On the structure of α-limit sets of backward trajectories for graph maps

Magdalena Foryś-Krawiec; Jana Hantáková; Piotr Oprocha

doi:10.3934/dcds.2021159

Article Contents

2022, Volume 42, Issue 3: 1435-1463. Doi: 10.3934/dcds.2021159 open access

This issue Previous Article Lower bounds for Orlicz eigenvalues Next Article Eternal solutions for a reaction-diffusion equation with weighted reaction

On the structure of α-limit sets of backward trajectories for graph maps

1.
AGH University of Science and Technology, Faculty of Applied Mathematics, al. Mickiewicza 30, 30-059 Kraków, Poland
2.
Mathematical Institute of the Silesian University in Opava, Na Rybníčku 1, 74601, Opava, Czech Republic
3.
Centre of Excellence IT4Innovations - Institute for Research and Applications of Fuzzy Modeling, University of Ostrava, 30. dubna 22, 701 03 Ostrava 1, Czech Republic

^* Corresponding author: Jana Hantáková
^* Corresponding author: Jana Hantáková

Received: July 2021

Revised: September 2021

Early access: November 4, 2021

Published online: November 4, 2021

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Abstract

In the paper we study what sets can be obtained as $ \alpha $-limit sets of backward trajectories in graph maps. We show that in the case of mixing maps, all those $ \alpha $-limit sets are $ \omega $-limit sets and for all but finitely many points $ x $, we can obtain every $ \omega $-limits set as the $ \alpha $-limit set of a backward trajectory starting in $ x $. For zero entropy maps, every $ \alpha $-limit set of a backward trajectory is a minimal set. In the case of maps with positive entropy, we obtain a partial characterization which is very close to complete picture of the possible situations.

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Mathematics Subject Classification: Primary: 37E25;Secondary: 37B20, 37B40.

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Figure 1. A map where the family $ \mathcal{A}(x) $ is uncountable

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Figure 2. A map mixing interval map $ g $ with a fixed point $ p $ such that every backward branch $ \{z_j\}_{j\leq 0} $ with $ \{p\} = \alpha_g(\{z_j\}_{j\leq 0}) $ converges to $ p $ only from the right side. Every backward branch $ \{z_j\}_{j\leq 0} $ converging to $ p $ from the left side or from both sides has $ \alpha_g(\{z_j\}_{j\leq 0})\cap \{q, r\}\neq\emptyset $, where $ q, r $ are preimages of $ p $

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