Pomeau-Manneville maps are global-local mixing

Bonanno, Claudio; Lenci, Marco

doi:10.3934/dcds.2020309

Cited by 6 publications

(4 citation statements)

References 12 publications

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“…We take into account a limit situation for Pomeau-Manneville maps with α = 2.05, since for α ≥ 2 they do not preserve an absolutely continuous probability measure, leading a not straightforward application of the Theorem. These maps are studied by the results of infinite ergodic theory, and it is known that the statistical properties of such maps are peculiar [8], [9].…”

Section: Numerical Analysismentioning

confidence: 99%

Estimation of Dynamical Noise Power in Unknown Systems

Scarciglia¹,

Gini²,

Catrambone³

et al. 2023

Preprint

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Noise can be modeled as a sequence of random variables defined on a probability space that may add to a given dynamical system T that is a map on a phase space. In the nontrivial case of dynamical noise {εn}n, where εn ∈ N(0, σ2) and the system output is xn = T(xn−1, xn−2, ..., x0)+εn, without any specific knowledge/assumption on T, the quantitative estimation of noise power σ2 is a challenge. Here, we introduce a formal method, based on nonlinear entropy profile, to estimate the dynamical noise power σ2 without knowledge on the specific T function. Time-series generated from Logistic maps and Pomeau-Manneville systems under different conditions are used to test the correctness of the method. Results demonstrate that the proposed estimation algorithm properly discerns different noise levels with no a priori information. Numerical simluations, including sereis generation, are performed with MATLAB.

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Section: Numerical Analysismentioning

confidence: 99%

Estimation of Dynamical Noise Power in Unknown Systems

Scarciglia¹,

Gini²,

Catrambone³

et al. 2023

Preprint

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“…exists and is finite (see [9,10] for more details). We first show that observables f ∈ L ∞ (X, µ) which are constant on the level sets (E n ) with f n := f | En and for which the induced function satisfies f E | An ∼ n as n → ∞, are global observables.…”

Section: 2mentioning

confidence: 99%

Almost sure asymptotic behaviour of Birkhoff sums for infinite measure-preserving dynamical systems

Bonanno

Schindler

2022

DCDS

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We consider a conservative ergodic measure-preserving transformation <inline-formula><tex-math id="M1">\begin{document}$ T $\end{document}</tex-math></inline-formula> of a <inline-formula><tex-math id="M2">\begin{document}$ \sigma $\end{document}</tex-math></inline-formula>-finite measure space <inline-formula><tex-math id="M3">\begin{document}$ (X, {\mathcal B},\mu) $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M4">\begin{document}$ \mu(X) = \infty $\end{document}</tex-math></inline-formula>. Given an observable <inline-formula><tex-math id="M5">\begin{document}$ f:X\to \mathbb R $\end{document}</tex-math></inline-formula>, we study the almost sure asymptotic behaviour of the Birkhoff sums <inline-formula><tex-math id="M6">\begin{document}$ S_Nf(x) : = \sum_{j = 1}^N\, (f\circ T^{j-1})(x) $\end{document}</tex-math></inline-formula>. In infinite ergodic theory it is well known that the asymptotic behaviour of <inline-formula><tex-math id="M7">\begin{document}$ S_Nf(x) $\end{document}</tex-math></inline-formula> strongly depends on the point <inline-formula><tex-math id="M8">\begin{document}$ x\in X $\end{document}</tex-math></inline-formula>, and if <inline-formula><tex-math id="M9">\begin{document}$ f\in L^1(X,\mu) $\end{document}</tex-math></inline-formula>, then there exists no real valued sequence <inline-formula><tex-math id="M10">\begin{document}$ (b(N)) $\end{document}</tex-math></inline-formula> such that <inline-formula><tex-math id="M11">\begin{document}$ \lim_{N\to\infty} S_Nf(x)/b(N) = 1 $\end{document}</tex-math></inline-formula> almost surely. In this paper we show that for dynamical systems with strong mixing assumptions for the induced map on a finite measure set, there exists a sequence <inline-formula><tex-math id="M12">\begin{document}$ (\alpha(N)) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M13">\begin{document}$ m\colon X\times \mathbb N\to \mathbb N $\end{document}</tex-math></inline-formula> such that for <inline-formula><tex-math id="M14">\begin{document}$ f\in L^1(X,\mu) $\end{document}</tex-math></inline-formula> we have <inline-formula><tex-math id="M15">\begin{document}$ \lim_{N\to\infty} S_{N+m(x,N)}f(x)/\alpha(N) = 1 $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M16">\begin{document}$ \mu $\end{document}</tex-math></inline-formula>-a.e. <inline-formula><tex-math id="M17">\begin{document}$ x\in X $\end{document}</tex-math></inline-formula>. Instead in the case <inline-formula><tex-math id="M18">\begin{document}$ f\not\in L^1(X,\mu) $\end{document}</tex-math></inline-formula> we give conditions on the induced observable such that there exists a sequence <inline-formula><tex-math id="M19">\begin{document}$ (G(N)) $\end{document}</tex-math></inline-formula> depending on <inline-formula><tex-math id="M20">\begin{document}$ f $\end{document}</tex-math></inline-formula>, for which <inline-formula><tex-math id="M21">\begin{document}$ \lim_{N\to\infty} S_{N}f(x)/G(N) = 1 $\end{document}</tex-math></inline-formula> holds for <inline-formula><tex-math id="M22">\begin{document}$ \mu $\end{document}</tex-math></inline-formula>-a.e. <inline-formula><tex-math id="M23">\begin{document}$ x\in X $\end{document}</tex-math></inline-formula>.

