Ergodic properties of skew products with Lasota-Yorke type maps in the base

Zbigniew Kowalski

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Ergodic properties of skew products withfibre maps of Lasota-Yorke type

Zbigniew Kowalski (1994)

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We consider the skew product transformation T(x,y)= (f(x), $T_{e (x)}$ ) where f is an endomorphism of a Lebesgue space (X,A,p), e : X → S and ${T_{s}}_{s \in S}$ is a family of Lasota-Yorke type maps of the unit interval into itself. We obtain conditions under which the ergodic properties of f imply the same properties for T. Consequently, we get the asymptotical stability of random perturbations of a single Lasota-Yorke type map. We apply this to some probabilistic model of the motion of cogged bits in the rotary...

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It is shown that in the group of invertible measurable nonsingular transformations on a Lebesgue probability space, endowed with the coarse topology, the transformations with infinite ergodic index are generic; they actually form a dense $G_{δ}$ set. (A transformation has infinite ergodic index if all its finite Cartesian powers are ergodic.) This answers a question asked by C. Silva. A similar result was proved by U. Sachdeva in 1971, for the group of transformations preserving an infinite...

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On a pointwise ergodic theorem for multiparameter semigroups.

Ryotaro Sato (1994)

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Let T_i (i = 1, 2, ..., d) be commuting null preserving transformations on a finite measure space (X, F, μ) and let 1 ≤ p < ∞. In this paper we prove that for every f ∈ L_p(μ) the averages A_nf(x) = (n + 1)^-d Σ_{0≤n_i≤n} f(T₁ ^n₁T₂ ^n_2...

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. ...

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

Pedro Ortega Salvador (1991)

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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...

The asymptotics of the Perron-Frobenius operator of a class of interval maps preserving infinite measures

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Infinite ergodic index $ℤ^{d}$ -actions in infinite measure

E. Muehlegger, A. Raich, C. Silva, M. Touloumtzis, B. Narasimhan, W. Zhao (1999)

Colloquium Mathematicae

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We construct infinite measure preserving and nonsingular rank one $ℤ^{d}$ -actions. The first example is ergodic infinite measure preserving but with nonergodic, infinite conservative index, basis transformations; in this case we exhibit sets of increasing finite and infinite measure which are properly exhaustive and weakly wandering. The next examples are staircase rank one infinite measure preserving $ℤ^{d}$ -actions; for these we show that the individual basis transformations have conservative...