A spectral gap for transfer operators of piecewise expanding maps

Thomine, Damien

doi:10.3934/dcds.2011.30.917

Cited by 35 publications

(74 citation statements)

References 22 publications

(55 reference statements)

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“…Nevertheless, the space BV in higher dimensions presents some drawbacks : it is not included in L ∞ and there exist some positive functions which are not bounded below on any ball, making the application of random covering difficult, in contrast to the the Quasi-Hölder space. Apart multidimensional BV, another possibility is to use fractional Sobolev spaces, as done in a deterministic setting by Thomine [66].…”

Section: Remark 24mentioning

confidence: 99%

Annealed and quenched limit theorems for random expanding dynamical systems

Aimino

Nicol

Vaienti

2014

Probab. Theory Relat. Fields

168

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In this paper, we investigate annealed and quenched limit theorems for random expanding dynamical systems. Making use of functional analytic techniques and more probabilistic arguments with martingales, we prove annealed versions of a central limit theorem, a large deviation principle, a local limit theorem, and an almost sure invariance principle. We also discuss the quenched central limit theorem, dynamical Borel-Cantelli lemmas, Erdös-Rényi laws and concentration inequalities.Appeared online on Probability Theory and Related Fields. The final publication is available at Springer via http://dx.

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Section: Remark 24mentioning

confidence: 99%

Annealed and quenched limit theorems for random expanding dynamical systems

Aimino

Nicol

Vaienti

2014

Probab. Theory Relat. Fields

168

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“…We now move on to proving the first statement of the theorem. First we note that by expansion, we may again rely on the results of [32] and [31]: for the uniqueness of the absolutely continuous invariant measure, it is enough to show that the map is locally eventually onto (l.e.o. for short).…”

Section: N =mentioning

confidence: 99%

Mean-Field Coupling of Identical Expanding Circle Maps

Sélley

Bálint

2016

J Stat Phys

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Globally coupled doubling maps are studied in this paper. In this setting and for finitely many sites, two distinct bifurcation values of the coupling strength have been identified in the literature, corresponding to the emergence of contracting directions ([24]) and, specifically for N = 3 sites, to the loss of ergodicity ([10]). On the one hand, we reconsider these results and provide an interpretation of the observed dynamical phenomena in terms of the synchronization of the sites. On the other hand, we initiate a new point of view which focuses on the evolution of distributions and allows to incorporate the investigation of a continuum of sites. In particular, we observe phenomena that is analogous to the limit states of the contracting regime of N = 3 sites.

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“…Of course in this case the symmetric images of B are also polyhedral domains and disjoint from B. Since the map is completely expanding (as ε < 1/2), we can use Theorem 1.7 in [15] for these sets independently, to obtain that on each of these sets an absolutely continuous invariant measure is supported. In conclusion, the existence of such an asymmetric invariant set means that multiple acims exist.…”

Section: Resultsmentioning

confidence: 99%

Symmetry breaking in a globally coupled map of four sites

Sélley¹

2018

Discrete &Amp; Continuous Dynamical Systems - A

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A system of four globally coupled doubling maps is studied in this paper. It is known that such systems have a unique absolutely continuous invariant measure (acim) for weak interaction, but the case of stronger coupling is still unexplored. As in the case of three coupled sites [14], we prove the existence of a critical value of the coupling parameter at which multiple acims appear. Our proof has several new ingredients in comparison to the one presented in [14]. We strongly exploit the symmetries of the dynamics in the course of the argument. This simplifies the computations considerably, and gives us a precise description of the geometry and symmetry properties of the arising asymmetric invariant sets. Some new phenomena are observed which are not present in the case of three sites. In particular, the asymmetric invariant sets arise in areas of the phase space which are transient for weaker coupling and a nontrivial symmetric invariant set emerges, shaped by an underlying centrally symmetric Lorenz map. We state some conjectures on further invariant sets, indicating that unlike the case of three sites, ergodicity breaks down in many steps, and not all of them are accompanied by symmetry breaking.

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A spectral gap for transfer operators of piecewise expanding maps

Cited by 35 publications

References 22 publications

Annealed and quenched limit theorems for random expanding dynamical systems

Annealed and quenched limit theorems for random expanding dynamical systems

Mean-Field Coupling of Identical Expanding Circle Maps

Symmetry breaking in a globally coupled map of four sites

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