Absolutely Continuous Invariant Measures Research Articles

Existence of Absolutely Continuous Invariant Measures for C1 Expanding Circle Maps

Existence of Absolutely Continuous Invariant Measures for C1 Expanding Circle 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.

Absolutely Continuous Invariant Measures for Piecewise Expanding Chaotic Transformations in R^n with Summable Oscillations of Derivative

This paper presents a detailed investigation of absolutely continuous invariant measures (ACIMs) for piecewise expanding chaotic transformations in , with particular attention paid to the case where the derivative has summable oscillations. ACIMs are important objects in the study of dynamical systems, as they provide a way to understand the long-term behavior of trajectories and the statistical properties of the system. The paper covers a range of important topics related to ACIMs, including the boundedness condition, distortion condition, localization condition, and Schmitt's condition. It also discusses the Perron-Frobenius operator, which plays a critical role in the existence and properties of ACIMs. The main result of the paper is the proof that the Perron-Frobenius operator is constrictive, which implies the existence of a finite number of ergodic ACIMs that satisfy Schmitt's condition and a condition dependent on the defining partition. This finding has significant implications for the understanding of complex systems and the advancement of research in this field. The paper also discusses the relationship between ACIMs and dynamical systems, highlighting the role of ACIMs in ergodic theory. Overall, this paper provides a valuable reference for researchers interested in the study of ACIMs and their significance in the analysis of dynamical systems and ergodic theory.

On the Absolutely Continuous Invariant Measures for Random Maps

We study the dynamics of a new family of transformations. We find a general formula for the invariant density of any transformation in our family. The properties of the family allow us to prove that the invariant density function [Formula: see text] of random map constructed from our family maps [Formula: see text] is the combination [Formula: see text] where [Formula: see text] are the invariant density functions of [Formula: see text] respectively. We also consider another family of transformations, and prove that the invariant density for any transformation of the family is [Formula: see text].

Absolutely continuous invariant measures for some piecewise hyperbolic affine maps – CORRIGENDUM

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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 measures for non-autonomous dynamical systems

We consider the non autonomous dynamical system $\{\tau_{n}\},$ where $\tau_{n}$ is a continuous map $X\rightarrow X,$ and $X$ is a compact metric space. We assume that $\{\tau_{n}\}$ converges uniformly to $\tau .$ The inheritance of chaotic properties as well as topological entropy by $\tau $ from the sequence $\{\tau_{n}\}$ has been studied in \cite{Can1, Can2, Li,Ste,Zhu}. In \cite{You} the generalization of SRB\ measures to non-autonomous systems has been considered. In this paper we study absolutely continuous invariant measures (acim) for non autonomous systems. After generalizing the Krylov-Bogoliubov Theorem \cite{KB} and Straube's Theorem \cite{Str} to the non autonomous setting, we prove that under certain conditions the limit map $\tau $ of a non autonomous sequence of maps $\{\tau_n\}$ with acims has an acim.

Absolutely continuous invariant measures for random non-uniformly expanding maps

We prove existence of (at most denumerable many) absolutely continuous invariant probability measures for random one-dimensional dynamical systems with asymptotic expansion. If the rate of expansion (Lyapunov exponents) is bounded away from zero, we obtain finitely many ergodic absolutely continuous invariant probability measures, describing the asymptotics of almost every point. We also prove a similar result for higher-dimensional random non-uniformly expanding dynamical systems. The results are consequences of the construction of such measures for skew-products with essentially arbitrary base dynamics and asymptotic expansion along the fibers. In both cases our method deals with either critical o singular points for the random maps.

Invariant measures for general induced maps and towers

Absolutely continuous invariant measures (acims) for general induced transformations are shown to be related, in a natural way, to popular tower constructions regardless of any particulars of the latter. When combined with (an appropriate generalization of) the known integrability criterion for the existence of such acims, this leads to necessary and sufficient conditions under which acims can be lifted to, or projected from, nonsingular extensions.

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.

Absolutely continuous invariant measures for expansive diffeomorphisms of the 2-torus

The aim of this paper is to establish an equivalent criterion for certain expansive diffeomorphisms of the 2-torus to admit an invariant Borel probability measure that is absolutely continuous with respect to the Riemannian volume. Our result is closely related to the well known Livsic-Sinai theorem for Anosov diffeomorphisms.

Absolutely Continuous Invariant Measures that Cannot be Observed Experimentally

We study an example of a random map where the component maps have absolutely continuous invariant measures (acims), but where computer experiments reveal the surprising fact that all orbits eventually fall into a stable periodic orbit. This is all the more surprising as we prove that this random map admits an acim, $\mu$. We study this phenomenon and explain why $\mu$ cannot be observed experimentally.

Optimal partition choice for invariant measure approximation for one-dimensional maps

The problem of finding absolutely continuous invariant measures (ACIMs) for a dynamical system can be formulated as a fixed point problem for a Markov operator (the Perron–Frobenius operator). This is an infinite-dimensional problem. Ulam's method replaces the Perron–Frobenius operator by a sequence of finite rank approximations whose fixed points are relatively easy to compute numerically. This paper concerns the optimal choice of Ulam approximations for one-dimensional maps; an adaptive partition selection is used to tailor the approximations to the structure of the invariant measure. The main idea is to select a partition which equally distributes the square root of the derivative of the invariant density amongst the bins of the partition. The results are illustrated for the logistic map where the ACIMs may have inverse square root singularities in their density functions. O(log n/n) convergence rates can be expected, whereas a non-adaptive algorithm yields O(n−1/2) at best. Studying the convergence of the adaptive algorithm allows an estimate to be made of the measure of the Jakobson parameter set (those logistic maps which admit an ACIM).

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 measures for most piecewise smooth expanding maps

We introduce transversality conditions for a piecewise smooth map in any dimension with a sufficiently regular discontinuity set. We prove that maps satisfying these conditions have certain nice properties and, among all maps with a common set of discontinuities, form an open and dense subset. This, in conjunction with other works of the author, implies the existence of stochastically stable invariant densities for an open and dense set of piecewise smooth expanding maps.

Absolutely continuous invariant measures for expanding piecewise linear maps

We prove the existence of absolutely continuous invariant measures for arbitrary expanding piecewise linear maps on bounded polyhedral domains in Euclidean spaces ℝ d .

Absolutely continuous invariant measures for multidimensional expanding maps

We investigate the existence and statistical properties of absolutely continuous invariant measures for multidimensional expanding maps with singularities. The key point is the establishment of a spectral gap in the spectrum of the transfer operator. Our assumptions appear quite naturally for maps with singularities. We allow maps that are discontinuous on some extremely wild sets, the shape of the discontinuities being completely ignored with our approach.

Absolutely continuous invariant probability measures for arbitrary expanding piecewise $\mathbb{R}$-analytic mappings of the plane

We prove that any expanding piecewise real-analytic map of a bounded region of the plane admits absolutely continuous invariant probability measures.

Absolutely Continuous Invariant Measures¶for Piecewise Real-Analytic Expanding Maps¶on the Plane

We prove the existence of absolutely continuous invariant measures for piecewise real-analytic expanding maps on bounded regions in the plane.