Universal skyscraper templates for infinite measure preserving transformations

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June 2006, 16(2): 343-360. doi: 10.3934/dcds.2006.16.343

Universal skyscraper templates for infinite measure preserving transformations

S. Eigen ^1, , A. B. Hajian ^1, and V. S. Prasad ^2,

1.	Northeastern University, Department of Mathematics, Boston, MA 02115, United States, United States
2.	University of Massachusetts Lowell, Department of Mathematics, One University Avenue, Lowell, MA 01854, United States

Received June 2005 Revised December 2005 Published July 2006

Abstract
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We call an ordered set $\mathbf{c} = (c(i): i \in \mathbb{N})$, of nonnegative extended real numbers $c(i)$, a universal skyscraper template if it is the distribution of first return times for every ergodic measure preserving transformation $T$ of an infinite Lebesgue measure space. If ∑ _i$ c(i)<\infty$, we give a family of examples of ergodic infinite measure preserving transformations that do not admit c as a skyscraper template.
If the distribution $\mathbf{c}$ satisfies $\gcd\{i: c(i) >0 \} = 1 $, and if either of the conditions $c(I) = \infty$ (for some integer $I$), or $i n f_i \{c(i) \} > 0$ is satisfied, then $\mathbf{c}$ is a universal skyscraper template.

Keywords: dyadic odometer., multitower, infinite measure preserving automorphism, weakly wandering, Skyscrapers, aperiodic.

Mathematics Subject Classification: Primary: 37A45, 28D0.

Citation: S. Eigen, A. B. Hajian, V. S. Prasad. Universal skyscraper templates for infinite measure preserving transformations. Discrete & Continuous Dynamical Systems - A, 2006, 16 (2) : 343-360. doi: 10.3934/dcds.2006.16.343

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