Several questions and hypotheses concerning the limit polynomials for Chacon transformation

Abstract

We study the weak closure $L$ of powers $T^k$ of the non-singular Chacon transformation $T$ with 2-cuts. This is still an open question does $L$ contain any Markov operator except an orthogonal projector to the constants $\Theta$ and some polynomials $P(T)$? In this paper we calculate a particular set of limit polynomials $P_m(T)={\lim }_{n\to \infty }{T}^{-m{h}_n}$, where $m$ is a fixed integer number and $h_n=(3^n-1)/2$ are the sequence of heights of towers in a standard rank one representation of the Chacon map. We show that for any $d\ge 2$ the family of limit polynomials contains infinitely many distinct polynomials of degree $d$. We also formulate hyposeses and open questions concerning the sequence $P_m$ and the entire set $L$.

14 pages, 9 figures, 3 tables