The existence of an a.c.i.p.m. for an expanding map of the interval; the study of a counterexample

B. Schmitt

Displaying similar documents to “The existence of an a.c.i.p.m. for an expanding map of the interval; the study of a counterexample”

On invariant measures for the tend map.

Francesc Bofill (1988)

Stochastica

Similarity:

The bifurcation structure of a one parameter dependent piecewise linear population model is described. An explicit formula is given for the density of the unique invariant absolutely continuous probability measure mu for each parameter value b. The continuity of the map b --> mu is established.

Invariant measures for piecewise convex transformations of an interval

Christopher Bose, Véronique Maume-Deschamps, Bernard Schmitt, S. Sujin Shin (2002)

Studia Mathematica

Similarity:

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...

Non-existence of absolutely continuous invariant probabilities for exponential maps

Neil Dobbs, Bartłomiej Skorulski (2008)

Fundamenta Mathematicae

Similarity:

We show that for entire maps of the form z ↦ λexp(z) such that the orbit of zero is bounded and Lebesgue almost every point is transitive, no absolutely continuous invariant probability measure can exist. This answers a long-standing open problem.

Pesin theory and equilibrium measures on the interval

Neil Dobbs (2015)

Fundamenta Mathematicae

Similarity:

We use Pesin theory to study possible equilibrium measures for a broad class of piecewise monotone maps of the interval and a broad class of potentials.

Absolutely continuous invariant measures for C2 unimodal maps satisfying the Collet-Eckmann conditions.

T. Nowicki, S. van Strien (1988)

Inventiones mathematicae

Similarity:

Physical measures for infinite-modal maps

Vítor Araújo, Maria José Pacifico (2009)

Fundamenta Mathematicae

Similarity:

We analyze certain parametrized families of one-dimensional maps with infinitely many critical points from the measure-theoretical point of view. We prove that such families have absolutely continuous invariant probability measures for a positive Lebesgue measure subset of parameters. Moreover, we show that both the density of such a measure and its entropy vary continuously with the parameter. In addition, we obtain exponential rate of mixing for these measures and also show that they...

Smoothness of invariant density for expanding transformations in higher dimensions.

Adl-Zarabi, Kourosh, Proppe, Harald (2000)

Journal of Applied Mathematics and Stochastic Analysis

Similarity:

On the lifting of invariant measures

Antoni Leon Dawidowicz (1990)

Annales Polonici Mathematici

Similarity:

Most expanding maps have no absolutely continuous invariant measure

Anthony Quas (1999)

Studia Mathematica

Similarity:

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.