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Poisson Limit Theorems in Dynamical ...

Auer, Max,

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Poisson Limit Theorems in Dynamical Systems and Ergodic Sums of Non-Integrable Observables /

紀錄類型:	書目-電子資源 : Monograph/item
正題名/作者:	Poisson Limit Theorems in Dynamical Systems and Ergodic Sums of Non-Integrable Observables // Max Auer.
作者:	Auer, Max,
面頁冊數:	1 electronic resource (104 pages)
附註:	Source: Dissertations Abstracts International, Volume: 86-12, Section: A.
Contained By:	Dissertations Abstracts International86-12A.
標題:	Mathematics. -
電子資源:	https://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=31936714
ISBN:	9798286434107

Poisson Limit Theorems in Dynamical Systems and Ergodic Sums of Non-Integrable Observables /
Auer, Max,

Poisson Limit Theorems in Dynamical Systems and Ergodic Sums of Non-Integrable Observables /Max Auer. - 1 electronic resource (104 pages)

Source: Dissertations Abstracts International, Volume: 86-12, Section: A.

This thesis investigates statistical properties of rare events in dynamical systems, with a particular focus on recurrence statistics and limit laws for ergodic sums of heavy-tailed observables. A central theme is the Poisson Limit Theorem (PLT), which describes the limiting distribution of return times to small sets, and its role in establishing asymptotics for extreme events. Concretely, this thesis addresses the following topics:First, we construct examples of non-mixing dynamical systems that still satisfy the PLT, demonstrating that strong statistical properties can persist even in the absence of mixing. This is significant because, until now, all known systems satisfying the PLT have been mixing, with existing proofs relying on mixing assumptions.Second, we establish strong limit laws for ergodic sums of non-integrable observables. In such cases, large fluctuations due to close visits to the singularity typically obstruct strong limit laws. To address this, we employ trimming, a well-known probabilistic technique, by systematically removing the closest visits to the singularity. We prove trimmed strong laws for irrational rotations, emphasizing once more that our results seem to be the first of their kind that do not rely on mixing.We also present in the conclusion some natural questions and directions a future research, including hitting time statistics, similar to the PLT setting, and distributional limit theorems for ergodic sums of non-integrable observables.

English

ISBN: 9798286434107Subjects--Topical Terms:

515831
Mathematics.
Subjects--Index Terms:

Dynamical systems

Poisson Limit Theorems in Dynamical Systems and Ergodic Sums of Non-Integrable Observables /
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500 $a Source: Dissertations Abstracts International, Volume: 86-12, Section: A.
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502 $b Ph.D. $c University of Maryland, College Park $d 2025.
520 $a This thesis investigates statistical properties of rare events in dynamical systems, with a particular focus on recurrence statistics and limit laws for ergodic sums of heavy-tailed observables. A central theme is the Poisson Limit Theorem (PLT), which describes the limiting distribution of return times to small sets, and its role in establishing asymptotics for extreme events. Concretely, this thesis addresses the following topics:First, we construct examples of non-mixing dynamical systems that still satisfy the PLT, demonstrating that strong statistical properties can persist even in the absence of mixing. This is significant because, until now, all known systems satisfying the PLT have been mixing, with existing proofs relying on mixing assumptions.Second, we establish strong limit laws for ergodic sums of non-integrable observables. In such cases, large fluctuations due to close visits to the singularity typically obstruct strong limit laws. To address this, we employ trimming, a well-known probabilistic technique, by systematically removing the closest visits to the singularity. We prove trimmed strong laws for irrational rotations, emphasizing once more that our results seem to be the first of their kind that do not rely on mixing.We also present in the conclusion some natural questions and directions a future research, including hitting time statistics, similar to the PLT setting, and distributional limit theorems for ergodic sums of non-integrable observables.
546 $a English
590 $a School code: 0117
650 4 $a Mathematics. $3 515831
650 4 $a Theoretical mathematics. $3 3173530
650 4 $a Mathematics education. $3 641129
653 $a Dynamical systems
653 $a Ergodic sums
653 $a Ergodic theory
653 $a Limit theorems
653 $a Return times
690 $a 0405
690 $a 0642
690 $a 0280
710 2 $a University of Maryland, College Park. $b Mathematics. $e degree granting institution. $3 3816279
720 1 $a Dolgopyat, Dmitry $e degree supervisor.
720 1 $a Kanigowski, Adam $e degree supervisor.
773 0 $t Dissertations Abstracts International $g 86-12A.
790 $a 0117
791 $a Ph.D.
792 $a 2025
856 4 0 $u https://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=31936714