Article Contents

2019, Volume 24, Issue 2: 657-670. Doi: 10.3934/dcdsb.2018201

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Persistent two-dimensional strange attractors for a two-parameter family of Expanding Baker Maps

Dep. de Matemáticas, Universidad de Oviedo., c\ Federico García Lorca 18, 33007, Oviedo. Spain

* Corresponding author: Enrique Vigil
* Corresponding author: Enrique Vigil

Received: June 2017

Revised: November 2017

Early access: June 2018

Published: February 2019

This work has been supported by project MINECO-15-MTM2014-56953-P

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Abstract

We characterize the attractors for a two-parameter class of two-dimensional piecewise affine maps. These attractors are strange attractors, probably having finitely many pieces, and coincide with the support of an ergodic absolutely invariant probability measure. Moreover, we demonstrate that every compact invariant set with non-empty interior contains one of these attractors. We also prove the existence, for each natural number $ n, $ of an open set of parameters in which the respective transformation exhibits at least $ 2^n $ non connected two-dimensional strange attractors each one of them formed by $ 4^n $ pieces.

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Mathematics Subject Classification: Primary: 37C70, 37D45; Secondary: 37G35.

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Figure 1. The smoothness domains for a map in $ \mathbb{F} .$

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Figure 2. Two numerically obtained attractors for $\Psi_{a, b}$ when $a = 1.12$ and $b = 1.35.$

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Figure 3. (a) Filled in black, the set $ \mathcal{P}_3 ;$ encircled in a dashed black line, the set $ H_{\Delta}(\mathcal{P}_3) .$ (b) Filled in black, the set $ \mathcal{P}_3 ;$ encircled in a dashed black line, the set $ H_{\Pi}(\mathcal{P}_3) .$

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