Abstract
In order to characterize the complexity of a system with zero entropy we introduce the notions of scaled topological and metric entropies. We allow asymptotic rates of the general form \(e^{\alpha a(n)}\) determined by an arbitrary monotonically increasing “scaling” sequence \(a(n)\). This covers the standard case of exponential scale corresponding to \(a(n)=n\) as well as the cases of zero and infinite entropy. We describe some basic properties of the scaled entropy including the inverse variational principle for the scaled metric entropy. Furthermore, we present some examples from symbolic and smooth dynamics that illustrate that systems with zero entropy may still exhibit various levels of complexity.
Similar content being viewed by others
References
Ahn, Y., Dou, D., Park, K.: Entropy dimension and variational principle. Stud. Math. 199(3), 295–309 (2010)
Brin, M., Katok, A.: On local entropy. In: Geometric Dynamics, Rio de Janeiro, 1981. Lecture notes in mathematics, vol. 1007, pp. 30–38. Springer, Berlin (1983)
Carvalho, M.: Entropy dimension of dynamical systems. Port. Math. 54(1), 19–40 (1997)
Cassaigne, J.: Constructing infinite words of intermediate complexity. In: Developments in Language Theory. Lecture notes in computer science, vol. 2450, pp. 173–184. Springer, Berlin (2013).
Cheng, W., Li, B.: Zero entropy systems. J. Stat. Phys. 140(5), 1006–1021 (2010)
Dou, D., Huang, W., Park, K.: Entropy dimension of topological dynamics. Trans. Am. Math. Soc. 363, 659–680 (2011)
Dou, D., Huang, W., Park, K.: Entropy dimension of measure preserving systems. ar**v:1312.7225 (2013).
Ferenczi, S., Park, K.: Entropy dimensions and a class of constructive examples. Discret. Contin. Dyn. Syst. 17(1), 133–141 (2007)
Fomenko, A.T.: Integrability and Nonintegrability in Geometry and Mechanics, Mathematics and Its Applications (Soviet Series), vol. 31. Kluwer, Dordrecht (1988)
Kim, D., Park, K.: The first return time properties of an irrational rotation. Proc. Am. Math. Soc. 136(11), 3941–3951 (2008)
Kuang, R., Cheng, W., Li, B.: Fractal entropy of nonautonomous systems. Pac. J. Math. 262(2), 421–436 (2013)
Labrousse, C.: Flat metrics are strict local minimizers for the polynomial entropy. Regul. Chaotic Dyn. 17(6), 479–491 (2012)
Labrousse, C., Marco, J. P.: Polynomial entropies for Bott non-degenerate Hamiltonian systems. ar**v:1207.4937. (2012)
Ma, D., Kuang, R., Li, B.: Topological entropy dimension for noncompact sets. Dyn. Syst. 27(3), 303–316 (2012)
Ma, J., Wen, Z.: A Billingsley type theorem for Bowen entropy. C. R. Acad. Sci. Paris Ser. I 346, 503–507 (2008)
Marco, J.P.: Obstructions topologiques à l’intégrabilité des flots géodésiques en classe de Bott. Bull. Sci. Math. 117(2), 185–209 (1993)
Milnor, J.: On the entropy geometry of cellular automata. Complex Syst. 2, 357–385 (1988)
Pesin, Y.: Dimension Theory in Dynamical Systems, Contemporary Views and Applications. University of Chicago Press, Chicago (1997)
Young, L.S.: Dimension, entropy and Lyapunov exponents. Ergod. Theory Dyn. Syst. 2, 109–129 (1982)
Acknowledgments
Yun Zhao is partially supported by CSC and NSFC 11371271. Yakov Pesin is partially supported by NSF Grant Nos. 1101165 and 1400027.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zhao, Y., Pesin, Y. Scaled Entropy for Dynamical Systems. J Stat Phys 158, 447–475 (2015). https://doi.org/10.1007/s10955-014-1133-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-014-1133-5