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Rank-one ℤd actions and directional entropy

Published online by Cambridge University Press:  11 December 2009

E. ARTHUR ROBINSON JR  and
AYŞE A. ŞAHİN
E. ARTHUR ROBINSON JR
Affiliation:
Department of Mathematics, George Washington University, 2115 G St. NW, Washington, DC 20052, USA (email: robinson@gwu.edu)
AYŞE A. ŞAHİN
Affiliation:
Department of Mathematical Sciences, DePaul University, 2320 North Kenmore Ave., Chicago, IL 60614, USA (email: asahin@depaul.edu)
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Abstract

We study the dynamic properties of rank-one ℤd actions as a function of the geometry of the shapes of the towers generating the action. Some basic properties require only minimal restrictions on the geometry of the towers. Our main results concern the directional entropy of rank-one ℤd actions with rectangular tower shapes, where we show that the geometry of the rectangles plays a significant role. We show that for each n≤d there is an n-dimensional direction with entropy zero. We also show that if the growth in eccentricity of the rectangular towers is sub-exponential, then all directional entropies are zero. An example of D. Rudolph shows that, without a restriction on eccentricity, a positive entropy direction is possible.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 31 , Issue 1 , February 2011 , pp. 285 - 299
Copyright
Copyright © Cambridge University Press 2009

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References

[1]Baxter, J. R.. A class of ergodic transformations having simple spectrum. Proc. Amer. Math. Soc. 27 (1971), 275–279.CrossRefGoogle Scholar
[2]Boyle, M. and Lind, D.. Expansive subdynamics. Trans. Amer. Math. Soc. 349(1) (1997), 55–102.CrossRefGoogle Scholar
[3]Chacon, R. V.. A geometric construction of measure preserving transformations. Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, CA, 1965/66), Vol. II: Contributions to Probability Theory, Part 2. University of California Press, Berkeley, CA, 1967, pp. 335–360.Google Scholar
[4]Danilenko, A. I. and Silva, C. E.. Mixing rank-one actions of locally compact abelian groups. Ann. Inst. H. Poincaré Probab. Statist. 43(4) (2007), 375–398.CrossRefGoogle Scholar
[5]Ferenczi, S.. Systèmes de rang un gauche. Ann. Inst. H. Poincaré Probab. Statist. 21(2) (1985), 177–186.Google Scholar
[6]Ferenczi, S.. Systems of finite rank. Colloq. Math. 73(1) (1997), 35–65.CrossRefGoogle Scholar
[7]Friedman, N. A.. Introduction to Ergodic Theory (Van Nostrand Reinhold Mathematical Studies, 29). Van Nostrand Reinhold, New York, 1970.Google Scholar
[8]Johnson, A. S. A. and Şahin, A. A.. Rank one and loosely Bernoulli actions in Zd. Ergod. Th. & Dynam. Sys. 18(5) (1998), 1159–1172.CrossRefGoogle Scholar
[9]Kamiński, B. and Park, K. K.. On the directional entropy for Z2-actions on a Lebesgue space. Studia Math. 133(1) (1999), 39–51.Google Scholar
[10]Keane, M. and Yassawi, R.. Personal communication.Google Scholar
[11]King, J.. The commutant is the weak closure of the powers, for rank-one transformations. Ergod. Th. & Dynam. Sys. 6(3) (1986), 363–384.CrossRefGoogle Scholar
[12]Milnor, J.. Directional entropies of cellular automaton-maps. Disordered Systems and Biological Organization (Les Houches, 1985) (NATO Advanced Science Institutes Series F: Computer and Systems Sciences). Springer, Berlin, 1986, pp. 113–115.CrossRefGoogle Scholar
[13]Milnor, J.. On the entropy geometry of cellular automata. Complex Systems 2(3) (1988), 357–385.Google Scholar
[14]Ornstein, D. S. and Weiss, B.. Entropy and isomorphism theorems for actions of amenable groups. J. Anal. Math. 48 (1987), 1–141.CrossRefGoogle Scholar
[15]Ornstein, D. S.. On the root problem in ergodic theory. Proc. Sixth Berkeley Sympos. Math. Statist. and Probability (Berkeley, CA, 1970/1971), Vol. II: Probability Theory. University of California Press, Berkeley, CA, 1972, pp. 347–356.Google Scholar
[16]Ornstein, D. S., Rudolph, D. J. and Weiss, B.. Equivalence of measure preserving transformations. Mem. Amer. Math. Soc. 37(262) (1982).Google Scholar
[17]Park, K. K.. Continuity of directional entropy. Osaka J. Math. 31(3) (1994), 613–628.Google Scholar
[18]Park, K. K. and Robinson, E. A. Jr. The joinings within a class of Z2 actions. J. Anal. Math. 57 (1991), 1–36.CrossRefGoogle Scholar
[19]Park, K. K.. On directional entropy functions. Israel J. Math. 113 (1999), 243–267.CrossRefGoogle Scholar
[20]Robinson, E. A. Jr and Şahin, A. A.. ℤd rotations are square rank 1. Preprint, 2005.Google Scholar
[21]Rudolph, D. J.. The second centralizer of a Bernoulli shift is just its powers. Israel J. Math. 29(2–3) (1978), 167–178.CrossRefGoogle Scholar
[22]Rudolph, D. J.. An example of a measure preserving map with minimal self-joinings, and applications. J. Anal. Math. 35 (1979), 97–122.CrossRefGoogle Scholar
[23]Shields, P. C.. The Ergodic Theory of Discrete Sample Paths (Graduate Studies in Mathematics, 13). American Mathematical Society, Providence, RI, 1996.CrossRefGoogle Scholar
[24]Sinaĭ, Ya. G.. An answer to a question by J. Milnor. Comment. Math. Helv. 60(2) (1985), 173–178.CrossRefGoogle Scholar
[25]Tempelman, A.. Ergodic Theorems for Group Actions: Informational and Thermodynamical Aspects (Mathematics and its Applications, 78). Kluwer Academic Publishers Group, Dordrecht, 1992, translated and revised from the 1986 Russian original.CrossRefGoogle Scholar