2010/08/25 by A. A. Prikhod’ko, A. A. Prikhod'ko, Prikhod'ko, A. A.
Computer Science · Mathematics · #28D05 #Algorithms and Data Compression #Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #math.CA #math.DS #msc:28D05 #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.1008.4301
23 pages, 17 figures
arxiv created 2010/08/25 · openalex publication_date 2010/08/25 · arxiv updated 2010/08/26 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
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 … \sqcup Th-1B, namely, an icelet is a sequence of disjoint measurable sets \B0, B1, …, Bh-1\ such that the levels are nested: Bj+1 ⊆ TBj. 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≪ \la where \la is the Lebesgue measure on the circle S1.