Dynamical systems with countably-multiple Lebesgue spectrum. I

In the fall of 1957 A. N. Kolmogorov started lecturing on the theory of dynamical systems and supervised a seminar on the same theme at the Mechanical–Mathematical Department of Moscow State University. He began his lectures with the theory of systems with a pure point spectrum, which he approached from a probabilistic point of view. This approach, undoubtedly, has many advantages. In the seminar we studied Ito's theory of multiple stochastic integrals and, under the supervision of A. N. Kolmogorov, I. V. Girsanov constructed an example of a Gaussian dynamical system with a simple continuous spectrum. At one of the meetings of the seminar, still before the advent of entropy, Kolmogorov suggested a proof of an assertion, which today would read: the unitary operator induced by a K-automorphism has a countable Lebesgue spectrum. At this point Kolmogorov was studying Shannon's theory of information and the concept of the capacity of functional spaces. Judging by his well known article, we can say that the first of these, and all that is related to it, played a big role in the development of information theory in our country. The investigation of capacity is connected with Kolmogorov's work on Hilbert's 13th problem and was summarized in his well-known survey co-authored by V. M. Tihomirov.

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Commutator methods for the spectral analysis of uniquely ergodic dynamical systems

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2013.76 ◽

2014 ◽

Vol 35 (3) ◽

pp. 944-967 ◽

Cited By ~ 6

Author(s):

R. TIEDRA DE ALDECOA

Keyword(s):

Spectral Analysis ◽

Dynamical Systems ◽

Absolute Continuity ◽

Unitary Operators ◽

Skew Products ◽

Lebesgue Spectrum ◽

The Absolute ◽

Time Changes ◽

Continuity Of The Spectrum ◽

Uniquely Ergodic

AbstractWe present a method, based on commutator methods, for the spectral analysis of uniquely ergodic dynamical systems. When applicable, it leads to the absolute continuity of the spectrum of the corresponding unitary operators. As an illustration, we consider time changes of horocycle flows, skew products over translations and Furstenberg transformations. For time changes of horocycle flows we obtain absolute continuity under assumptions weaker than those to be found in the literature, and for skew products over translations and Furstenberg transformations we obtain countable Lebesgue spectrum under assumptions not previously covered in the literature.

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