Open Conformal Systems and Perturbations of Transfer Operators

Pollicott, Mark; Urbanski, Mariuski

doi:10.1007/978-3-319-72179-8

Cited by 13 publications

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“…where K(L) was defined by (23). Thus, N = O As λ ε = 1 − μ 0 (H ε )(θ + o(1)) by (30) we also have lim ε→0 ξ ε = θ > 0, in particular ξ ε > 0 for ε small enough.…”

Section: Osc(pmentioning

confidence: 91%

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Map Lattices Coupled by Collisions: Hitting Time Statistics and Collisions Per Lattice Unit

Bahsoun

Sélley

2022

Ann. Henri Poincaré

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We study map lattices coupled by collision and show how perturbations of transfer operators associated with the spatially periodic approximation of the model can be used to extract information about collisions per lattice unit. More precisely, we study a map on a finite box of L sites with periodic boundary conditions, coupled by collision. We derive, via a non-trivial first-order approximation for the leading eigenvalue of the rare event transfer operator, a formula for the first collision rate and a corresponding first hitting time law. For the former we show that the formula scales at the order of $$L\cdot \varepsilon ^2$$ L · ε 2 , where $$\varepsilon $$ ε is the coupling strength, and for the latter, by tracking the L dependency in our arguments, we show that the error in the law is of order $$O\left( C(L)\frac{L\varepsilon ^2}{\zeta (L)}\cdot \left| \ln \frac{L\varepsilon ^2}{\zeta (L)}\right| \right) $$ O C ( L ) L ε 2 ζ ( L ) · ln L ε 2 ζ ( L ) , where $$\zeta (L)$$ ζ ( L ) is given in terms of the spectral gap of the rare event transfer operator, and C(L) has an explicit expression. Finally, we derive an explicit formula for the first collision rate per lattice unit.

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“…where K(L) was defined by (23). Thus, N = O As λ ε = 1 − μ 0 (H ε )(θ + o(1)) by (30) we also have lim ε→0 ξ ε = θ > 0, in particular ξ ε > 0 for ε small enough.…”

Section: Osc(pmentioning

confidence: 91%

“…For more information on systems with holes we refer the reader to[13] for a basic introduction and a review, to[30] for a recent account and to[7,19] for their connections with hitting time statistics.…”

mentioning

confidence: 99%

Map Lattices Coupled by Collisions: Hitting Time Statistics and Collisions Per Lattice Unit

Bahsoun

Sélley

2022

Ann. Henri Poincaré

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“…7 There is an intimate analogy between the systems we are studying and maps with holes. For such systems Hε are viewed as holes in the phase space and Xn is the set of points which survived landing, under the dynamics in the hole, up to time n. for more information on systems with holes we refer the reader to [11] for a basic introduction and a review, to [28] for a recent account and to [6,17] for their connections with hitting time statistics.…”

Section: Transfer Operators First Hitting Times and First Collision R...mentioning

confidence: 99%

Map lattices coupled by collisions: hitting time statistics and collisions per lattice unit

Bahsoun¹,

Sélley²

2021

Preprint

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We study map lattices coupled by collision and show how perturbations of transfer operators associated with the spatially periodic approximation of the model can be used to extract information about the complete system. More precisely, we first study a map on a finite box of L sites with periodic boundary conditions, coupled by collision. We derive, via a non-trivial first order approximation for the leading eigenvalue of the rare event transfer operator, a formula for the first collision rate and a corresponding first hitting time law. For the former we show that the formula scales at the order of L • ε 2 , where ε is the coupling strength, and for the latter we show that the error in the law is of order O Lε 2 • | ln Lε 2 | . Using this information, we then derive an explicit formula for the first collision rate per lattice unit in the infinite dimensional system.

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“…Such results have been obtained primarily for uniformly hyperbolic systems, beginning with expanding maps [PY, CMS, LM], Anosov diffeomorphisms [CM, CMT], finite [FP] and countable [DIMMY] state topological Markov chains, and dispersing billiards [DWY,D2]. Their extension to non-uniformly hyperbolic systems has been primarily restricted to unimodal and multimodal interval maps [BDM,DT1,PU] and intermittent maps [DT2].…”

Section: Introductionmentioning

confidence: 99%

Asymptotic escape rates and limiting distributions for multimodal maps

Demers¹,

Todd²

2020

Ergod. Th. Dynam. Sys.

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We consider multimodal maps with holes and study the evolution of the open systems with respect to equilibrium states for both geometric and Hölder potentials. For small holes, we show that a large class of initial distributions share the same escape rate and converge to a unique absolutely continuous conditionally invariant measure; we also prove a variational principle connecting the escape rate to the pressure on the survivor set, with no conditions on the placement of the hole. Finally, introducing a weak condition on the centre of the hole, we prove scaling limits for the escape rate for holes centred at both periodic and nonperiodic points, as the diameter of the hole goes to zero.

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Open Conformal Systems and Perturbations of Transfer Operators

Cited by 13 publications

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Map Lattices Coupled by Collisions: Hitting Time Statistics and Collisions Per Lattice Unit

Map Lattices Coupled by Collisions: Hitting Time Statistics and Collisions Per Lattice Unit

Map lattices coupled by collisions: hitting time statistics and collisions per lattice unit

Asymptotic escape rates and limiting distributions for multimodal maps

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