Counterexamples in ergodic theory and number theory

Junco, Andrés del; Rosenblatt, Joseph

doi:10.1007/bf01673506

Cited by 101 publications

(56 citation statements)

References 10 publications

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“…(C n )>0 for every n, lim ^ ( C ) = 0, and lim ju (C n AgC n )//a(C n ) = 0 for every ge G n n (cf. [4] If the action of G is not strongly ergodic, it admits many a.i. sequences.…”

Section: Strong Ergodicity and Invariant Meansmentioning

confidence: 99%

“…The situation is quite different for groups satisfying Kazhdan's condition T (like SL (n, Z), n > 3 , for example): they can be characterized as precisely those groups for which every finite measure preserving ergodic action is strongly ergodic [2]. In three recent papers [4], [7], [12], another concept has been introduced which is fairly closely related to strong ergodicity: the uniqueness of G-invariant means on V°(X, SF, /JL) (this means that integration is the unique G-invariant mean on L°°CY, Sf, fi)). Again one can characterize amenable groups as those groups for which no finite measure preserving action admits a unique G-invariant mean, and groups with property T as those for which every finite measure preserving ergodic action has a unique invariant mean.…”

Section: Introductionmentioning

confidence: 99%

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Amenability, Kazhdan's property T, strong ergodicity and invariant means for ergodic group-actions

Schmidt

1981

Ergod. Th. Dynam. Sys.

122

110

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Abstract. This paper discusses the relations between the following properties of finite measure preserving ergodic actions of a countable group G: strong ergodicity (i.e. the non-existence of almost invariant sets), uniqueness of G-invariant means on the measure space carrying the group action, and certain cohomological properties. Using these properties one can characterize all actions of amenable groups and of groups with Kazhdan's property T. For groups which fall in between these two definitions these notions lead to some interesting examples.

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“…(C n )>0 for every n, lim ^ ( C ) = 0, and lim ju (C n AgC n )//a(C n ) = 0 for every ge G n n (cf. [4] If the action of G is not strongly ergodic, it admits many a.i. sequences.…”

Section: Strong Ergodicity and Invariant Meansmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Amenability, Kazhdan's property T, strong ergodicity and invariant means for ergodic group-actions

Schmidt

1981

Ergod. Th. Dynam. Sys.

122

110

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“…The question is whether I is the only invariant mean. In del Junco and Rosenblatt [2], it is shown that when G is a countable amenable semigroup, and (X, ß, p) is nonatomic, there are other G-invariant means. This is an abstract analogue for countable semigroups of [3] and [13].…”

mentioning

confidence: 99%

“…They use [2], but apparently were unaware of [14] which inspired us. Although obtained independently of [5], both Proposition 3.5 and Theorem 3.6 have been written here to show more clearly their similarity to the main theorem in [5] as well as their dependence on § 1.…”

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confidence: 99%

Uniqueness of invariant means for measure-preserving transformations

Rosenblatt¹

1981

Trans. Amer. Math. Soc.

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For some compact abelian groups X X (e.g. T n T^n , n ⩾ 2 n \geqslant 2 , and ∏ n = 1 ∞ Z 2 \prod \nolimits _{n = 1}^\infty {{Z_2}} ), the group G G of topological automorphisms of X X has the Haar integral as the unique G G -invariant mean on L ∞ ( X , λ X ) {L_\infty }(X,{\lambda _X}) . This gives a new characterization of Lebesgue measure on the bounded Lebesgue measurable subsets β \beta of R n {R^n} , n ⩾ 3 n \geqslant 3 ; it is the unique normalized positive finitely-additive measure on β \beta which is invariant under isometries and the transformation of R n : ( x 1 , … , x n ) ↦ ( x 1 + x 2 , x 2 , … , x n ) {R^n}:({x_1}, \ldots ,{x_n}) \mapsto ({x_1} + {x_2},{x_2}, \ldots ,{x_n}) . Other examples of, as well as necessary and sufficient conditions for, the uniqueness of a mean on L ∞ ( X , β , p ) {L_\infty }(X,\beta ,p) , which is invariant by some group of measure-preserving transformations of the probability space ( X , β , p ) (X,\beta ,p) , are described.

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“…When G is a countable amenable semigroup, del Junco and Rosenblatt [3] proved LIM(X, S) contains more than one element. Chou [2] showed that the cardinality of LIM(X, G) is at least 2C for any countable amenable group, where c is the cardinality of the continuum.…”

Section: Exposed Points Of Lim(x G)mentioning

confidence: 99%

The exposed points of the set of invariant means

Miao¹

1995

Trans. Amer. Math. Soc.

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Let G G be a σ \sigma -compact infinite locally compact group, and let L I M LIM be the set of left invariant means on L ∞ ( G ) {L^\infty }(G) . We prove in this paper that if G G is amenable as a discrete group, then L I M LIM has no exposed points. We also give another proof of the Granirer theorem that the set L I M ( X , G ) LIM(X,G) of G G -invariant means on L ∞ ( X , β , p ) {L^\infty }(X,\beta ,p) has no exposed points, where G G is an amenable countable group acting ergodically as measure-preserving transformations on a nonatomic probability space ( X , β , p ) (X,\beta ,p) .

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Counterexamples in ergodic theory and number theory

Cited by 101 publications

References 10 publications

Amenability, Kazhdan's property T, strong ergodicity and invariant means for ergodic group-actions

Amenability, Kazhdan's property T, strong ergodicity and invariant means for ergodic group-actions

Uniqueness of invariant means for measure-preserving transformations

The exposed points of the set of invariant means

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