On a class of stable conditional measures

2010 ◽  
Vol 31 (5) ◽  
pp. 1499-1515 ◽  
Author(s):  
EUGEN MIHAILESCU
Keyword(s):  
Invariant Measures ◽  
Reversible Systems ◽  
Absolutely Continuous ◽  
Stable Manifolds ◽  
Equilibrium Measures ◽  
Basic Set ◽  
Stable Dimension ◽  
Saddle Type ◽  
Local Stable Manifolds ◽  
Basic Sets

AbstractThe dynamics of endomorphisms (smooth non-invertible maps) presents many differences from that of diffeomorphisms or that of expanding maps; most methods from those cases do not work if the map has a basic set of saddle type with self-intersections. In this paper we study the conditional measures of a certain class of equilibrium measures, corresponding to a measurable partition subordinated to local stable manifolds. We show that these conditional measures are geometric probabilities on the local stable manifolds, thus answering in particular the questions related to the stable pointwise Hausdorff and box dimensions. These stable conditional measures are shown to be absolutely continuous if and only if the respective basic set is a non-invertible repeller. We find also invariant measures of maximal stable dimension, on folded basic sets. Examples are given, too, for such non-reversible systems.

Dynamical behavior of endomorphisms on certain invariant sets

2013 ◽  
Vol 63 (1) ◽  
Author(s):  
Diana Putan ◽  
Diana Stan
Keyword(s):  
Hausdorff Dimension ◽  
Dynamical Behavior ◽  
Invariant Sets ◽  
Hyperbolic Polynomial ◽  
Stable Manifolds ◽  
Basic Set ◽  
Polynomial Endomorphisms ◽  
Complex Projective ◽  
Local Stable Manifolds ◽  
Basic Sets

AbstractWe study the Hausdorff dimension of the intersection between local stable manifolds and the respective basic sets of a class of hyperbolic polynomial endomorphisms on the complex projective space ℙ2. We consider the perturbation (z 2 +ɛz +bɛw 2, w 2) of (z 2, w 2) and we prove that, for b sufficiently small, it is injective on its basic set Λɛ close to Λ:= {0} × S 1. Moreover we give very precise upper and lower estimates for the Hausdorff dimension of the intersection between local stable manifolds and Λɛ, in the case of these maps.

scholarly journals Degenerate random perturbations of Anosov diffeomorphisms

2009 ◽  
Vol 30 (3) ◽  
pp. 931-951 ◽  
Author(s):  
TATIANA YARMOLA
Keyword(s):  
Riemannian Manifolds ◽  
Invariant Measures ◽  
Random Perturbations ◽  
Absolutely Continuous ◽  
Stable Manifolds ◽  
Anosov Diffeomorphisms ◽  
Absolutely Continuous Invariant Measures ◽  
Generic Rank ◽  
Toral Automorphisms ◽  
Absolutely Continuous Invariant

AbstractThis paper deals with random perturbations of diffeomorphisms on n-dimensional Riemannian manifolds with distributions supported on k-dimensional disks, where k<n. First we demonstrate general but not very intuitive conditions which guarantee that all invariant measures for rank-k random perturbations of C2 diffeomorphisms are absolutely continuous with respect to the Riemannian measure on M. For two subclasses of Anosov diffeomorphisms, hyperbolic toral automorphisms and Anosov diffeomorphisms with codimension 1 stable manifolds, the above conditions are modified in order to relate k-dimensional disks that support the distributions to certain foliations that arise from Anosov diffeomorphisms. We conclude that generic rank-k random perturbations have absolutely continuous invariant measures.

CHAOTIC BEHAVIOR OF HIGHER DIMENSIONAL TRANSFORMATIONS DEFINED ON COUNTABLE PARTITIONS

1993 ◽  
Vol 03 (04) ◽  
pp. 1045-1049
Author(s):  
A. BOYARSKY ◽  
Y. S. LOU
Keyword(s):  
Invariant Measure ◽  
Additional Condition ◽  
Invariant Measures ◽  
Chaotic Behavior ◽  
Finite Collection ◽  
Continuous Invariant Measure ◽  
Absolutely Continuous ◽  
Higher Dimensional ◽  
Absolutely Continuous Invariant Measure ◽  
Absolutely Continuous Invariant

Jablonski maps are higher dimensional maps defined on rectangular partitions with each component a function of only one variable. It is well known that expanding Jablonski maps have absolutely continuous invariant measures. In this note we consider Jablonski maps defined on countable partitions. Such maps occur, for example, in multivariable number theoretic problems. The main result establishes the existence of an absolutely continuous invariant measure for Jablonski maps on a countable partition with the additional condition that the images of all the partition elements form a finite collection. An example is given.

