Continuous Invariant Measure Research Articles

Existence of Absolutely Continuous Invariant Measures for C1 Expanding Circle Maps

Existence of Absolutely Continuous Invariant Measures for C1 Expanding Circle Maps

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.

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.

Computing Invariant Measures of Weakly Convex Random Maps with Position Dependent Probabilities

A random map is closely related to a chaotic system consisting of a collection of measurable transformations. For a class of weakly convex random maps, we propose piecewise linear or quadratic Markov methods for numerically computing an absolutely continuous invariant measure and prove their convergence.

Dynamics of Ostrowski skew-product: 1. Limit laws and Hausdorff dimensions

We present a dynamical study of Ostrowski’s map based on the use of transfer operators. The Ostrowski dynamical system is obtained as a skew-product of the Gauss map (it has the Gauss map as a base and intervals as fibers) and produces expansions of real numbers with respect to an irrational base given by continued fractions. By studying spectral properties of the associated transfer operators, we show that the absolutely continuous invariant measure of the Ostrowski dynamical system has exponential mixing properties. We deduce a central limit theorem for random variables of an arithmetic nature, and motivated by applications in inhomogeneous Diophantine approximation, we also get Bowen–Ruelle type implicit estimates in terms of spectral elements for the Hausdorff dimension of a bounded digit set.

Memory Maps with Elliptical Trajectories

A family of maps with memory, parameterized by [Formula: see text], is shown to have either periodic trajectories or dense trajectories on ellipses which support absolutely continuous invariant measures. Furthermore, for [Formula: see text], i.e. [Formula: see text] with [Formula: see text] and [Formula: see text], all points except [Formula: see text] either go into a polygonal region centered at [Formula: see text] if [Formula: see text] is rational, or are attracted to an elliptical region having the same center, if [Formula: see text] is irrational. In the polygonal case, we examine a mechanism for the appearance of islands supporting absolutely continuous invariant measures.

Decay of correlations for critically intermittent systems

For a family of random intermittent dynamical systems with a superattracting fixed point we prove that a phase transition occurs for the existence of an absolutely continuous invariant probability measure depending on the randomness parameters and the orders of the maps at the superattracting fixed point. In case the systems have an absolutely continuous invariant probability measure, we show that the systems are mixing and that correlations decay polynomially even though some of the deterministic maps present in the system have exponential decay of correlations. This contrasts other known results, where a system maintains exponential decay of correlations under stochastic perturbations of a deterministic map with exponential rate of mixing, see e.g. Baladi and Viana (1996 Ann. Sci. l’Ecole Norm. Sup. 29 483–517).

Intermittency generated by attracting and weakly repelling fixed points

Recently for a class of critically intermittent random systems a phase transition was found for the finiteness of the absolutely continuous invariant measure. The systems for which this result holds are characterized by the interplay between a superexponentially attracting fixed point and an exponentially repelling fixed point. In this article we consider a closely related family of random systems with exponentially fast attraction to and polynomially fast repulsion from two fixed points, and show that such a phase transition still exists. The method of the proof however is different and relies on the construction of a suitable invariant set for the transfer operator.

The natural extension of the random beta-transformation

AbstractWe construct a geometrico-symbolic version of the natural extension of the random $\beta $ -transformation introduced by Dajani and Kraaikamp [Random $\beta $ -expansions. Ergod. Th. & Dynam. Sys.23(2) (2003) 461–479]. This construction provides a new proof of the existence of a unique absolutely continuous invariant probability measure for the random $\beta $ -transformation, and an expression for its density. We then prove that this natural extension is a Bernoulli automorphism, generalizing to the random case the result of Smorodinsky [ $\beta $ -automorphisms are Bernoulli shifts. Acta Math. Acad. Sci. Hungar.24 (1973), 273–278] about the greedy transformation.

