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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: [email protected])
AYŞE A. ŞAHİN
Affiliation:
Department of Mathematical Sciences, DePaul University, 2320 North Kenmore Ave., Chicago, IL 60614, USA (email: [email protected])
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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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