Genericity of nonsingular transformations with infinite ergodic index

J. Choksi; M. Nadkarni

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Displaying similar documents to “Genericity of nonsingular transformations with infinite ergodic index”

Pointwise ergodic theorems in Lorentz spaces L(p,q) for null preserving transformations

Ryotaro Sato (1995)

Studia Mathematica

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Let (X,ℱ,µ) be a finite measure space and τ a null preserving transformation on (X,ℱ,µ). Functions in Lorentz spaces L(p,q) associated with the measure μ are considered for pointwise ergodic theorems. Necessary and sufficient conditions are given in order that for any f in L(p,q) the ergodic average $n^{- 1} \sum_{i = 0}^{n - 1} f \circ τ^{i} (x)$ converges almost everywhere to a function f* in $L (p_{1}, q_{1}]$ , where (pq) and $(p_{1}, q_{1}]$ are assumed to be in the set $(r, s) : r = s = 1, o r 1 < r < \infty a n d 1 \leq s \leq \infty, o r r = s = \infty$ . Results due to C. Ryll-Nardzewski, S. Gładysz, and I. Assani and J. Woś are generalized...

Convergence of the averages and finiteness of ergodic power funtions in weighted L spaces.

Pedro Ortega Salvador (1991)

Publicacions Matemàtiques

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Let (X, F, μ) be a finite measure space. Let T: X → X be a measure preserving transformation and let Af denote the average of Tf, k = 0, ..., n. Given a real positive function v on X, we prove that {Af} converges in the a.e. sense for every f in L(v dμ) if and only if inf v(Tx) > 0 a.e., and the same condition is equivalent to the finiteness of a related ergodic power function Pf for every f in L(v dμ). We apply this result to characterize, being T null-preserving, the finite...

On the ratio maximal functions for an ergodic flow

Ryotaro Sato (1984)

Studia Mathematica

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Weighted L spaces and pointwise ergodic theorems.

Ryotaro Sato (1995)

Publicacions Matemàtiques

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In this paper we give an operator theoretic version of a recent result of F. J. Martín-Reyes and A. de la Torre concerning the problem of finding necessary and sufficient conditions for a nonsingular point transformation to satisfy the Pointwise Ergodic Theorem in L_p. We consider a positive conservative contraction T on L₁ of a σ-finite measure space (X, F, μ), a fixed function e in L₁ with

Weak almost periodicity of $L_{1}$ contractions and coboundaries of non-singular transformations

Isaac Kornfeld, Michael Lin (2000)

Studia Mathematica

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It is well known that a weakly almost periodic operator T in a Banach space is mean ergodic, and in the complex case, also λT is mean ergodic for every |λ|=1. We prove that a positive contraction on $L_{1}$ is weakly almost periodic if (and only if) it is mean ergodic. An example shows that without positivity the result is false. In order to construct a contraction T on a complex $L_{1}$ such that λT is mean ergodic whenever |λ|=1, but T is not weakly almost periodic, we prove the following: Let...

Appendix on return-time sequences

Jean Bourgain, Harry Furstenberg, Yitzhak Katznelson, Donald S. Ornstein (1989)

Publications Mathématiques de l'IHÉS

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An equivalent measure for some nonsingular transformations and application

I. Assam, J. Woś (1990)

Studia Mathematica

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A reverse maximal ergodic theorem

Ryotaro Sato (1983)

Studia Mathematica

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Almost everywhere convergence and boundedness of Cesàro-α ergodic averages in L-spaces.

Francisco J. Martín Reyes, María Dolores Sarrión Gavilán (1999)

Publicacions Matemàtiques

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Let (X, μ) be a σ-finite measure space and let τ be an ergodic invertible measure preserving transformation. We study the a.e. convergence of the Cesàro-α ergodic averages associated with τ and the boundedness of the corresponding maximal operator in the setting of L(wdμ) spaces.

On the distribution function of the majorant of ergodic means

Lasha Epremidze (1992)

Studia Mathematica

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Let T be a measure-preserving ergodic transformation of a measure space (X,,μ) and, for f ∈ L(X), let $f * = s u p_{N} 1 / N \sum_{m = 0}^{N - 1} f \circ T^{m}$ . In this paper we mainly investigate the question of whether (i) $ʃ_{a}^{\infty} | μ (f * > t) - 1 / t ʃ_{(f * > t)} f d μ | d t < \infty$ and whether (ii) $ʃ_{a}^{\infty} | μ (f * > t) - 1 / t ʃ_{(f > t)} f d μ | d t < \infty$ for some a > 0. It is proved that (i) holds for every f ≥ 0. (ii) holds if f ≥ 0 and f log log (f + 3) ∈ L(X) or if μ(X) = 1 and the random variables $f \circ T^{m}$ are independent. Related inequalities are proved. Some examples and counterexamples are constructed. Several known results are obtained as corollaries. ...