Structure of mixing and category of complete mixing for stochastic operators

Volume 56 / 1992

A. Iwanik, R. Rębowski Annales Polonici Mathematici 56 (1992), 233-242 DOI: 10.4064/ap-56-3-233-242

Abstract

Let T be a stochastic operator on a σ-finite standard measure space with an equivalent σ-finite infinite subinvariant measure λ. Then T possesses a natural "conservative deterministic factor" Φ which is the Frobenius-Perron operator of an invertible measure preserving transformation φ. Moreover, T is mixing ("sweeping") iff φ is a mixing transformation. Some stronger versions of mixing are also discussed. In particular, a notion of *L¹-s.o.t. mixing is introduced and characterized in terms of weak compactness. Finally, it is shown that most stochastic operators are completely mixing and that the same holds for convolution stochastic operators on l.c.a. groups.

Authors

  • A. Iwanik
  • R. Rębowski

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