Exploding Markov operators
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Positivity
Exploding Markov operators Bartosz Frej1 Received: 28 December 2018 / Accepted: 5 February 2020 © The Author(s) 2020
Abstract A special class of doubly stochastic (Markov) operators is constructed. In a sense these operators come from measure preserving transformations and inherit some of their properties, namely ergodicity and positivity of entropy, yet they may have no pointwise factors. Keywords Markov operator · Doubly stochastic operator · Measure-preserving transformation · Entropy · Ergodicity · Koopman operator Mathematics Subject Classification 28D20 · 37A30 · 47A35
1 Introduction The subject of the current paper lies in the border zone between ergodic theory and operator theory. The main motivation of study was the desire to increase the number of examples of doubly stochastic operators, which escape the scope of classical ergodic theory (because they are not induced by measure preserving maps as their Koopman operators), but they still reveal a nontrivial dynamical behavior. By a doubly stochastic or a Markov operator we understand an operator P : L p (μ) → L p (ν), where (X , μ) and (Y , ν) are probability spaces, which fulfills the following conditions: (i) P f is positive for every positive f ∈ L p (μ), (ii) P1 = 1 (where 1 is the function constantly equal to 1), (iii) P f dν = f dμ for every f ∈ L p (μ). For example, the well-studied class of quasi-compact doubly stochastic operators on L 2 lies pretty far from the theory of measure preserving maps. But the domain of a
Research is supported from resources for science in years 2013–2018 as research project (NCN Grant 2013/08/A/ST1/00275, Poland).
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Bartosz Frej [email protected] Faculty of Pure and Applied Mathematics, Wrocław University of Science and Technology, Wybrze˙ze Wyspia´nskiego 27, 50-370 Wrocław, Poland
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quasi-compact operator decomposes into the direct sum of two reducing subspaces, called reversible and almost weakly stable parts, respectively, such that the first one is finite dimensional, while on the other one orbits of functions converge to zero in L 2 norm. The restriction of such an operator to the reversible part is Markov isomorphic to a rotation of a compact abelian group (which is finite in this case). The transition probability associated to the operator forces points of the underlying space to ramble periodically through finitely many sets of states (in a fixed order), randomly choosing the succeeding state from a set which is next in the queue. These operators are null, meaning that their sequence entropy is always zero (see [7] for details). As another example one may think of a convex combination of finitely many measure preserving maps, which leads to studying a rich class of iterated function systems. Unfortunately, such operators are hard to handle by the entropy theory as defined in [3]—e.g., it is possible that the combination of maps with positive entropy has entropy equal to zero. In the current paper another class of examples which stem from pointwise maps is proposed and
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