Countably piecewise expanding transformations without absolutely continuous invariant measure
Paweł Góra (1989)
Banach Center Publications
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Paweł Góra (1989)
Banach Center Publications
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P. Kasprowski (1983)
Annales Polonici Mathematici
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M. Jabłoński (1976)
Annales Polonici Mathematici
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Christopher Bose, Véronique Maume-Deschamps, Bernard Schmitt, S. Sujin Shin (2002)
Studia Mathematica
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We investigate the existence and ergodic properties of absolutely continuous invariant measures for a class of piecewise monotone and convex self-maps of the unit interval. Our assumption entails a type of average convexity which strictly generalizes the case of individual branches being convex, as investigated by Lasota and Yorke (1982). Along with existence, we identify tractable conditions for the invariant measure to be unique and such that the system has exponential decay of correlations...
Giulio Pianigiani (1981)
Annales Polonici Mathematici
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E. J. Wilezynski (1923)
Journal de Mathématiques Pures et Appliquées
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Adl-Zarabi, Kourosh, Proppe, Harald (2000)
Journal of Applied Mathematics and Stochastic Analysis
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Tomasz Komorowski (1991)
Annales Polonici Mathematici
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Abstract. The paper concerns the problem of the existence of a finite invariant absolutely continuous measure for piecewise -regular and convex transformations T: [0, l]→[0,1]. We show that in the case when T’(0) = 1 and T"(0) exists T does not admit such a measure. This result is complementary to the ones contained in [3] and [5].
Michael C. Mackey, Marta Tyran-Kamińska (2008)
Colloquium Mathematicae
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Using the Perron-Frobenius operator we establish a new functional central limit theorem for non-invertible measure preserving maps that are not necessarily ergodic. We apply the result to asymptotically periodic transformations and give a specific example using the tent map.
Zbigniew Kowalski (1993)
Studia Mathematica
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We consider skew products 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 , 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....
Amos Koeller, Rodney Nillsen, Graham Williams (2007)
Colloquium Mathematicae
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Let 𝕋 denote the set of complex numbers of modulus 1. Let v ∈ 𝕋, v not a root of unity, and let T: 𝕋 → 𝕋 be the transformation on 𝕋 given by T(z) = vz. It is known that the problem of calculating the outer measure of a T-invariant set leads to a condition which formally has a close resemblance to Carathéodory's definition of a measurable set. In ergodic theory terms, T is not weakly mixing. Now there is an example, due to Kakutani, of a transformation ψ̃ which is weakly mixing but...
M. Jabłoński, J. Malczak (1984)
Colloquium Mathematicae
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Zbigniew Kowalski (1990)
Studia Mathematica
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M. Pal (1972)
Publications de l'Institut Mathématique
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F. Schweiger (1989)
Banach Center Publications
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Gerhard Keller (1978)
Publications mathématiques et informatique de Rennes
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Roland Zweimüller (2008)
Colloquium Mathematicae
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For infinite measure preserving transformations with a compact regeneration property we establish a central limit theorem for visits to good sets of finite measure by points from Poissonian ensembles. This extends classical results about (noninteracting) infinite particle systems driven by Markov chains to the realm of systems driven by weakly dependent processes generated by certain measure preserving transformations.