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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 on...

Invariant measures for position dependent random maps with continuous random parameters

Tomoki Inoue (2012)

Studia Mathematica

We consider a family of transformations with a random parameter and study a random dynamical system in which one transformation is randomly selected from the family and applied on each iteration. The parameter space may be of cardinality continuum. Further, the selection of the transformation need not be independent of the position in the state space. We show the existence of absolutely continuous invariant measures for random maps on an interval under some conditions.

Invariant measures for random dynamical systems [Book]

Katarzyna Horbacz (2008)

Invariant measures for the stable foliation on negatively curved periodic manifolds

François Ledrappier (2008)

Annales de l’institut Fourier

We classify reversible measures for the stable foliation on manifolds which are infinite covers of compact negatively curved manifolds. We extend the known results from hyperbolic surfaces to varying curvature and to all dimensions.

Invariant Measures of C...-Diffeomorphisms of Markov Type in R...

Piotr Bugiel (1992)

Monatshefte für Mathematik

Invariant measures of the geodesic flow and measures at infinity on negatively curved manifolds

Vadim A. Kaimanovich (1990)

Annales de l'I.H.P. Physique théorique

Invariant sets with zero measure and full Hausdorff dimension.

Barreira, Luis, Schmeling, Jörg (1997)

Electronic Research Announcements of the American Mathematical Society [electronic only]

Invariants l2 de relations d’équivalence et de groupes

Damien Gaboriau (2002)

Publications Mathématiques de l'IHÉS

IP-Dirichlet measures and IP-rigid dynamical systems: an approach via generalized Riesz products

Sophie Grivaux (2013)

Studia Mathematica

If ${(n_{k})}_{k \geq 1}$ is a strictly increasing sequence of integers, a continuous probability measure σ on the unit circle is said to be IP-Dirichlet with respect to ${(n_{k})}_{k \geq 1}$ if $σ ̂ (\sum_{k \in F} n_{k}) \to 1$ as F runs over all non-empty finite subsets F of ℕ and the minimum of F tends to infinity. IP-Dirichlet measures and their connections with IP-rigid dynamical systems have recently been investigated by Aaronson, Hosseini and Lemańczyk. We simplify and generalize some of their results, using an approach involving generalized Riesz products.

Isometric extensions, 2-cocycles and ergodicity of skew products

Alexandre Danilenko, Mariusz Lemańczyk (1999)

Studia Mathematica

We establish existence and uniqueness of a canonical form for isometric extensions of an ergodic non-singular transformation T. This is applied to describe the structure of commutors of the isometric extensions. Moreover, for a compact group G, we construct a G-valued T-cocycle α which generates the ergodic skew product extension $T_{α}$ and admits a prescribed subgroup in the centralizer of $T_{α}$ .

Isomorphic random Bernoulli shifts

V. Gundlach, G. Ochs (2000)

Colloquium Mathematicae

We develop a relative isomorphism theory for random Bernoulli shifts by showing that any random Bernoulli shifts are relatively isomorphic if and only if they have the same fibre entropy. This allows the identification of random Bernoulli shifts with standard Bernoulli shifts.

Isomorphism of regular Morse dynamical systems

J. Kwiatkowski (1982)

Studia Mathematica

Iterated function systems with continuous place dependent probabilities.

Jaroszewska, Joanna (2002)

Zeszyty Naukowe Uniwersytetu Jagiellońskiego. Universitatis Iagellonicae Acta Mathematica

Iterations of the Frobenius-Perron operator for parabolic random maps

Zbigniew S. Kowalski (2009)

Fundamenta Mathematicae

We describe totally dissipative parabolic extensions of the one-sided Bernoulli shift. For the fractional linear case we obtain conservative and totally dissipative families of extensions. Here, the property of conservativity seems to be extremely unstable.

Jordan tori and polynomial endomorphisms in $ℂ^{2}$

Manfred Denker, Stefan Heinemann (1998)

Fundamenta Mathematicae

For a class of quadratic polynomial endomorphisms $f : ℂ^{2} \to ℂ^{2}$ close to the standard torus map $(x, y) \to (x^{2}, y^{2})$ , we show that the Julia set J(f) is homeomorphic to the torus. We identify J(f) as the closure ℛ of the set of repelling periodic points and as the Shilov boundary of the set K(f) of points with bounded forward orbit. Moreover, it turns out that (J(f),f) is a mixing repeller and supports a measure of maximal entropy for f which is uniquely determined as the harmonic measure for K(f).

Jumps of entropy for $C^{r}$ interval maps

David Burguet (2015)

Fundamenta Mathematicae

We study the jumps of topological entropy for $C^{r}$ interval or circle maps. We prove in particular that the topological entropy is continuous at any $f \in C^{r} ([0, 1])$ with $h_{t o p} (f) > (l o g ⁺ | | f^{'} {| |}_{\infty}) / r$ . To this end we study the continuity of the entropy of the Buzzi-Hofbauer diagrams associated to $C^{r}$ interval maps.

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