Entry and return times for semi-flows

Marklof, Jens

doi:10.1088/1361-6544/aa518b

Cited by 9 publications

(8 citation statements)

References 49 publications

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“…If the underlying dynamics is Gibbs-Markov, and under additional constraints on the step time and step function, he got distributional asymptotics for the local time, and an equivalent of the tail of the first return time. In [38], Marklof explained how entry and return times can be interpreted in terms of stationary point processes and their Palm distribution and generalized the results by Haydn et al to non-ergodic maps and continuous-time dynamical systems. From [1, 3, 14-17, 34, 38, 48, 49] we can see that the generalizations of the results obtained for continuous maps over a compact metric space to semiflows on a compact metric space and the study on the dynamical properties of semi-flows on a compact metric space are interesting.…”

Section: Introductionmentioning

confidence: 89%

Multi-sensitivity, syndetical sensitivity and the asymptotic average- shadowing property for continuous semi-flows

Li¹,

Lu²,

Yu³

et al. 2017

J. Nonlinear Sci. Appl.

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In this paper, for a continuous semi-flow θ on a compact metric space E with the asymptotic average-shadowing property (AASP), we show that if the almost periodic points of θ are dense in E then θ is multi-sensitive and syndetically sensitive. Also, we show that if θ is a Lyapunov stable semi-flow with the AASP, then the space E is trivial. Consequently, a Lyapunov stable semi-flow with the AASP is minimal. Furthermore, we prove that for a syndetically transitive continuous semi-flow on a compact metric space, sensitivity is equivalent to syndetical sensitivity. As an application, we show that for a continuous semi-flow θ on a compact metric space E with the AASP, if the almost periodic points of ϕ are dense in E then θ is syndetically sensitive. Moreover, we prove that for any continuous semi-flow θ on a compact metric space, it has the AASP if and only if so does its inverse limit ( E, θ), and if only if so does its lifting continuous semi-flow ( E, θ). Also, an example which contains two numerical experiments is given. Our results extend some corresponding and existing ones.

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Section: Introductionmentioning

confidence: 89%

Multi-sensitivity, syndetical sensitivity and the asymptotic average- shadowing property for continuous semi-flows

Li¹,

Lu²,

Yu³

et al. 2017

J. Nonlinear Sci. Appl.

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“…This is similar to the approach in [1] for the statistics of Farey sequences, but with a different section, see section 4.5 for details. We refer the reader to [20] for more background on the connection between statistics of entry and return times for a given Poincaré section on one hand, and point processes and their Palm distributions on the other.…”

Section: A One-dimensional Point Processmentioning

confidence: 99%

“…Furthermore the fact that Ξ({0}) = 1 almost surely (this follows from (4.95)) implies that Ξ 0 and hence Ξ are simple. For further details on this connection see [20] and references therein. We will explore a dynamical interpretation of this in the next section.…”

Section: 96)mentioning

confidence: 99%

Fine-scale distribution of roots of quadratic congruences

Marklof¹,

Welsh²

2021

Preprint

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We establish limit laws for the distribution in small intervals of the roots of the quadratic congruence µ 2 ≡ D mod m, with D > 0 square-free and D ≡ 1 mod 4. This is achieved by translating the problem to convergence of certain geodesic random line processes in the hyperbolic plane. This geometric interpretation allows us in particular to derive an explicit expression for the pair correlation density of the roots.

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“…Theorem 1 is stated for the convergence of entry time distributions. It is a general fact that the convergence of entry time distributions implies the convergence of return time distributions and vice versa, with a simple formula relating the two [33].…”

Section: (29)mentioning

confidence: 99%

Universal Hitting Time Statistics for Integrable Flows

2016

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The perceived randomness in the time evolution of "chaotic" dynamical systems can be characterized by universal probabilistic limit laws, which do not depend on the fine features of the individual system. One important example is the Poisson law for the times at which a particle with random initial data hits a small set. This was proved in various settings for dynamical systems with strong mixing properties. The key result of the present study is that, despite the absence of mixing, the hitting times of integrable flows also satisfy universal limit laws which are, however, not Poisson. We describe the limit distributions for "generic" integrable flows and a natural class of target sets, and illustrate our findings with two examples: the dynamics in central force fields and ellipse billiards. The convergence of the hitting time process follows from a new equidistribution theorem in the space of lattices, which is of independent interest. Its proof exploits Ratner's measure classification theorem for unipotent flows, and extends earlier work of Elkies and McMullen.

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Entry and return times for semi-flows

Cited by 9 publications

References 49 publications

Multi-sensitivity, syndetical sensitivity and the asymptotic average- shadowing property for continuous semi-flows

Multi-sensitivity, syndetical sensitivity and the asymptotic average- shadowing property for continuous semi-flows

Fine-scale distribution of roots of quadratic congruences

Universal Hitting Time Statistics for Integrable Flows

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