On systems with finite ergodic degree

Advances in Mathematics, 1985

We show that one-dimensional maps f with strictly positive Lyapunov exponents almost everywhere admit an absolutely continuous invariant measure. If f is topologically transitive, some power of f is mixing and, in particular, the correlation of Hölder continuous observables decays to zero. The main objective of this paper is to show that the rate of decay of correlations is determined, in some situations, by the average rate at which typical points start to exhibit exponential growth of the derivative. http://journals.cambridge.org Downloaded: 07 Dec 2012 IP address: 200.128.60.106 638 J. F. Alves et al 1.2. Decay of correlations. Positive Lyapunov exponents are known to be a cause of sensitive dependence on initial conditions and other dynamical features which give rise to a degree of chaoticity or stochasticity in the dynamics. We can formalize this idea through the notion of mixing with respect to some invariant measure. Definition 2. A probability measure µ defined on the Borel sets of I is said to be f-invariant if µ(f −1 (A)) = µ(A) for every Borel set A ⊂ I. Definition 3. A map f is said to be mixing with respect to some f-invariant probability measure µ if |µ(f −n (A) ∩ B) − µ(A)µ(B)| → 0, when n → ∞, for any measurable sets A, B. One interpretation of this property is that the conditional probability of B given f −n (A), i.e. the probability that the event A is a consequence of the event B having occurred at some time in the past, is asymptotically the same as if the two events were completely independent. This is sometimes referred to as a property of loss of memory, and thus in some sense of stochasticity, of the system. A natural question of interest both for application and for intrinsic reasons, therefore, is the speed at which such loss of memory occurs. Standard counterexamples show that, in general, there is no specific rate: it is always possible to choose sets A and B for which mixing is arbitrarily slow. However, this notion can be generalized in the following way. Definition 4. For a map f : I → I preserving a probability measure µ and functions ϕ, ψ ∈ L 1 (µ), we define the correlation function C n = C n (ϕ, ψ) = (ϕ • f n)ψ dµ − ϕ dµ ψ dµ .

Encyclopaedia of Mathematical Sciences, 1989

Journal of Pure & Applied Sciences, 2023

This paper presents a detailed investigation of absolutely continuous invariant measures (ACIMs) for piecewise expanding chaotic transformations in ℝ , 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.

Acta Mathematica Hungarica, 1996

2023

This paper presents a detailed investigation of absolutely continuous invariant measures (ACIMs) for piecewise expanding chaotic transformations in ℝ 𝒏 , 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.

1999

There is studied an invariant measure structure of a class of ergodical discrete dynamical systems by means of the measure generating function method.

arXiv: Dynamical Systems, 2020

In this paper, we consider finitely many interval maps simultaneously acting on the unit interval $I = [0, 1]$ in the real line $\mathbb{R}$; each with utmost finitely many jump discontinuities and study certain important statistical properties. Even though we use the symbolic space on $N$ letters to reduce the case of simultaneous dynamics to maps on an appropriate space, our aim in this paper remains to resolve ergodicity, rates of recurrence, decay of correlations and invariance principles leading upto the central limit theorem for the dynamics that evolves through simultaneous action. In order to achieve our ends, we define various Ruelle operators, normalise them by various means and exploit their spectra.

Proceedings of the International Congress of Mathematicians 2010 (ICM 2010) - Vol. I: Plenary Lectures and Ceremonies, Vols. II-IV: Invited Lectures, 2011

Encyclopedia of Complexity and Systems Science, 2009

SpringerReference, 2011