A scaling entropy sequence of an automorphism is an entropy-type metric invariant suggested by A. M. Vershik. We confirm his conjecture that it does not depend on the choice of a semimetric. This means that it is indeed a metric invariant. We also calculate this invariant for several classical dynamical systems.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. N. Kolmogorov, “A new metric invariant of transitive dynamical systems and automorphisms of Lebesgue spaces,” Dokl. Akad. Nauk SSSR, 119, 861–864 (1958).
A. N. Kolmogorov, “Entropy per unit time as a metric invariant of automorphisms,” Dokl. Akad. Nauk SSSR, 124, 754–755 (1959).
Ya. G. Sinai, “On the concept of entropy for a dynamic system,” Dokl. Akad. Nauk SSSR, 124, 768–771 (1959).
V. A. Rokhlin, “Lectures on the entropy theory of measure-preserving transformations,” Russian Math. Surveys, 22, No. 5, 1–52 (1967).
S. D. Ornstein, Ergodic Theory, Randomness, and Dynamical Systems, Yale Univ. Press, New Haven–London (1974).
I. P. Kornfeld, Ya. G. Sinai, and S. V. Fomin, Ergodic Theory, Nauka, Moscow (1980).
A. M. Vershik, “Information, entropy, dynamics,” in: Mathematics of the XXth Century: A View from St.Petersburg [in Russian], MCCME, Moscow (2010), pp. 47–76.
A. M. Vershik, “Dynamics of metrics in measure spaces and their asymptotic invariants,” Markov Process. Related Fields, 16, No. 1, 169–184 (2010).
A. M. Vershik, “Scailing entropy and automorphisms with pure point spectrum,” St. Petersburg Math. J., 23, No. 1, 75–91 (2011).
S. Ferenczi, “Measure-theoretic complexity of ergodic systems,” Israel J. Math., 100, 189–207 (1997).
A. Katok and J.-P. Thouvenot, “Slow entropy type invariants and smooth realization of commuting measure-preserving transformations,” Ann. Inst. H. Poincaré Probab. Statist., 33, No. 3, 323–338 (1997).
A. M. Vershik, P. B. Zatitskiy, and F. V. Petrov, “Geometry and dynamics of admissible metrics in measure spaces,” Cent. Eur. J. Math., 11, No. 3, 379–400 (2013).
P. B. Zatitskiy and F. V. Petrov, “Correction of metrics,” J. Math. Sci., 181, No. 6, 867–870 (2012).
A. M. Vershik, P. B. Zatitskiy, and F. V. Petrov, “Virtual continuity of measurable functions and its applications,” Uspekhi Mat. Nauk, 69, No. 6, 81–114 (2014).
P. B. Zatitskiy, “On a scaling entropy sequence of a dynamical system,” Funct. Anal. Appl., 48, No. 4, 291–294 (2014).
V. A. Rokhlin, “On the fundamental ideas of measure theory,” Mat. Sb., 25, No. 67, 107–150 (1949).
A. M. Vershik and A. D. Gorbulsky, “Scaled entropy of filtrations of σ-fields,” Theory Probab. Appl., 52, No. 3, 493–508 (2007).
M. Queffélec, Substitution Dynamical Systems. Spectral Analysis, 2nd edition, Springer-Verlag, Berlin (2010).
K. Teturo, “A topological invariant of substitution minimal sets,” J. Math. Soc. Japan, 24, 285–306 (1972).
J. C. Martin, “Substitution minimal flows,” Amer. J. Math., 93, 503–526 (1971).
F. M. Dekking, “The spectrum of dynamical systems arising from substitutions of constant length,” Z. Wahrscheinlichkeitstheor. Verw. Geb., 41, No. 3, 221–239 (1977/78).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 432, 2015, pp. 128–161.
Rights and permissions
About this article
Cite this article
Zatitskiy, P.B. Scaling Entropy Sequence: Invariance and Examples. J Math Sci 209, 890–909 (2015). https://doi.org/10.1007/s10958-015-2536-9
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-015-2536-9