scholarly journals W-LIKE MAPS WITH VARIOUS INSTABILITIES OF ACIM'S

2013 ◽  
Vol 23 (05) ◽  
pp. 1350079
Author(s):  
ZHENYANG LI
Keyword(s):  
Lower Bound ◽  
Continuous Measure ◽  
Singular Measure ◽  
Absolutely Continuous ◽  
Expanding Map ◽  
Absolutely Continuous Invariant Measure ◽  
Absolutely Continuous Measure ◽  
Invariant Densities ◽  
Absolutely Continuous Invariant ◽  
Piecewise Expanding Maps

This paper generalizes the results of [Li et al., 2011] and then provides an interesting example. We construct a family of W-like maps {Wa} with a turning fixed point having slope s1 on one side and –s2 on the other. Each Wa has an absolutely continuous invariant measure μa. Depending on whether [Formula: see text] is larger, equal or smaller than 1, we show that the limit of μa is a singular measure, a combination of singular and absolutely continuous measure or an absolutely continuous measure, respectively. It is known that the invariant density of a single piecewise expanding map has a positive lower bound on its support. In Sec. 4 we give an example showing that in general, for a family of piecewise expanding maps with slopes larger than 2 in modulus and converging to a piecewise expanding map, their invariant densities do not necessarily have a uniform positive lower bound on the support.

Uniqueness of Invariant Densities for Certain Random Maps of The Interval

1987 ◽  
Vol 30 (3) ◽  
pp. 301-308 ◽  
Author(s):  
Abraham Boyarsky
Keyword(s):  
Piecewise Linear ◽  
Sufficient Conditions ◽  
Time Process ◽  
Continuous Invariant Measure ◽  
Absolutely Continuous ◽  
Random Maps ◽  
Absolutely Continuous Invariant Measure ◽  
Invariant Densities ◽  
Absolutely Continuous Invariant

AbstractA random map is a discrete time process in which one of a number of maps, 𝓜, is chosen at random at each stage and applied. In this note we study a random map, where 𝓜 is a set of piecewise linear Markov maps on [0, 1]. Sufficient conditions are presented which allow the determination of the unique absolutely continuous invariant measure of the process.

scholarly journals Un exemple de transformation dilatante et C1 par morceaux de l'intervalle, sans probabilité absolument continue invariante

1989 ◽  
Vol 9 (1) ◽  
pp. 101-113 ◽  
Author(s):  
P. Gora ◽  
B. Schmitt
Keyword(s):  
Probability Measure ◽  
Modulus Of Continuity ◽  
Invariant Probability Measure ◽  
Negative Answer ◽  
Continuous Measure ◽  
Absolutely Continuous ◽  
Absolutely Continuous Measure ◽  
Absolutely Continuous Invariant

AbstractWe construct a transformation on the interval [0, 1] into itself, piecewiseC1 and expansive, which doesn't admit any absolutely continuous invariant probability measure (a.c.i.p.).So in this case we give a negative answer to a question by Anosov: is C1 character sufficient for the existence of absolutely continuous measure?Moreover, in our example,ƒ' has a modulus of type K/(|1+|log|x‖); it is known that a modulus of continuity of type K/(1+|log|x‖)1+γ, γ>0 implies the existence of a.c.i.p..

Absolutely continuous invariant measures for piecewise expanding C2 transformations in Rn on domains with cusps on the boundaries

1996 ◽  
Vol 16 (1) ◽  
pp. 1-18 ◽  
Author(s):  
Kourosh Adl-Zarabi
Keyword(s):  
Lower Bound ◽  
Invariant Measure ◽  
Finite Number ◽  
Invariant Measures ◽  
Bounded Region ◽  
Continuous Invariant Measure ◽  
Absolutely Continuous ◽  
Absolutely Continuous Invariant Measure ◽  
Derivative Matrix ◽  
Absolutely Continuous Invariant

AbstractLet Ω be a bounded region in Rn and let be a partition of Ω into a finite number of subsets having piecewise C2 boundaries. The boundaries may contain cusps. Let τ: Ω → Ω be piecewise C2 on and expanding in the sense that there exists α > 1 such that for any i = 1, 2,…,m, where is the derivative matrix of and ‖·‖ is the euclidean matrix norm. The main result provides a lower bound on α which guarantees the existence of an absolutely continuous invariant measure for τ.

Non-ergodicity for C1 expanding maps and g-measures

1996 ◽  
Vol 16 (3) ◽  
pp. 531-543 ◽  
Author(s):  
Anthony N. Quasf
Keyword(s):  
Invariant Measure ◽  
Lebesgue Measure ◽  
Probability Measures ◽  
Continuous Invariant Measure ◽  
Expanding Maps ◽  
Absolutely Continuous ◽  
Expanding Map ◽  
Absolutely Continuous Invariant Measure ◽  
Absolutely Continuous Invariant

AbstractWe introduce a procedure for finding C1 Lebesgue measure-preserving maps of the circle isomorphic to one-sided shifts equipped with certain invariant probability measures. We use this to construct a C1 expanding map of the circle which preserves Lebesgue measure, but for which Lebesgue measure is non-ergodic (that is there is more than one absolutely continuous invariant measure). This is in contrast with results for C1+e maps. We also show that this example answers in the negative a question of Keane's on uniqueness of g-measures, which in turn is based on a question raised by an incomplete proof of Karlin's dating back to 1953.

scholarly journals Family of piecewise expanding maps having singular measure as a limit of ACIMs

