A note on double rotations of infinite type

Artigiani, Mauro; Fougeron, Charles; Hubert, Pascal; Skripchenko, Alexandra

doi:10.1090/mosc/311

Cited by 3 publications

(1 citation statement)

References 11 publications

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“…As a byproduct of the proof of this statement, in [6] we constructed a probability measure µ ′ whose support set is A D . This measure is invariant with respect to the renormalization that we introduced there (it is different from SIA induction) and induces a maximum entropy measure for the suspension flow associated with an interval translation map.…”

Section: Bruin-troubetzkoy Interval Translation Mapsmentioning

confidence: 99%

Renormalization in one-dimensional dynamics

Skripchenko

2023

Russian Math. Surveys

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The study of the dynamical and topological properties of interval exchange transformations and their natural generalizations is an important problem, which lies at the intersection of several branches of mathematics, including dynamical systems, low-dimensional topology, algebraic geometry, number theory, and geometric group theory. The purpose of the survey is to make a systematic presentation of the existing results on the ergodic and geometric characteristics of the one-dimensional maps under consideration, as well as on the measured foliations on surfaces and two-dimensional complexes that one can associate with these maps. These results are based on the research of the ergodic properties of the renormalization process - an algorithm that takes an original dynamical system and builds a sequence of equivalent dynamical systems with a smaller support set. For all dynamical systems considered in the paper these renormalization algorithms can be viewed as multidimensional fraction algorithms. Bibiliography: 74 titles.

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Section: Bruin-troubetzkoy Interval Translation Mapsmentioning

confidence: 99%

Renormalization in one-dimensional dynamics

Skripchenko

2023

Russian Math. Surveys

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Dynamics of Systems with a Discontinuous Hysteresis Operator and Interval Translation Maps

Kryzhevich

Аврутин

Begun

et al. 2021

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We studied topological and metric properties of the so-called interval translation maps (ITMs). For these maps, we introduced the maximal invariant measure and demonstrated that an ITM, endowed with such a measure, is metrically conjugated to an interval exchange map (IEM). This allowed us to extend some properties of IEMs (e.g., an estimate of the number of ergodic measures and the minimality of the symbolic model) to ITMs. Further, we proved a version of the closing lemma and studied how the invariant measures depend on the parameters of the system. These results were illustrated by a simple example or a risk management model where interval translation maps appear naturally.

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Ренормализация В Одномерной Динамике

Skripchenko

2023

Uspekhi Mat. Nauk

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Изучение динамических и топологических свойств перекладываний отрезков и их естественных обобщений является важной задачей, находящейся на стыке нескольких разделов математики: теории динамических систем, маломерной топологии, алгебраической геометрии, теории чисел и геометрической теории групп. Целью настоящего обзора является систематическое изложение результатов, касающихся эргодических и геометрических характеристик орбит рассматриваемых одномерных отображений, а также ассоциированных с ними измеримых слоений на поверхностях и двумерных комплексах. Эти результаты опираются на изучение свойств процесса ренормализации - алгоритма, строящего по исходной динамической системе последовательность эквивалентных ей с меньшим множеством-носителем. Эти алгоритмы для всех рассмотренных в работе классов динамических систем являются многомерными цепными дробями. Библиография: 74 названия.

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A note on double rotations of infinite type

Cited by 3 publications

References 11 publications

Renormalization in one-dimensional dynamics

Renormalization in one-dimensional dynamics

Dynamics of Systems with a Discontinuous Hysteresis Operator and Interval Translation Maps

Ренормализация В Одномерной Динамике

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