Some examples of sequence entropy as an isomorphism invariant
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- by F. M. Dekking PDF
- Trans. Amer. Math. Soc. 259 (1980), 167-183 Request permission
Abstract:
With certain geometrically diverging sequences A and the shift T on dynamical systems arising from substitutions we associate a Markov shift S such that the A-entropy of T equals the usual entropy of S. We present examples to demonstrate the following results. Sequence entropy can distinguish between an invertible ergodic transformation and its inverse. A-entropy does not depend monotonically on A. The variational principle for topological sequence entropy need not hold.References
- Ethan M. Coven and Michael S. Keane, The structure of substitution minimal sets, Trans. Amer. Math. Soc. 162 (1971), 89–102. MR 284995, DOI 10.1090/S0002-9947-1971-0284995-1
- F. M. Dekking, The spectrum of dynamical systems arising from substitutions of constant length, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 41 (1977/78), no. 3, 221–239. MR 461470, DOI 10.1007/BF00534241
- Manfred Denker, Christian Grillenberger, and Karl Sigmund, Ergodic theory on compact spaces, Lecture Notes in Mathematics, Vol. 527, Springer-Verlag, Berlin-New York, 1976. MR 0457675
- Ernst Eberlein, On topological entropy of semigroups of commuting transformations, International Conference on Dynamical Systems in Mathematical Physics (Rennes, 1975) Astérisque, No. 40, Soc. Math. France, Paris, 1976, pp. 17–62. MR 0453976
- F. R. Gantmacher, Matrizenrechnung. II. Spezielle Fragen und Anwendungen, Hochschulbücher für Mathematik, Band 37, VEB Deutscher Verlag der Wissenschaften, Berlin, 1959 (German). MR 0107647
- T. N. T. Goodman, Topological sequence entropy, Proc. London Math. Soc. (3) 29 (1974), 331–350. MR 356009, DOI 10.1112/plms/s3-29.2.331
- W. H. Gottschalk, An irreversible minimal set, Ergodic Theory (Proc. Internat. Sympos., Tulane Univ., New Orleans, La., 1961) Academic Press, New York, 1963, pp. 135–150. MR 0160199 P. Hulse, On the sequence entropy of transformations with quasi-discrete spectrum, University of Warwick, 1978 (preprint).
- M. Keane, Generalized Morse sequences, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 10 (1968), 335–353. MR 239047, DOI 10.1007/BF00531855
- Michael Keane, Strongly mixing $g$-measures, Invent. Math. 16 (1972), 309–324. MR 310193, DOI 10.1007/BF01425715
- John G. Kemeny and J. Laurie Snell, Finite Markov chains, The University Series in Undergraduate Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR 0115196
- Madga Komorníkova and Jozef Komorník, On sequential entropy of measure preserving transformation, Dynamical systems, Vol. I—Warsaw, Astérisque, No. 49, Soc. Math. France, Paris, 1977, pp. 141–144. MR 0486416
- E. Krug and D. Newton, On sequence entropy of automorphisms of a Lebesgue space, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 24 (1972), 211–214. MR 322136, DOI 10.1007/BF00532532
- A. G. Kušnirenko, Metric invariants of entropy type, Uspehi Mat. Nauk 22 (1967), no. 5 (137), 57–65 (Russian). MR 0217257
- Nelson G. Markley, Substitution-like minimal sets, Israel J. Math. 22 (1975), no. 3-4, 332–353. MR 391057, DOI 10.1007/BF02761597
- D. Newton, On sequence entropy. I, II, Math. Systems Theory 4 (1970), 119–125; ibid. 4 (1970), 126–128. MR 274714, DOI 10.1007/BF01691095
- William Parry, Intrinsic Markov chains, Trans. Amer. Math. Soc. 112 (1964), 55–66. MR 161372, DOI 10.1090/S0002-9947-1964-0161372-1
- B. S. Pickel′, Certain properties of $A$-entropy, Mat. Zametki 5 (1969), 327–334 (Russian). MR 241603 J. Kwiatkowski, Isomorphism of some Morse dynamical systems, Nicholas Copernicus University, Torun, 1978 (preprint).
Additional Information
- © Copyright 1980 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 259 (1980), 167-183
- MSC: Primary 28D20
- DOI: https://doi.org/10.1090/S0002-9947-1980-0561831-2
- MathSciNet review: 561831