2011 ◽  
Vol 33 (1) ◽  
pp. 158-167 ◽  
Author(s):  
ZHENYANG LI ◽  
PAWEŁ GÓRA ◽  
ABRAHAM BOYARSKY ◽  
HARALD PROPPE ◽  
PEYMAN ESLAMI
Keyword(s):  
Piecewise Linear ◽  
Dynamical Behavior ◽  
Singular Measure ◽  
Expanding Maps ◽  
Linear Maps ◽  
Absolutely Continuous Invariant Measure ◽  
Piecewise Linear Maps ◽  
Point Measure ◽  
Invariant Densities ◽  
Absolutely Continuous Invariant

AbstractKeller [Stochastic stability in some chaotic dynamical systems. Monatsh. Math.94(4) (1982), 313–333] introduced families of W-shaped maps that can have a great variety of behaviors. As a family approaches a limit W map, he observed behavior that was either described by a probability density function (PDF) or by a singular point measure. Based on this, Keller conjectured that instability of the absolutely continuous invariant measure (ACIM) can result only from the existence of small invariant neighborhoods of the fixed critical point of the limit map. In this paper, we show that the conjecture is not true. We construct a very simple family of W-maps with ACIMs supported on the whole interval, whose limiting dynamical behavior is captured by a singular measure. Key to the analysis is the use of a general formula for invariant densities of piecewise linear and expanding maps [P. Góra. Invariant densities for piecewise linear maps of interval. Ergod. Th. & Dynam. Sys. 29(5) (2009), 1549–1583].

scholarly journals THE DIFFUSION COEFFICIENT FOR PIECEWISE EXPANDING MAPS OF THE INTERVAL WITH METASTABLE STATES

2012 ◽  
Vol 12 (01) ◽  
pp. 1150005 ◽  
Author(s):  
DMITRY DOLGOPYAT ◽  
PAUL WRIGHT
Keyword(s):  
Diffusion Coefficient ◽  
Markov Chain ◽  
Bounded Variation ◽  
Continuous Time ◽  
Probability Measures ◽  
Continuous Time Markov Chain ◽  
Absolutely Continuous ◽  
Expanding Map ◽  
Absolutely Continuous Invariant ◽  
Piecewise Expanding Maps

Consider a piecewise smooth expanding map of the interval possessing several invariant subintervals and the same number of ergodic absolutely continuous invariant probability measures (ACIMs). After this system is perturbed to make the subintervals lose their invariance in such a way that there is a unique ACIM, we show how to approximate the diffusion coefficient for an observable of bounded variation by the diffusion coefficient of a related continuous time Markov chain.

scholarly journals Approximating invariant densities of metastable systems

2010 ◽  
Vol 31 (5) ◽  
pp. 1345-1361 ◽  
Author(s):  
CECILIA GONZÁLEZ-TOKMAN ◽  
BRIAN R. HUNT ◽  
PAUL WRIGHT
Keyword(s):  
Lebesgue Measure ◽  
Convex Combination ◽  
Invariant Sets ◽  
Positive Lebesgue Measure ◽  
Probability Measures ◽  
Absolutely Continuous ◽  
Expanding Map ◽  
Invariant Densities ◽  
Absolutely Continuous Invariant ◽  
Metastable Systems

AbstractWe consider a piecewise smooth expanding map on an interval which has two invariant subsets of positive Lebesgue measure and exactly two ergodic absolutely continuous invariant probability measures (ACIMs). When this system is perturbed slightly to make the invariant sets merge, we describe how the unique ACIM of the perturbed map can be approximated by a convex combination of the two initial ergodic ACIMs. The result is generalized to the case of finitely many invariant components.

scholarly journals Nice sets and invariant densities in complex dynamics

2010 ◽  
Vol 150 (1) ◽  
pp. 157-165 ◽  
Author(s):  
NEIL DOBBS
Keyword(s):  
Complex Dynamics ◽  
Julia Set ◽  
Integrability Condition ◽  
Singular Set ◽  
Absolutely Continuous ◽  
Absolutely Continuous Invariant Measure ◽  
Recent Theorem ◽  
Meromorphic Maps ◽  
Invariant Densities ◽  
Absolutely Continuous Invariant

AbstractIn complex dynamics, we construct a so-called nice set (one for which the first return map is Markov) around any point which is in the Julia set but not in the post-singular set, adapting a construction of Rivera–Letelier. This simplifies the study of absolutely continuous invariant measures. We prove a converse to a recent theorem of Kotus and Świątek, so for a certain class of meromorphic maps the absolutely continuous invariant measure is finite if and only if an integrability condition is satisfied.

scholarly journals THE SMOOTHNESS OF ORBITAL MEASURES ON NONCOMPACT SYMMETRIC SPACES

2021 ◽  
pp. 1-20
Author(s):  
SANJIV KUMAR GUPTA ◽  
KATHRYN E. HARE
Keyword(s):  
Symmetric Space ◽  
Symmetric Spaces ◽  
Continuous Measure ◽  
Convolution Product ◽  
Absolutely Continuous ◽  
Regular Elements ◽  
Connected Subgroup ◽  
Decay Properties ◽  
Absolutely Continuous Measure ◽  
Special Case

Abstract Let $G/K$ be an irreducible symmetric space, where G is a noncompact, connected Lie group and K is a compact, connected subgroup. We use decay properties of the spherical functions to show that the convolution product of any $r=r(G/K)$ continuous orbital measures has its density function in $L^{2}(G)$ and hence is an absolutely continuous measure with respect to the Haar measure. The number r is approximately the rank of $G/K$ . For the special case of the orbital measures, $\nu _{a_{i}}$ , supported on the double cosets $Ka_{i}K$ , where $a_{i}$ belongs to the dense set of regular elements, we prove the sharp result that $\nu _{a_{1}}\ast \nu _{a_{2}}\in L^{2},$ except for the symmetric space of Cartan class $AI$ when the convolution of three orbital measures is needed (even though $\nu _{a_{1}}\ast \nu _{a_{2}}$ is absolutely continuous).

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