Absolutely Continuous Invariant Measure Research Articles

Ulam’s Method for Computing Stationary Densities of Invariant Measures for Piecewise Convex Maps with Countably Infinite Number of Branches

Let [Formula: see text] be a piecewise convex map with countably infinite number of branches. In [Góra et al., 2022], the existence of Absolutely Continuous Invariant Measure (ACIM) [Formula: see text] for [Formula: see text] and the exactness of the system [Formula: see text] have been proven. In this paper, we develop an Ulam method for approximation of [Formula: see text], the density of ACIM [Formula: see text]. We construct a sequence [Formula: see text] of maps [Formula: see text] s.t. [Formula: see text] has a finite number of branches and the sequence [Formula: see text] converges to [Formula: see text] almost uniformly. Using supremum norms and Lasota–Yorke-type inequalities, we prove the existence of ACIMs [Formula: see text] for [Formula: see text] with the densities [Formula: see text]. For a fixed [Formula: see text], we apply Ulam’s method with [Formula: see text] subintervals to [Formula: see text] and compute approximations [Formula: see text] of [Formula: see text]. We prove that [Formula: see text] as [Formula: see text] both a.e. and in [Formula: see text]. We provide examples of piecewise convex maps [Formula: see text] with countably infinite number of branches and their approximations by [Formula: see text]’s with finite number of branches. For the increasing values of parameter [Formula: see text] we calculate the errors [Formula: see text].

Absolutely Continuous Invariant Measure for Generalized Horseshoe Maps

Absolutely Continuous Invariant Measure for Generalized Horseshoe Maps

Absolutely continuous invariant measures for random dynamical systems of beta-transformations

We consider an independent and identically distributed (i.i.d.) random dynamical system of simple linear transformations on the unit interval Tβ(x)=βx (mod 1), x∈[0,1] , β > 0, which are the so-called beta-transformations. For such a random dynamical system, including the case that it is generated by uncountably many maps, we give an explicit formula for the density function of a unique stationary measure under the assumption that the random dynamics is expanding in mean. As an application, in the case that the random dynamics is generated by finitely many maps and the maps are chosen according to a Bernoulli measure, we show that the density function is analytic as a function of parameter in the Bernoulli measure and give its derivative explicitly. Furthermore, for a non-i.i.d. random dynamical system of beta-transformations, we also give an explicit formula for the random densities of a unique absolutely continuous invariant measure under a certain strong expanding condition or under the assumption that the maps randomly chosen are close to the beta-transformation for a non-simple number in the sense of parameter β.

Piecewise Convex Deterministic Dynamical Systems and Weakly Convex Random Dynamical Systems and Their Invariant Measures

Absolutely continuous invariant measures of deterministic dynamical systems and random dynamical systems respectively are studied via a general spline maximum entropy optimization method. In the first part of this paper, we consider piecewise convex deterministic dynamical systems (maps) $$\tau : [0, 1]\rightarrow [0, 1]$$ and we study their absolutely continuous invariant measures. We assume that the deterministic piecewise convex transformation $$\tau $$ has a unique absolutely continuous invariant measure (acim) $$\mu ^*$$ with density $$f^*.$$ We present a general spline maximum entropy optimization method for the approximation of $$f^*.$$ The proof of convergence of our general spline maximum entropy optimization method is presented. A numerical example is presented for the general spline (linear, quadratic and cubic respectively) maximum entropy numerical scheme for the approximation of $$f^*.$$ In the second part of this paper, we generalize above results for weakly convex position dependent random map $$T=\{\tau _1(x),\tau _2(x),\ldots , \tau _K(x); p_1(x),p_2(x),\ldots ,p_K(x)\}$$ on $$I=[0, 1],$$ where $$\tau _k: [0, 1] \rightarrow [0, 1]), k=1, 2, \dots , K$$ is a piecewise convex map and $$\{p_1(x), p_2(x),\ldots ,p_K(x) \}$$ is a set of position dependent probabilities on [0, 1]. We assume that T has a unique acim $$\nu ^*$$ with density $$h^*.$$ We present a general spline maximum entropy optimization method for the approximation of $$h^*.$$ The proof of convergence of our numerical schemes is presented. Also, we present a numerical example of the general spline maximum entropy method for the approximation of $$h^*.$$

Absolutely continuous invariant measure of a map from grazing-impact oscillators

Abstract In this paper we investigate a one-dimensional map with unbounded derivative. The map is the limit of the Nordmark map which is the normal form of a discrete time representation of impact oscillators near grazing, i.e. when the dissipation of the systems is large, the Nordmark map can be viewed as a perturbation of the one-dimensional map. We prove that the map has an ergodic absolutely continuous invariant probability measure in a region of parameter space by constructing an induced Markov map.

ACIM for random intermittent maps: existence, uniqueness and stochastic stability

We study a random map T which consists of intermittent maps and a position-dependent probability distribution . We prove existence of a unique absolutely continuous invariant measure (ACIM) for the random map T. Moreover, we show that, as ϵ goes to zero, the invariant density of the random system T converges in the L 1-norm to the invariant density of the deterministic intermittent map T 1. The outcome of this article contains a first result on stochastic stability, in the strong sense, of intermittent maps.

