Exactness of skew products with expanding fibre maps

Thomas Bogenschütz; Zbigniew Kowalski

Displaying similar documents to “Exactness of skew products with expanding fibre maps”

An equivalent measure for some nonsingular transformations and application

I. Assam, J. Woś (1990)

Studia Mathematica

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

Zbigniew Kowalski (1994)

Applicationes Mathematicae

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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...

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

Zbigniew Kowalski (1993)

Studia Mathematica

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We consider skew products $T (x, y) = (f (x), T_{e (x)} y)$ preserving a measure which is absolutely continuous with respect to the product measure. Here f is a 1-sided Markov shift with a finite set of states or a Lasota-Yorke type transformation and $T_{i}$ , i = 1,..., max e, are nonsingular transformations of some probability space. We obtain the description of the set of eigenfunctions of the Frobenius-Perron operator for T and consequently we get the conditions ensuring the ergodicity, weak mixing and exactness of T....

Genericity of nonsingular transformations with infinite ergodic index

J. Choksi, M. Nadkarni (2000)

Colloquium Mathematicae

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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...

Absolutely continuous, invariant measures for dissipative, ergodic transformations

Jon Aaronson, Tom Meyerovitch (2008)

Colloquium Mathematicae

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We show that a dissipative, ergodic measure preserving transformation of a σ-finite, non-atomic measure space always has many non-proportional, absolutely continuous, invariant measures and is ergodic with respect to each one of these.

On non-ergodic versions of limit theorems

Dalibor Volný (1989)

Aplikace matematiky

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The author investigates non ergodic versions of several well known limit theorems for strictly stationary processes. In some cases, the assumptions which are given with respect to general invariant measure, guarantee the validity of the theorem with respect to ergodic components of the measure. In other cases, the limit theorem can fail for all ergodic components, while for the original invariant measure it holds.

On the Pointwise Ergodic Theorem in $L_{p}$

F. Martín-Reyes, A. de la Torre (1994)

Studia Mathematica

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Ergodic theory approach to chaos: Remarks and computational aspects

Paweł J. Mitkowski, Wojciech Mitkowski (2012)

International Journal of Applied Mathematics and Computer Science

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We discuss basic notions of the ergodic theory approach to chaos. Based on simple examples we show some characteristic features of ergodic and mixing behaviour. Then we investigate an infinite dimensional model (delay differential equation) of erythropoiesis (red blood cell production process) formulated by Lasota. We show its computational analysis on the previously presented theory and examples. Our calculations suggest that the infinite dimensional model considered possesses an attractor...

On a pointwise ergodic theorem for multiparameter semigroups.

Ryotaro Sato (1994)

Publicacions Matemàtiques

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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...

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...