Ergodic Theory and Maximal Abelian Subalgebras of the Hyperfinite Factor

AbstractWe introduce a notion of absorption operators in the context of quantum Markov processes. The absorption problem in invariant domains (enclosures) is treated for a quantum Markov evolution on a separable Hilbert space, both in discrete and continuous times: We define a well-behaving set of positive operators which can correspond to classical absorption probabilities, and we study their basic properties, in general, and with respect to accessibility structure of channels, transience and recurrence. In particular, we can prove that no accessibility is allowed between the null and positive recurrent subspaces. In the case, when the positive recurrent subspace is attractive, ergodic theory will allow us to get additional results, in particular about the description of fixed points.

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Ergodic theory for energetically open compressible fluid flows

Physica D Nonlinear Phenomena ◽

10.1016/j.physd.2021.132914 ◽

2021 ◽

Vol 423 ◽

pp. 132914

Author(s):

Francesco Fanelli ◽

Eduard Feireisl ◽

Martina Hofmanová

Keyword(s):

Ergodic Theory ◽

Compressible Fluid ◽

Fluid Flows ◽

Compressible Fluid Flows

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Category Theory for Autonomous and Networked Dynamical Systems

Entropy ◽

10.3390/e21030302 ◽

2019 ◽

Vol 21 (3) ◽

pp. 302 ◽

Cited By ~ 4

Author(s):

Jean-Charles Delvenne

Keyword(s):

Dynamical Systems ◽

Systems Theory ◽

Ergodic Theory ◽

Category Theory ◽

Open Systems ◽

Topological Dynamics ◽

Control Problems ◽

Natural Conditions ◽

Discussion Paper ◽

Open Systems Theory

In this discussion paper we argue that category theory may play a useful role in formulating, and perhaps proving, results in ergodic theory, topogical dynamics and open systems theory (control theory). As examples, we show how to characterize Kolmogorov–Sinai, Shannon entropy and topological entropy as the unique functors to the nonnegative reals satisfying some natural conditions. We also provide a purely categorical proof of the existence of the maximal equicontinuous factor in topological dynamics. We then show how to define open systems (that can interact with their environment), interconnect them, and define control problems for them in a unified way.

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Rokhlin's School in ergodic theory

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385700005241 ◽

1989 ◽

Vol 9 (4) ◽

pp. 609-618

Author(s):

Sergey Yuzvinsky

Keyword(s):

Ergodic Theory ◽

General Development

Ergodic theory is one of several fields of mathematics where the name Vladimir Abramovich Rokhlin (spelled also ‘Rochlin’ and ‘Rohlin’) is well known to the specialists. That name is attached to some very often used theorems, but the goal of this paper is not just to remind the reader of these theorems. I put them in the context of the general development of ergodic theory during the thirty years 1940–70. Most of all, I want to emphasize that Rokhlin was not only a researcher producing powerful results but also a founder of schools at the two best Universities in the USSR. For at least 10 years (1958–68) these schools dominated ergodic theory. This paper is not biographical. Rokhlin's life certainly deserves a better biographer. However, I mention certain circumstances of a non-mathematical nature where it seems to be appropriate.

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