Theory of dynamical systems and general transformation groups with invariant measure

Katok, Anatole; Sinaĭ, Ya. G.; Stëpin, A. M.

doi:10.1007/bf01223133

Cited by 47 publications

(19 citation statements)

References 387 publications

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“…In [2] it was proved that for a wide class of "big" piecewise linear perturbations of the twist map the whole system exibits stochastic behaviour. A similar result was obtained much earlier by Osetedec ( [4,5]), but to my knowledge, the details of his results remained unpublished. In this paper we take smaller perturbations and study the appearance of islands of periodic (quasiperiodic) motions.…”

Section: Introductionsupporting

confidence: 81%

A model problem with the coexistence of stochastic and integrable behaviour

Wojtkowski

1981

Commun.Math. Phys.

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Section: Introductionsupporting

confidence: 81%

A model problem with the coexistence of stochastic and integrable behaviour

Wojtkowski

1981

Commun.Math. Phys.

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“…1.1. m. were introduced by Easton [4]. It seems however that the basic phenomena were observed earlier by Oseledec, see [5]. Chap.…”

Section: If^>0(a;<0) What Implies Df/dy>q(<0) Define the Slope Of Fmentioning

confidence: 78%

Ergodicity of toral linked twist mappings

Przytycki¹

1983

Ann. Sci. École Norm. Sup.

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“…One says an ergodic T is rigid if Ide WC1 + (T). In [7] is the handy observation (later made by del Junco, independently) that any rigid transformation 'l-z |r>|>o.…”

Section: We Call This Quotient the Essential Commutant Of T And Denotmentioning

confidence: 98%

The commutant is the weak closure of the powers, for rank-1 transformations

King

1986

Ergod. Th. Dynam. Sys.

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Abstract. In the class of rank-1 transformations, there is a strong dichotomy. For such a T, the commutant is either trivial, consisting only of the powers of T, or is uncountable. In addition, the commutant semigroup, C{T), is in fact a group. As a consequence, the notion of weak isomorphism between two transformations is equivalent to isomorphism, if at least one of the transformations is rank-1. In § 2, we show that any proper factor of a rank-1 must be rigid. Hence, neither Ornstein's rank-1 mixing nor Chacon's transformation, can be a factor of a rank-1. N.B. In this paper 'transformation' shall mean a measure preserving transformation on a Lebesgue probability space. In general, we assume our measure spaces nonatomic; however the spaces that factors live on may be atomic. A transformation T is invertible if T" 1 exists and is a transformation. All sets mentioned are assumed measurable. In particular, S will be invertible if the T t are various powers of a single transformation.

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Theory of dynamical systems and general transformation groups with invariant measure

Cited by 47 publications

References 387 publications

A model problem with the coexistence of stochastic and integrable behaviour

A model problem with the coexistence of stochastic and integrable behaviour

Ergodicity of toral linked twist mappings

The commutant is the weak closure of the powers, for rank-1 transformations

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