# American Institute of Mathematical Sciences

February  2019, 24(2): 657-670. doi: 10.3934/dcdsb.2018201

## 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

Received  June 2017 Revised  November 2017 Published  June 2018

Fund Project: This work has been supported by project MINECO-15-MTM2014-56953-P.

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.

Citation: Antonio Pumariño, José Ángel Rodríguez, Enrique Vigil. Persistent two-dimensional strange attractors for a two-parameter family of Expanding Baker Maps. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 657-670. doi: 10.3934/dcdsb.2018201
##### References:
 [1] J. Buzzi, Absolutely continuous invariant probability measures for arbitrary expanding piecewise $\mathbb{R}$ analytic mappings of the plane, Ergodic Theory and Dynamical Systems, 20 (2000), 697-708.  doi: 10.1017/S0143385700000377.  Google Scholar [2] A. Pumariño, J. A. Rodríguez, J. C. Tatjer and E. Vigil, Expanding Baker maps as models for the dynamics emerging from 3-D homoclinic bifurcations, Discrete Continuous Dynam. Systems - B, 19 (2014), 523-541.  doi: 10.3934/dcdsb.2014.19.523.  Google Scholar [3] A. Pumariño, J. A. Rodríguez, J. C. Tatjer and E. Vigil, Chaotic dynamics for 2-d tent maps, Nonlinearity, 28 (2015), 407-434.  doi: 10.1088/0951-7715/28/2/407.  Google Scholar [4] A. Pumariño, J. A. Rodríguez and E. Vigil, Renormalizable Expanding Baker Maps: Coexistence of strange attractors, Discrete Continuous Dynam. Systems - A, 37 (2017), 1651-1678.  doi: 10.3934/dcds.2017068.  Google Scholar [5] A. Pumariño, J. A. Rodríguez and E. Vigil, Renormalization of two-dimensional piecewise linear maps: Abundance of 2-D strange attractors, Discrete Continuous Dynam. Systems - A, 38 (2018), 941-966.  doi: 10.3934/dcds.2018040.  Google Scholar [6] A. Pumariño and J. C. Tatjer, Dynamics near homoclinic bifurcations of three-dimensional dissipative diffeomorphisms, Nonlinearity, 19 (2006), 2833-2852.  doi: 10.1088/0951-7715/19/12/006.  Google Scholar [7] A. Pumariño and J. C. Tatjer, Attractors for return maps near homoclinic tangencies of three-dimensional dissipative diffeomorphism, Discrete Continuous Dynam. Systems - B, 8 (2007), 971-1005.  doi: 10.3934/dcdsb.2007.8.971.  Google Scholar [8] B. Saussol, Absolutely continuous invariant measures for multidimensional expanding maps, Israel Journal Mathematics, 116 (2000), 223-248.  doi: 10.1007/BF02773219.  Google Scholar [9] J. C. Tatjer, Three-dimensional dissipative diffeomorphisms with homoclinic tangencies, Ergodic Theory and Dynamical Systems, 21 (2001), 249-302.  doi: 10.1017/S0143385701001146.  Google Scholar [10] M. Tsujii, Absolutely continuous invariant measures for expanding piecewise linear maps, Invent. Math., 143 (2001), 349-373.  doi: 10.1007/PL00005797.  Google Scholar

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##### References:
 [1] J. Buzzi, Absolutely continuous invariant probability measures for arbitrary expanding piecewise $\mathbb{R}$ analytic mappings of the plane, Ergodic Theory and Dynamical Systems, 20 (2000), 697-708.  doi: 10.1017/S0143385700000377.  Google Scholar [2] A. Pumariño, J. A. Rodríguez, J. C. Tatjer and E. Vigil, Expanding Baker maps as models for the dynamics emerging from 3-D homoclinic bifurcations, Discrete Continuous Dynam. Systems - B, 19 (2014), 523-541.  doi: 10.3934/dcdsb.2014.19.523.  Google Scholar [3] A. Pumariño, J. A. Rodríguez, J. C. Tatjer and E. Vigil, Chaotic dynamics for 2-d tent maps, Nonlinearity, 28 (2015), 407-434.  doi: 10.1088/0951-7715/28/2/407.  Google Scholar [4] A. Pumariño, J. A. Rodríguez and E. Vigil, Renormalizable Expanding Baker Maps: Coexistence of strange attractors, Discrete Continuous Dynam. Systems - A, 37 (2017), 1651-1678.  doi: 10.3934/dcds.2017068.  Google Scholar [5] A. Pumariño, J. A. Rodríguez and E. Vigil, Renormalization of two-dimensional piecewise linear maps: Abundance of 2-D strange attractors, Discrete Continuous Dynam. Systems - A, 38 (2018), 941-966.  doi: 10.3934/dcds.2018040.  Google Scholar [6] A. Pumariño and J. C. Tatjer, Dynamics near homoclinic bifurcations of three-dimensional dissipative diffeomorphisms, Nonlinearity, 19 (2006), 2833-2852.  doi: 10.1088/0951-7715/19/12/006.  Google Scholar [7] A. Pumariño and J. C. Tatjer, Attractors for return maps near homoclinic tangencies of three-dimensional dissipative diffeomorphism, Discrete Continuous Dynam. Systems - B, 8 (2007), 971-1005.  doi: 10.3934/dcdsb.2007.8.971.  Google Scholar [8] B. Saussol, Absolutely continuous invariant measures for multidimensional expanding maps, Israel Journal Mathematics, 116 (2000), 223-248.  doi: 10.1007/BF02773219.  Google Scholar [9] J. C. Tatjer, Three-dimensional dissipative diffeomorphisms with homoclinic tangencies, Ergodic Theory and Dynamical Systems, 21 (2001), 249-302.  doi: 10.1017/S0143385701001146.  Google Scholar [10] M. Tsujii, Absolutely continuous invariant measures for expanding piecewise linear maps, Invent. Math., 143 (2001), 349-373.  doi: 10.1007/PL00005797.  Google Scholar
The smoothness domains for a map in $\mathbb{F} .$
Two numerically obtained attractors for $\Psi_{a, b}$ when $a = 1.12$ and $b = 1.35.$
(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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