Mixing but not exact quasi-Markovian process

• Autorzy: Sachse P.
• Języki publikacji: EN

Abstrakty

EN

Mixing Smorodinsky-Adams map [2] with q_n = n^2, is shown to satisfy, with respect to the time-zero partition, the d+ > 1/16 property. Basing on a general theorem from [7], an explicite example of a mixing but not exact quasi-Markovian process is given. The d+ > O property is studied: although defined for a process, it is shown to be a property of the transformation itself. All mixing rank one transformations are shown to possess this property, which strenghtens the result from [3]. Several other transformations are observed not to satisfy this property (von Neumann-Kakutani's map, irrational rotations, Feldman's map). A rank one weakly mixing example with d+ = O is also given.

Słowa kluczowe

EN

quasi-Markovian process; rank-1; mixing; d+>0 property

PL

proces quasi-Markowa; mieszanie

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2003
• Tom: Vol. 51, no 4
• Strony: 365--379
• Opis fizyczny: Bibliogr. 7 poz., rys.

Twórcy

autor

Sachse P.

• Wrocław University of Technology, Institute of Mathematics, Janiszewskiego 14, 50-370 Wrocław, Poland,

Bibliografia

• [1] J. Aaronson, M. Denker, Local limit theorems for partial sums of stationary sequences generated by Gibbs-Markov maps, Stochastic Dynam., 1 (2) (2001) 193-237.
• [2] T. M. Adams, Smorodinsky’s conjecture on rank-one mixing, Proc. Amer. Math. Soc., 126 (3) (1998) 739-744.
• [3] C. Bose, Mixing examples in the class of piecewise monotone and continuous maps of the unit interval, Israel J. Math., 83 (1-2) (1993) 129-152.
• [4] J. Feldman, New K-automorphisms and a problem of Kakutani, Israel J. Math., 24 (1) (1976) 16-38.
• [5] N. A. Friedman, Introduction to ergodic theory, van Nostrand, New York 1970.
• [6] Z. S. Kowalski, Quasi-Markovian transformations, Ergodic Theory Dynam. Systems, 17 (1997) 885-897.
• [7] Z. S. Kowalski, Weakly mixing but not mixing quasi-markovian processes, Studia Math., 142 (3) (2000) 235-244.