Symbolic dynamics of piecewise contractions

Pires, Benito

doi:10.1088/1361-6544/ab4361

Cited by 8 publications

(12 citation statements)

References 26 publications

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“…This proves that ω(x) coincides with some set of the decomposition (13) and it also proves that the sets of the decomposition ( 13) are all pairwise X -disjoint. We conclude that, for any x ∈ X , either there exists i ∈ {1, .…”

Section: 3mentioning

confidence: 52%

“…It is in particular the case for the half-closed unit interval map x → λx + µ mod 1, for which the properties of the rotation number as a function of λ and µ ∈ [0, 1) have been studied in detail [2,3,6,8]. For injective PCIMs with N 2 contraction pieces, it has been proved that the complexity of the itinerary of any orbit is an eventually affine function [4,13]. The growth rate of the complexity is at most equal to N − 1 and there are some examples of PCIMs with such a maximal complexity [4].…”

Section: Introductionmentioning

confidence: 99%

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A spectral decomposition of the attractor of piecewise-contracting maps of the interval

Calderon¹,

Catsigeras²,

Guiraud³

2020

Ergod. Th. Dynam. Sys.

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We study the asymptotic dynamics of piecewise-contracting maps defined on a compact interval. For maps that are not necessarily injective, but have a finite number of local extrema and discontinuity points, we prove the existence of a decomposition of the support of the asymptotic dynamics into a finite number of minimal components. Each component is either a periodic orbit or a minimal Cantor set and such that the $\unicode[STIX]{x1D714}$ -limit set of (almost) every point in the interval is exactly one of these components. Moreover, we show that each component is the $\unicode[STIX]{x1D714}$ -limit set, or the closure of the orbit, of a one-sided limit of the map at a discontinuity point or at a local extremum.

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Section: 3mentioning

confidence: 52%

Section: Introductionmentioning

confidence: 99%

A spectral decomposition of the attractor of piecewise-contracting maps of the interval

Calderon¹,

Catsigeras²,

Guiraud³

2020

Ergod. Th. Dynam. Sys.

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“…which shows that (H3) is true. Applying (17) to the third column in Table 5 Table 4 i r i = N (c i ) i 0 i 1 . .…”

Section: Piecewise Contractions and The Proof Of Theorem 22mentioning

confidence: 99%

“…In this article, we provide an example of a switched server system with atypical non-trivial dynamics. Our approach benefits from recent advances in the understanding of the topological dynamics of piecewise contractions (see [17]). The switched server system we consider here consists of 3 buffers (tanks) numbered 1, 2, 3, and a server.…”

mentioning

confidence: 99%

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A switched server system semiconjugate to a minimal interval exchange

Fernandes

Pires

2019

Eur. J. Appl. Math

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Switched server systems are mathematical models of manufacturing, traffic and queueing systems that have being studied since the early 1990s. In particular, it is known that typically the dynamics of such systems is asymptotically periodic: each orbit of the system converges to one of its finitely many limit cycles. In this article, we provide an explicit example of a switched server system with exotic behaviour: each orbit of the system converges to the same Cantor attractor. To accomplish this goal, we bring together recent advances in the understanding of the topological dynamics of piecewise contractions and interval exchange transformations (IETs) with flips. The ultimate result is a switched server system whose Poincaré map is semiconjugate to a minimal and uniquely ergodic IET with flips.

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Rotation number of 2-interval piecewise affine maps

Gaivão,

Laurent,

Nogueira

2024

Aequat. Math.

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We study maps of the unit interval whose graph is made up of two increasing segments and which are injective in an extended sense. Such maps $$f_{\varvec{p}}$$ f p are parametrized by a quintuple $$\varvec{p}$$ p of real numbers satisfying inequations. Viewing $$f_{\varvec{p}}$$ f p as a circle map, we show that it has a rotation number $$\rho (f_{\varvec{p}})$$ ρ ( f p ) and we compute $$\rho (f_{\varvec{p}})$$ ρ ( f p ) as a function of $$\varvec{p}$$ p in terms of Hecke–Mahler series. As a corollary, we prove that $$\rho (f_{\varvec{p}})$$ ρ ( f p ) is a rational number when the components of $$\varvec{p}$$ p are algebraic numbers.

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Symbolic dynamics of piecewise contractions

Cited by 8 publications

References 26 publications

A spectral decomposition of the attractor of piecewise-contracting maps of the interval

A spectral decomposition of the attractor of piecewise-contracting maps of the interval

A switched server system semiconjugate to a minimal interval exchange

Rotation number of 2-interval piecewise affine maps

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