The homogeneous spectrum problem in ergodic theory

2005 ◽  
Vol 160 (2) ◽  
pp. 417-446 ◽  
Author(s):  
Oleg Ageev
Keyword(s):  
Ergodic Theory ◽  
Homogeneous Spectrum

Group actions: Entropy, mixing, spectra, and generic properties

2021 ◽  
pp. 311-324
Author(s):  
Anatoly Stepin ◽  
Sergey Tikhonov
Keyword(s):  
Ergodic Theory ◽  
Group Actions ◽  
Generic Properties ◽  
Finite Multiplicity ◽  
Further Development ◽  
Homogeneous Spectrum

We talk about several directions of V. Rokhlin’s heritage in ergodic theory: ideas that influenced the further development of investigations (genericity, approximations), problems put forward by V. Rokhlin in his papers, problems that V. Rokhlin put forward verbally (in particular, the question about homogeneous spectrum of finite multiplicity). We touch upon the directions close to the authors of this text and their school. Many of the questions raised by Rokhlin have analogs for different classes of transformations, for group actions, and versions about the genericity of properties appearing in these formulations. We will consider the corresponding topics in such a generalized sense.

scholarly journals Absorption in Invariant Domains for Semigroups of Quantum Channels

2021 ◽  
Author(s):  
Raffaella Carbone ◽  
Federico Girotti
Keyword(s):  
Hilbert Space ◽  
Ergodic Theory ◽  
Markov Processes ◽  
Separable Hilbert Space ◽  
Quantum Channels ◽  
Quantum Markov Processes ◽  
Positive Recurrent ◽  
Markov Evolution ◽  
Absorption Probabilities ◽  
Invariant Domains

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.

scholarly journals Ergodic theory for energetically open compressible fluid flows

2021 ◽  
Vol 423 ◽  
pp. 132914
Author(s):  
Francesco Fanelli ◽  
Eduard Feireisl ◽  
Martina Hofmanová
Keyword(s):  
Ergodic Theory ◽  
Compressible Fluid ◽  
Fluid Flows ◽  
Compressible Fluid Flows

scholarly journals Category Theory for Autonomous and Networked Dynamical Systems

Entropy ◽  
2019 ◽  
Vol 21 (3) ◽  
pp. 302 ◽  
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.

scholarly journals Rokhlin's School in ergodic theory

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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