On Coupling Lemma and Stochastic Properties with Unbounded Observables for 1-d Expanding Maps

In this paper, we establish a coupling lemma for standard families in the setting of piecewise expanding interval maps with countably many branches. Our coupling method only requires that the expanding map satisfies three assumptions: (H1) Chernov’s one-step expansion at q-scale; (H2) dynamically Hölder continuity of log Jacobian (on each branch); (H3) eventually covering over a magnet interval, which are much weaker than the standard assumptions for uniformly expanding maps. We further conclude the existence of an absolutely continuous invariant probability measure and establish the regularity of its density function. Moreover, we obtain the exponential decay of correlations and the almost sure invariance principle (which is a functional version of the central limit theorem) with respect to a large class of unbounded observables that we call “dynamically Hölder series". Our approach is particularly powerful for the piecewise expanding maps which do not satisfy the “big image property" and the maps that have inverse Jacobian of low regularity. As few is known on the statistical properties of such maps in the literature, we demonstrate our results for a specific class of piecewise expanding maps of this kind.

On absolutely continuous invariant measures and Krieger-type of Markov subshifts

It is shown that for a nonsingular conservative shift on a topologically mixing Markov subshift with the Doeblin condition the only possible absolutely continuous shift-invariant measure is a Markov measure. Moreover, if it is not equivalent to a homogeneous Markov measure then the shift is of Krieger-type III1. A criteria for equivalence of Markov measures is included.

Distribution of the reduced quadratic irrationals arising from the odd continued fraction expansion

This paper investigates the quadratic irrationals that arise as periodic points of the Gauss type shift associated to the odd continued fraction expansion. It is shown that these numbers, which we call O-reduced, when ordered by the length of the associated closed primitive geodesic on some modular surface Γ∖H, are equidistributed with respect to the Lebesgue absolutely continuous invariant probability measure of the Odd Gauss shift.

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].

Invariant densities for random continued fractions

We continue the study of random continued fraction expansions, generated by random application of the Gauss and the Rényi backward continued fraction maps. We show that this random dynamical system admits a unique absolutely continuous invariant measure with smooth density.

Lower bound in Pesin formula of C 1 interval maps* *This research was partially supported by NSFC (12071328, 11790274, 12122113), IEPJ, BK20211595 and Tang Scholar.

We prove that the lower bound in Pesin formula holds for any integrable absolutely continuous invariant measure of a piecewise C 1 interval map, i.e., the metric entropy is bounded from below by the Lyapunov exponent. Consequently, Pesin formula holds whenever Ruelle inequality is valid.

Invariant measures for random expanding on average Saussol maps

In this paper, we investigate the existence of random absolutely continuous invariant measures (ACIP) for random expanding on average Saussol maps in higher dimensions. This is done by the establishment of a random Lasota–Yorke inequality for the transfer operators on the space of bounded oscillation. We prove that the number of ergodic skew product ACIPs is finite and will provide an upper bound for the number of these ergodic ACIPs. This work can be seen as a generalization of the work in [F. Batayneh and C. González-Tokman, On the number of invariant measures for random expanding maps in higher dimensions, Discrete Contin. Dyn. Syst. 41 (2021) 5887–5914] on admissible random Jabłoński maps to a more general class of higher-dimensional random maps.

Phase transition in piecewise linear random maps in the interval.

We consider a simple example of a one-parameter family of random maps in the interval that exhibits a phase transition phenomenon in the sense of a spontaneous transition from non-existence to existence of an absolutely continuous invariant measure by changing the parameter. For this example of random maps, we estimate numerically the critical value at which the so-called transition occurs. This is done through the numerical computation of its invariant densities and the Lyapunov exponent. We further investigate the behavior of the rate of decay of correlations for the different values of the parameter, changing from power-law-like to exponential-like behavior. We also study a family of random maps with a non-expansive branch having no phase transition. For this family of random maps, we compute the invariant densities for all values of the parameter.

There are no σ-finite absolutely continuous invariant measures for multicritical circle maps * *The first author has been supported by ‘Projeto Temático Dinâmica em Baixas Dimensões’ FAPESP Grant 2016/25053-8, while the second author has been supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) and by Coordenação de Aperfeiçoamento de Pessoal de Nível Superior-Brasil (CAPES) Grant 23038.009189/2013-05.

It is well-known that every multicritical circle map without periodic orbits admits a unique invariant Borel probability measure which is purely singular with respect to Lebesgue measure. Can such a map leave invariant an infinite, σ-finite invariant measure which is absolutely continuous with respect to Lebesgue measure? In this paper, using an old criterion due to Katznelson, we show that the answer to this question is no.