CONVERGENCE OF INVARIANT MEASURES FOR DEGENERATING SEQUENCES OF HYPERBOLIC MAPPINGS WITH SINGULARITIES

1996 ◽  
Vol 06 (06) ◽  
pp. 1143-1151
Author(s):  
E. A. SATAEV
Keyword(s):  
Invariant Measure ◽  
Invariant Measures ◽  
Continuous Invariant Measure ◽  
Absolutely Continuous ◽  
Absolutely Continuous Invariant Measure ◽  
Absolutely Continuous Invariant ◽  
Expanding Mapping

This paper is devoted to presenting and giving a sketch of the proof of the theorem which states that, if the sequence of hyperbolic mappings with singularities converges to degenerating piecewise expanding mapping, then the corresponding sequence of measures of a Sinai-Bowen-Ruelle type converges to an absolutely continuous invariant measure.

Quenched stochastic stability for eventually expanding-on-average random interval map cocycles

2018 ◽  
Vol 39 (10) ◽  
pp. 2769-2792
Author(s):  
GARY FROYLAND ◽  
CECILIA GONZÁLEZ-TOKMAN ◽  
RUA MURRAY
Keyword(s):  
Invariant Measures ◽  
Single Step ◽  
Absolutely Continuous ◽  
Random Maps ◽  
Driving Mechanisms ◽  
The Family ◽  
Ulam’S Method ◽  
Ulam's Method ◽  
Invariant Densities ◽  
Absolutely Continuous Invariant

The paper by Froyland, González-Tokman and Quas [Stability and approximation of random invariant densities for Lasota–Yorke map cocycles.Nonlinearity27(4) (2014), 647] established fibrewise stability of random absolutely continuous invariant measures (acims) for cocycles of random Lasota–Yorke maps under a variety of perturbations, including ‘Ulam’s method’, a popular numerical method for approximating acims. The expansivity requirements of Froylandet alwere that the cocycle (or powers of the cocycle) should be ‘expanding on average’ before applying a perturbation, such as Ulam’s method. In the present work, we make a significant theoretical and computational weakening of the expansivity hypotheses of Froylandet al, requiring only that the cocycle be eventually expanding on average, and importantly,allowing the perturbation to be applied after each single step of the cocycle. The family of random maps that generate our cocycle need not be close to a fixed map and our results can handle very general driving mechanisms. We provide a detailed numerical example of a random Lasota–Yorke map cocycle with expanding and contracting behaviour and illustrate the extra information carried by our fibred random acims, when compared to annealed acims or ‘physical’ random acims.

Invariant measures for interval maps with different one-sided critical orders

2013 ◽  
Vol 35 (3) ◽  
pp. 835-853 ◽  
Author(s):  
HONGFEI CUI ◽  
YIMING DING
Keyword(s):  
Probability Measure ◽  
Critical Points ◽  
Mild Condition ◽  
Invariant Measures ◽  
Invariant Probability Measure ◽  
Absolutely Continuous ◽  
Interval Maps ◽  
Jump Discontinuities ◽  
Point Set ◽  
Invent Math

AbstractFor an interval map whose critical point set may contain critical points with different one-sided critical orders and jump discontinuities, under a mild condition on critical orbits, we prove that it has an invariant probability measure which is absolutely continuous with respect to Lebesgue measure by using the methods of Bruin et al [Invent. Math. 172(3) (2008), 509–533], together with ideas from Nowicki and van Strien [Invent. Math. 105(1) (1991), 123–136]. We also show that it admits no wandering intervals.

scholarly journals Certain semigroups embeddabable in topological groups

1982 ◽  
Vol 33 (1) ◽  
pp. 30-39 ◽  
Author(s):  
Heneri A. M. Dzinotyiweyi
Keyword(s):  
Invariant Measures ◽  
Locally Compact Group ◽  
Continuous Measure ◽  
Topological Groups ◽  
Empty Interior ◽  
Locally Compact ◽  
Absolutely Continuous ◽  
Topological Semigroups ◽  
Absolutely Continuous Measure ◽  
Locally Compact Topological Group

AbstractIn this paper we study commutative topological semigroups S admitting an absolutely continuous measure. When S is cancellative we show that S admits a weaker topology J with respect to which (S, J) is embeddable as a subsemigroup with non-empty interior in some locally compact topological group. As a consequence, we deduce certain results related to the existence of invariant measures on S and for a large class of locally compact topological semigroups S, we associate S with some useful topological subsemigroup of a locally compact group.

Refined scales of weak-mixing dynamical systems: typical behaviour

2019 ◽  
Vol 40 (12) ◽  
pp. 3296-3309
Author(s):  
SILAS L. CARVALHO ◽  
CÉSAR R. DE OLIVEIRA
Keyword(s):  
Invariant Measures ◽  
Dynamical Behaviour ◽  
Lebesgue Spaces ◽  
Weak Mixing ◽  
Axiom A ◽  
Typical Behaviour ◽  
Topologically Mixing ◽  
Continuous Measures ◽  
Basic Sets ◽  
Measure Preserving Transformations

We study sets of measure-preserving transformations on Lebesgue spaces with continuous measures taking into account extreme scales of variations of weak mixing. It is shown that the generic dynamical behaviour depends on subsequences of time going to infinity. We also present corresponding generic sets of (probability) invariant measures with respect to topological shifts over finite alphabets and Axiom A diffeomorphisms over topologically mixing basic sets.

Sign in / Sign up

Export Citation Format

Share Document