Approximations in Ergodic Theory

Katok, Anatole; Stëpin, A. M.

doi:10.1070/rm1967v022n05abeh001227

Cited by 171 publications

(177 citation statements)

References 10 publications

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“…'s are minimal (this is guaranteed by a condition due to Keane [Ke1], which is automatically dealt with if the interval lengths are rationally independent) but note that ergodic properties of minimal i.e.m. 's can differ substantially from those of circle rotations: first they need not be uniquely ergodic [Ke2,KN,Co], and second, being ergodic they can be weakly mixing [KS,V3,V4]. On the other hand uniquely ergodic i.e.m.…”

Section: Introductionmentioning

confidence: 99%

The cohomological equation for Roth-type interval exchange maps

Marmi

Moussa

Yoccoz³

2005

J. Amer. Math. Soc.

124

235

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We exhibit an explicit class of minimal interval exchange maps (i.e.m.’s) T T for which the cohomological equation \[ Ψ − Ψ ∘ T = Φ \Psi -\Psi \circ T=\Phi \] has a bounded solution Ψ \Psi provided that the datum Φ \Phi belongs to a finite codimension subspace of the space of functions having on each interval a derivative of bounded variation. The proof is purely dynamical and is based on a renormalization argument and on Gottshalk-Hedlund’s theorem. If the datum is more regular the loss of differentiability in solving the cohomological equation will be the same. The class of interval exchange maps is characterized in terms of a diophantine condition of Roth type imposed to an acceleration of the Rauzy-Veech-Zorich continued fraction expansion associated to T T . More precisely one must impose a growth rate condition for the matrices appearing in the continued fraction algorithm together with a spectral gap condition (which guarantees unique ergodicity) and a coherence condition. We also prove that the set of Roth-type interval exchange maps has full measure. In the appendices we construct concrete examples of Roth-type i.e.m.’s and we show how the growth rate condition alone does not imply unique ergodicity.

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Section: Introductionmentioning

confidence: 99%

The cohomological equation for Roth-type interval exchange maps

Marmi

Moussa

Yoccoz³

2005

J. Amer. Math. Soc.

124

235

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“…Halmos proved in [12] that for every aperiodic automorphism T and every natural number p there exists an automorphism 5 of period p such that d{ T, S) < 4/p. Rokhlin showed that instead of 4/p one can put \/p + e for every positive e. This result became fundamental in the approximation theory developed by A. Katok and A. Stepin [18].…”

Section: Bementioning

confidence: 90%

Rokhlin's School in ergodic theory

Yuzvinsky

1989

Ergod. Th. Dynam. Sys.

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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. This paper is divided into three sections corresponding to three easily distinguishable periods in Rokhlin's work in ergodic theory. The first section called BE (before entropy) corresponds to the first period which lasted from 1940 to 1949. It was devoted to axiomatization, spectral invariants, and mixing properties. The following 9 years were a quiet period in ergodic theory. In particular, Rokhlin was working in topology and apparently did not plan to return to ergodic theory. In 1958, A. Kolmogorov introduced a new invariant: the entropy of a measure preserving transformation. This started a new era in ergodic theory that I will call ET (entropy theory) and attracted Rokhlin back to the area. The second section of this paper is devoted to the results of the seminar organized by Rokhlin at the Moscow State University and covers approximately 1958-1959. We also follow the later history of several important problems formulated in this seminar. Finally, the third section is devoted to Rokhlin's seminar at the Leningrad State University in 1960 on the isomorphism of Bernoulli automorphisms with equal entropy started yet another period in ergodic theory. This period can be characterized by the development of methods of more combinatorial nature that were harder to formalize. Many old problems were solved by these methods. At this point the center of pure ergodic theory or rather Bernoulli theory moved to the West. Perhaps this was one of the reasons why Rokhlin stopped doing active research in the area. Besides, he was busy proving his famous results in 4-dimensional topology.

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“…In §4 we show that the spectrum is generically zero dimensional. The proof uses a result of Strichartz in Fourier theory [23], new spectral decompositions of self-adjoint operators [16], and elements of the theory of periodic approximations [10,1]. These two results show that the spectrum is generically singular continuous and zero dimensional.…”

Section: A Hof and O Knillmentioning

confidence: 99%

Zero-dimensional singular continuous spectrum for smooth differential equations on the torus

Hof

Knill

1998

Ergod. Th. Dynam. Sys.

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Abstract. We study spectral properties of the flowẋ = 1/F (x, y),ẏ = 1/λF (x, y) on the 2-torus. We show that, in general, the speed of approximation in cyclic approximation gives an upper bound on the Hausdorff dimension of the supports of spectral measures. We use this to prove that for generic pairs (F, λ) the spectrum of the flow on the torus is singular continuous with all spectral measures supported on sets of zero Hausdorff dimension.

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Approximations in Ergodic Theory

Cited by 171 publications

References 10 publications

The cohomological equation for Roth-type interval exchange maps

The cohomological equation for Roth-type interval exchange maps

Rokhlin's School in ergodic theory

Zero-dimensional singular continuous spectrum for smooth differential equations on the torus

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