Invariant densities for C¹ maps

Anthony Quas

Displaying similar documents to “Invariant densities for C¹ maps”

Most expanding maps have no absolutely continuous invariant measure

Anthony Quas (1999)

Studia Mathematica

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We consider the topological category of various subsets of the set of expanding maps from a manifold to itself, and show in particular that a generic $C^{1}$ expanding map of the circle has no absolutely continuous invariant probability measure. This is in contrast with the situation for $C^{2}$ or $C^{1 + ε}$ expanding maps, for which it is known that there is always a unique absolutely continuous invariant probability measure.

Addition to the paper “Translation invariant subspaces of $L_{p} (G)$ ” Studia Mathematica 48 (1973), pp. 245-250

Aharon Atzmon (1975)

Studia Mathematica

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Piecewise convex transformations with no finite invariant measure

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 $C^{2}$ -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].

On systems of imprimitivity on locally compact abelian groups with dense actions

J. Mathew, M. G. Nadkarni (1978)

Annales de l'institut Fourier

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Consider the four pairs of groups $(Γ, R)$ , $(Γ / Γ_{0}, R / Γ_{0})$ , $(K \cap S, P)$ and $(S, B)$ , where $Γ$ , $R$ are locally compact second countable abelian groups, $Γ$ is a dense subgroup of $R$ with inclusion map from $Γ$ to $R$ continuous; $Γ_{0} \subseteq Γ \subseteq R$ is a closed subgroup of $R$ ; $S$ , $B$ are the duals of $R$ and $Γ$ respectively, and $K$ is the annihilator of $Γ_{0}$ in $B$ . Let the first co-ordinate of each pair act on the second by translation. We connect, by a commutative diagram, the systems of imprimitivity which arise in a natural fashion on each pair, starting...

Metric transitivity and integer valued functions

Solomon Schwartzman (1960)

Annales de l'institut Fourier

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Soit $X$ un espace mesurable de mesure $μ$ finie ; $φ : X \to X$ une application vérifiant $μ (φ^{- 1} (S)) = μ (S)$ pour chaque ensemble mesurable $S \subseteq X$ . On donne des conditions nécessaires et suffisantes pour que $X$ soit un ensemble ergodique.

Incompressible transformations

D. Maharam (1964)

Fundamenta Mathematicae

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

Maximilian Thaler (2000)

Studia Mathematica

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We determine the asymptotic behaviour of the iterates of the Perron-Frobenius operator for specific interval maps with an indifferent fixed point which gives rise to an infinite invariant measure.

On concentrated probabilities on non locally compact groups

Wojciech Bartoszek (1996)

Commentationes Mathematicae Universitatis Carolinae

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Let $G$ be a Polish group with an invariant metric. We characterize those probability measures $μ$ on $G$ so that there exist a sequence $g_{n} \in G$ and a compact set $A \subseteq G$ with $μ^{* n} (g_{n} A) \equiv 1$ for all $n$ .