Topological dynamics of piecewise -affine maps

Nogueira, Arnaldo; Pires, Benito; Rosales, Rafael

doi:10.1017/etds.2016.104

Cited by 15 publications

(26 citation statements)

References 12 publications

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“…A general result, which holds in any compact metric phase space, is that the attractor of a piecewise contracting map consists of a finite number of periodic orbits, whenever it does not intersect the boundary of a contraction piece (see [5]). For PCIM defined on a half-closed interval, Nogueira, Pires and Rosales proved moreover that this periodic asymptotic behavior is generic in a metric sense and with a number of periodic orbits which is bounded above by the number of contraction pieces [10,11,12]. This generalizes and refines a previous result by Brémont obtained in [1].…”

Section: Introductionsupporting

confidence: 83%

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

Calderón,

Catsigeras,

Guiraud

2019

Preprint

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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 ω-limit set of (almost) every point in the interval is exactly one of these components. Moreover, we show that each component is the ω-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: Introductionsupporting

confidence: 83%

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

Calderón,

Catsigeras,

Guiraud

2019

Preprint

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show abstract

“…A general result, which holds in any compact metric phase space, is that the attractor of a piecewise-contracting map consists of a finite number of periodic orbits whenever it does not intersect the boundary of a contraction piece (see [5]). Moreover, for PCIMs defined on a half-closed interval, Nogueira, Pires and Rosales proved that this periodic asymptotic behavior is generic in a metric sense and with a number of periodic orbits which is bounded above by the number of contraction pieces [10][11][12]. This generalizes and refines a previous result obtained by Brémont in [1].…”

Section: Introductionsupporting

confidence: 82%

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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“…On the one hand, as already mentioned, recent advances (see [13,14]) in the understanding of the topological dynamics of piecewise contractions show that generically piecewise contractions have finitely many limit cycles that attracts all orbits. Hence, an affirmative answer to (Q) is very unlikely.…”

mentioning

confidence: 91%

“…By [14,Theorem 1.4], we have that for Lebesgue almost every vector (d 1 , d 2 , d 3 ) with positive entries, any switched server system with parameters d ij satisfying (1) is structurally stable and admits finitely many limit cycles that attract all the orbits. The same result was obtained in [4,Theorem 4.1] under the additional restrictions: d 21 = d 31 , d 12 = d 32 and d 13 = d 23 .…”

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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Topological dynamics of piecewise -affine maps

Abstract: Let −1 < λ < 1 and f : [0, 1) → R be a piecewise λ-affine map, that is, there exist points 0 = c 0 < c 1

Cited by 15 publications

References 12 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 spectral decomposition of the attractor of piecewise-contracting maps of the interval

A switched server system semiconjugate to a minimal interval exchange

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