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“…A particular class of observables studied in the context of infinite mixing theory is that of global observables. For maps in T , we recall that a global observable is a function f ∈ L ∞ ([0, 1], µ) such that the limit lim a→0 + 1 µ(a, 1) 1 a f dµ exists and is finite (see [BGL18,BL21] for more details). We first show that observables f ∈ L ∞ (X, µ) which are constant on the level sets (E n ) with f n := f | En and for which the induced function satisfies f E | An ∼ n as n → ∞, are global observables.…”

Section: 34]) and So µ(Amentioning

confidence: 99%

Almost sure asymptotic behaviour of Birkhoff sums for infinite measure-preserving dynamical systems

Bonanno¹,

Schindler²

2021

Preprint

Self Cite

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We consider a conservative ergodic measure-preserving transformation T of a σ-finite measure space (X, B, µ) with µ(X) = ∞. Given an observable f : X → R we study the almost sure asymptotic behaviour of the Birkhoff sums SN f (x) := N j=1 (f •T j−1 )(x). In infinite ergodic theory it is well known that the asymptotic behaviour of SN f (x) strongly depends on the point x ∈ X, and if f ∈ L 1 (X, µ), then there exists no real valued sequence (b(N )) such that limN→∞ SN f (x)/b(N ) = 1 almost surely. In this paper we show that for dynamical systems with strong mixing assumptions for the induced map on a finite measure set, there exists a sequence (α(N )) and m : X × N → N such that for f ∈ L 1 (X, µ) we have limN→∞ S N+m(x,N) f (x)/α(N ) = 1 for µ-a.e. x ∈ X. Moreover if f ∈ L 1 (X, µ) we give conditions on the induced observable such that there exists a sequence (G(N )) depending on f , for which limN→∞ SN f (x)/G(N ) = 1 holds for µ-a.e. x ∈ X.

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Pomeau-Manneville maps are global-local mixing

Cited by 6 publications

References 12 publications

Estimation of Dynamical Noise Power in Unknown Systems

Estimation of Dynamical Noise Power in Unknown Systems

Almost sure asymptotic behaviour of Birkhoff sums for infinite measure-preserving dynamical systems

Almost sure asymptotic behaviour of Birkhoff sums for infinite measure-preserving dynamical systems

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