors of Markov chains, Conference in modern analysis and probability (New Haven, Conn., 1982) Contemp. Math., vol. 26, Amer. Math. Soc., Providence, RI, 1984, pp. 279–293. MR 737408, DOI 10.1090/conm/026/737408
- William Parry, Intrinsic Markov chains, Trans. Amer. Math. Soc. 112 (1964), 55–66. MR 161372, DOI 10.1090/S0002-9947-1964-0161372-1
- Karl Petersen, Information compression and retention in dynamical processes, Dynamics and randomness (Santiago, 2000) Nonlinear Phenom. Complex Systems, vol. 7, Kluwer Acad. Publ., Dordrecht, 2002, pp. 147–217. MR 1975578, DOI 10.1007/978-94-010-0345-2_{6}
- Karl Petersen, Anthony Quas, and Sujin Shin, Measures of maximal relative entropy, Ergodic Theory Dynam. Systems 23 (2003), no. 1, 207–223. MR 1971203, DOI 10.1017/S0143385702001153
- Daniel J. Rudolph, Fundamentals of measurable dynamics, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1990. Ergodic theory on Lebesgue spaces. MR 1086631
- David Ruelle, Statistical mechanics on a compact set with $Z^{v}$ action satisfying expansiveness and specification, Trans. Amer. Math. Soc. 187 (1973), 237–251. MR 417391, DOI 10.1090/S0002-9947-1973-0417391-6
- C. E. Shannon, A mathematical theory of communication, Bell System Tech. J. 27 (1948), 379–423, 623–656. MR 26286, DOI 10.1002/j.1538-7305.1948.tb01338.x
- Sujin Shin, Measures that maximize weighted entropy for factor maps between subshifts of finite type, Ergodic Theory Dynam. Systems 21 (2001), no. 4, 1249–1272. MR 1849609, DOI 10.1017/S0143385701001584
- Peter Walters, An introduction to ergodic theory, Graduate Texts in Mathematics, vol. 79, Springer-Verlag, New York-Berlin, 1982. MR 648108
- Peter Walters, Relative pressure, relative equilibrium states, compensation functions and many-to-one codes between subshifts, Trans. Amer. Math. Soc. 296 (1986), no. 1, 1–31. MR 837796, DOI 10.1090/S0002-9947-1986-0837796-8
- Peter Walters, Convergence of the Ruelle operator for a function satisfying Bowen’s condition, Trans. Amer. Math. Soc. 353 (2001), no. 1, 327–347. MR 1783787, DOI 10.1090/S0002-9947-00-02656-8
- Peter Walters, Regularity conditions and Bernoulli properties of equilibrium states and $g$-measures, J. London Math. Soc. (2) 71 (2005), no. 2, 379–396. MR 2122435, DOI 10.1112/S0024610704006076
Additional Information
- Mahsa Allahbakhshi
- Affiliation: Department of Mathematics and Statistics, University of Victoria, PO Box 3045 STN CSC, Victoria British Columbia, Canada V8W 3P4
- Address at time of publication: Centro de Modelamiento Matemático, Universidad de Chile, Av. Blanco Encalada 2120 Piso 7, Santiago, Chile
- Anthony Quas
- Affiliation: Department of Mathematics and Statistics, University of Victoria, PO Box 3045 STN CSC, Victoria British Columbia, Canada V8W 3P4
- MR Author ID: 317685
- Received by editor(s): April 28, 2010
- Received by editor(s) in revised form: April 13, 2011
- Published electronically: August 9, 2012
- Additional Notes: The authors thank the referee for detailed and helpful comments.
This research was supported by NSERC and the University of Victoria. - © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 365 (2013), 1347-1368
- MSC (2010): Primary 37B10; Secondary 37A35
- DOI: https://doi.org/10.1090/S0002-9947-2012-05637-6
- MathSciNet review: 3003267