Structure of dendrites admitting an existence of an arc horseshoe

Makhrova, E. N.

doi:10.3103/s1066369x1508006x

Russ Math.

2015

DOI: 10.3103/s1066369x1508006x

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Structure of dendrites admitting an existence of an arc horseshoe

E. N. Makhrova

Abstract: We say that a dendrite X admits an existence of an arc horseshoe if for any continuous map f of X into itself which has a horseshoe (A, B) one can find a natural number n such that nth iteration of f has an arc horseshoe. We investigate the structure of dendrites admitting the existence of an arc horseshoe.

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Cited by 3 publications

References 12 publications

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One-dimensional dynamical systems

Ефремова¹,

Makhrova²

2021

Russ. Math. Surv.

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The survey is devoted to the topological dynamics of maps defined on one-dimensional continua such as a closed interval, a circle, finite graphs (for instance, finite trees), or dendrites (locally connected continua without subsets homeomorphic to a circle). Connections between the periodic behaviour of trajectories, the existence of a horseshoe and homoclinic trajectories, and the positivity of topological entropy are investigated. Necessary and sufficient conditions for entropy chaos in continuous maps of an interval, a circle, or a finite graph, and sufficient conditions for entropy chaos in continuous maps of dendrites are presented. Reasons for similarities and differences between the properties of maps defined on the continua under consideration are analyzed. Extensions of Sharkovsky’s theorem to certain discontinuous maps of a line or an interval and continuous maps on a plane are considered. Bibliography: 207 titles.

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One-dimensional dynamical systems

Ефремова¹,

Makhrova²

2021

Russ. Math. Surv.

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Remarks on minimal sets on dendrites and finite graphs

Makhrova

2023

Journal of Difference Equations and Applications

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Одномерные Динамические Системы

Ефремова¹,

Efremova

Makhrova³

2021

Uspekhi Mat. Nauk

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Обзор посвящен топологической динамике отображений, заданных на одномерных континуумах, таких как отрезок, окружность, конечные графы (в том числе и конечные деревья), дендриты (локально связные континуумы, не содержащие подмножеств, гомеоморфных окружности). Исследуются взаимосвязи между периодическим поведением траекторий, существованием подковы и гомоклинических траекторий, положительностью топологической энтропии. Приведены необходимые и достаточные условия энтропийного хаоса в непрерывных отображениях отрезка, окружности, конечных графов и достаточные условия энтропийного хаоса в непрерывных отображениях дендритов. Проанализированы причины сходства и различия в свойствах отображений, заданных на указанных континуумах. Рассмотрены обобщения теоремы А. Н. Шарковского на случай некоторых разрывных отображений прямой или отрезка и непрерывных отображений в плоскости. Библиография: 207 названий.

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Structure of dendrites admitting an existence of an arc horseshoe

Cited by 3 publications

References 12 publications

One-dimensional dynamical systems

One-dimensional dynamical systems

Remarks on minimal sets on dendrites and finite graphs

Одномерные Динамические Системы

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