Generic spectral properties of measure-preserving maps and applications

Abstract

Abstract.Let 3? denote the group of all automorphisms of a finite Lebesgue space equipped with the weak topology. For T e S£ , let 177-denote its maximal spectral type.

A note on the isomorphism of Cartesian products of ergodic flows

“…Similar problems to what we consider here were taken up in [5]. It was shown that for a typical automorphism and any k 1 , .…”

“…(ii) If α + β > 1, then T ⊥ S. 5 Let α ∈ (0, 1) and let S 1 , S 2 be ergodic flows on (Y 1 , C 1 , ν 1 ) and (Y 2 , C 2 , ν 2 ), respectively.…”

2. Orthonectida

“…The disjointness of the convolution powers follows from α-weakly mixing property when 0 < α < 1 (see [6] and [7]). This property is widely used in ergodic theory to build counterexamples ( [8]). The question of α-weak mixing for Chacon's automorphism can be considered as a special case of the general problem of description of the weak limits of powers ofT .…”

Generic behavior of a measure-preserving transformation

“…The main spectral property of a generic T ∈ Aut(X, µ) is given by Del Junco-Lemańczyk [3] where Aut(X, µ) is the set of all measure preserving transformations on a Borel measure space (X, µ). They proved that for a generic T ∈ Aut(X, µ), for every k(1), k(2), .…”

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Del Junco-Lemańczyk [3] showed that a generic measure preserving transformation satisfies a certain orthogonality conditions. More precisely,

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