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# Universal skyscraper templates for infinite measure preserving transformations

• 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.
Mathematics Subject Classification: Primary: 37A45, 28D05.

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