Measurable dynamics of -unimodal maps of the interval
A. M. Blokh, M. Yu. Lyubich (1991)
Annales scientifiques de l'École Normale Supérieure
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A. M. Blokh, M. Yu. Lyubich (1991)
Annales scientifiques de l'École Normale Supérieure
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Zbigniew Kowalski (1994)
Applicationes Mathematicae
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We consider the skew product transformation T(x,y)= (f(x), ) where f is an endomorphism of a Lebesgue space (X,A,p), e : X → S and 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...
C. Ryll-Nardzewski (1951)
Studia Mathematica
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Sébastien Ferenczi, Luca Q. Zamboni (2011)
Annales scientifiques de l'École Normale Supérieure
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For a class of -interval exchange transformations, which we call the symmetric class, we define a new self-dual induction process in which the system is successively induced on a union of sub-intervals. This algorithm gives rise to an underlying graph structure which reflects the dynamical behavior of the system, through the Rokhlin towers of the induced maps. We apply it to build a wide assortment of explicit examples on four intervals having different dynamical properties: these include...
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...
John Coffey (1988)
Mathematische Zeitschrift
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Kim, Jeong H. (1995)
International Journal of Mathematics and Mathematical Sciences
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R. Sato (1990)
Colloquium Mathematicae
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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.
F. Martín-Reyes, A. de la Torre (1994)
Studia Mathematica
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Antoni Leon Dawidowicz (1983)
Annales Polonici Mathematici
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Roland Zweimüller (2004)
Fundamenta Mathematicae
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We consider S-unimodal Misiurewicz maps T with a flat critical point c and show that they exhibit ergodic properties analogous to those of interval maps with indifferent fixed (or periodic) points. Specifically, there is a conservative ergodic absolutely continuous σ-finite invariant measure μ, exact up to finite rotations, and in the infinite measure case the system is pointwise dual ergodic with many uniform and Darling-Kac sets. Determining the order of return distributions to suitable...
Franz Hofbauer, Peter Raith, Thomas Steinberger (2003)
Fundamenta Mathematicae
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The multifractal generalizations of Hausdorff dimension and packing dimension are investigated for an invariant subset A of a piecewise monotonic map on the interval. Formulae for the multifractal dimension of an ergodic invariant measure, the essential multifractal dimension of A, and the multifractal Hausdorff dimension of A are derived.
W. Bułatek, M. Lemańczyk, D. Rudolph (1997)
Studia Mathematica
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We construct a coboundary cocycle which is of bounded variation, is homotopic to the identity and is Hölder continuous with an arbitrary Hölder exponent smaller than 1.
Roland Zweimüller (2008)
Fundamenta Mathematicae
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We determine, for certain ergodic infinite measure preserving transformations T, the asymptotic behaviour of the distribution of the waiting time for an excursion (from some fixed reference set of finite measure) of length larger than l as l → ∞, generalizing a renewal-theoretic result of Lamperti. This abstract distributional limit theorem applies to certain weakly expanding interval maps, where it clarifies the distributional behaviour of hitting times of shrinking neighbourhoods of...