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

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].

Absolutely continuous invariant measures for non-uniformly expanding maps

AbstractFor a large class of non-uniformly expanding maps of ℝm, with indifferent fixed points and unbounded distortion and that are non-necessarily Markovian, we construct an absolutely continuous invariant measure. We extend previously used techniques for expanding maps on quasi-Hölder spaces to our case. We give general conditions and provide examples to which our results apply.

Absolutely continuous invariant measures for some piecewise hyperbolic affine maps

AbstractA class of piecewise affine hyperbolic maps on a bounded subset of the plane is considered. It is shown that if a map from this class is sufficiently area-expanding then almost surely this map has an absolutely continuous invariant measure.

Differentiating the Absolutely Continuous Invariant Measure of an Interval Map f with Respect to f

Let the map f:[−1,1]→[−1,1] have a.c.i.m. ρ (absolutely continuous f-invariant measure with respect to Lebesgue). Let δρ be the change of ρ corresponding to a perturbation X=δf○f−1 of f. Formally we have, for differentiable A, but this expression does not converge in general. For f real-analytic and Markovian in the sense of covering (−1,1) m times, and assuming an analytic expanding condition, we show that is meromorphic in C, and has no pole at λ=1. We can thus formally write δρ(A)=Ψ(1).

Absolutely continuous invariant measures for random maps with position dependent probabilities

A random map is discrete-time dynamical system in which one of a number of transformations is randomly selected and applied at each iteration of the process. Usually the map τ k is chosen from a finite collection of maps with constant probability p k . In this note we allow the p k 's to be functions of position. In this case, the random map cannot be considered to be a skew product. The main result provides a sufficient condition for the existence of an absolutely continuous invariant measure for position dependent random maps on [0,1]. Geometrical and topological properties of sets of absolutely continuous invariant measures, attainable by means of position dependent random maps, are studied theoretically and numerically.

Absolutely continuous invariant measure of a piecewise concave mapping of [0, 1

Absolutely continuous invariant measure of a piecewise concave mapping of [0, 1

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

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.

Importance of Bounded Variation Condition for Stability of an Absolutely Continuous Invariant Measure

We give an example showing that the uniformly bounded variation condition in the compactness theorem of Góra and Boyarsky [ Canad. J. Math. 41 (1989), 855-869] is crucial for L 1 stability of an absolutely continuous invariant measure (acim) of a piecewise expanding transformation. The same example shows that a simple method of approximating an invariant density of a piecewise expanding transformation can give the wrong results.

Absolutely continuous invariant measures and random perturbations for certain one-dimensional maps

AbstractWe study the quadratic family and show that for a positive measure set of parameters the map has an absolutely continuous invariant measure that is stable under small random perturbations.

Absolutely continuous invariant measures for the family of maps x → rxe-bxwith application to the Belousov-Zhabotinski reaction

For the family of transformations τ(r,b,x)=rxe-bx which models the Poincare sections of the Belousov-Zhabotinski reaction we prove that there is an uncountable set A such that for each τξA,τ(r,b,˙) has an absolutely continuous invariant measure. We also approximate the density of this measure and thereby the Lyapunov exponent of τ(r,b,˙) .This is accomplished by proving that τ(r,b,˙) is conjugate, via an absolutely continuous homeomorphism, to a transormation T, where Tx is piecewise expanding, for some integers

Absolutely continuous invariant measures for piecewise expandingC 2 transformations inR N

LetS be a bounded region inR N and let ℊ={S i} i =1/m be a partition ofS into a finite number of subsets having piecewiseC 2 boundaries. We assume that whereC 2 segments of the boundaries meet, the angle subtended by tangents to these segments at the point of contact is bounded away from 0. Letτ:S →S be piecewiseC 2 on ℊ and expanding in the sense that there exists 0<σ< 1 such that for anyi=1, 2, ...,m, ‖Dτ i −1 ‖<σ, whereDτ i −1 is the derivative matrix ofτ i −1 and ‖ ‖ is the euclidean matrix norm. The main result provides an upper bound onσ which guarantees the existence of an absolutely continuous invariant measure forτ.

Absolutely Continuous Invariant Measures that are Maximal

Let $A$ be a certain irreducible $0{\text {-}}1$ matrix and let $\tau$ denote the family of piecewise linear Markov maps on $[0,1]$ which are consistent with $A$. The main result of this paper characterizes those maps in $\tau$ whose (unique) absolutely continuous invariant measure is maximal, and proves that for "most" of the maps that are consistent with $A$, the absolutely continuous invariant measure is not maximal.

Absolutely continuous invariant measures that are maximal

Let A be a certain irreducible 0-1 matrix and let T denote the family of piecewise linear Markov maps on [0, 1] which are consistent with A. The main result of this paper characterizes those maps in T whose (unique) absolutely continuous invariant measure is maximal, and proves that for most of the maps that are consistent with A, the absolutely continuous invariant measure is not maximal.