On ergodic properties of iceberg transformations. I: Approximation and spectral multiplicity

Abstract

We investigate a class of mixing dynamical systems around the concept of iceberg transformation. In brief, an iceberg transformation is defined using symbolic language as follows. We build a sequence of words such that the next word is a concatenation of rotated copies of the previous word. For example, a word CAT can turn into CAT.ATC.TCA.TCA.CAT.ATC, then we repeat the procedure applying it to this new word and so on. Geometrically, given an invertible measure preserving transformation $T$ an iceberg is a union of two icelets for the map $T$, one direct and one reverse with common base set, where icelet is defined in a similar way as Rokhlin tower $B \sqcup TB \sqcup \ldots \sqcup T^{h-1}B$, namely, an icelet is a sequence of disjoint measurable sets $\{B_0, B_1, \ldots, B_{h-1}\}$ such that the levels are nested: $B_{j+1} \subseteq TB_j$. Reverse icelet is defined as icelet for $T^{-1}$, and it grows towards the past. Iceberg transformation is approximated by a sequence of icebergs, resembling the behaviour of rank one ergodic maps. It is show that a class of random iceberg transformations almost surely has simple spectrum, $1/4$-local rank property and spectral type $σ$ such that $σ\conv σ\ll \la$ where $\la$ is the Lebesgue measure on the circle $S^1$.