# American Institute of Mathematical Sciences

February  2018, 38(2): 941-966. doi: 10.3934/dcds.2018040

## Renormalization of two-dimensional piecewise linear maps: Abundance of 2-D strange attractors

 Dep. de Matemáticas, Universidad de Oviedo, Calvo Sotelo s/n, 33007, Oviedo, Spain

* Corresponding author: Enrique Vigil

Received  January 2017 Revised  September 2017 Published  February 2018

Fund Project: The authors are supported by project MINECO-15-MTM2014-56953-P

For a two parameter family of two-dimensional piecewise linear maps and for every natural number $n$, we prove not only the existence of intervals of parameters for which the respective maps are n times renormalizable but also we show the existence of intervals of parameters where the coexistence of at least $2^n$ strange attractors takes place. This family of maps contains the two-dimensional extension of the classical one-dimensional family of tent maps.

Citation: Antonio Pumariño, José Ángel Rodríguez, Enrique Vigil. Renormalization of two-dimensional piecewise linear maps: Abundance of 2-D strange attractors. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 941-966. doi: 10.3934/dcds.2018040
##### References:
 [1] M. Benedicks and L. Carleson, On iterations of $1-ax^2\;on\;(-1, 1)$, Annals of Mathematics, 122 (1985), 1-25. doi: 10.2307/1971367. Google Scholar [2] M. Benedicks and L. Carleson, The dynamics of the Hénon map, Annals of Mathematics, 133 (1991), 73-169. doi: 10.2307/2944326. Google Scholar [3] K. M. Brucks and H. Bruin, Topics from One-Dimensional Dynamics Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511617171. Google Scholar [4] 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 [5] W. de Melo and S. van Strien, One-Dimensional Dynamics Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-642-78043-1. Google Scholar [6] W. de Melo, Renormalization in one-dimensional dynamics, Journal of Difference Equations and Applications, 17 (2011), 1185-1197. doi: 10.1080/10236190902998016. Google Scholar [7] L. Mora and M. Viana, Abundance of strange attractors, Acta Mathematica, 171 (1993), 1-71. doi: 10.1007/BF02392766. Google Scholar [8] A. Pumariño and J. A. Rodríguez, Coexistence and Persistence of Strange Attractors Lecture Notes in Mathematics, 1658, Springer-Verlag, Berlin, 1997. doi: 10.1007/BFb0093337. Google Scholar [9] A. Pumariño, J. A. Rodríguez, J. C. Tatjer and E. Vigil, Piecewise linear bidimensional maps as models of return maps for 3D-diffeomorphisms, in Progress and Challenges in Dynamical Systems, Springer Proc. Math. Stat., 54, Springer, Heidelberg, (2013), 351-366. doi: 10.1007/978-3-642-38830-9_22. Google Scholar [10] A. Pumariño, J. A. Rodríguez, J. C. Tatjer and E. Vigil, Expanding Baker Maps as models for the dynamics emerging from 3D-homoclinic bifurcations, Discrete and Continuous Dynamical Systems -Series B, 19 (2014), 523-541. doi: 10.3934/dcdsb.2014.19.523. Google Scholar [11] 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 [12] A. Pumariño, J. A. Rodríguez and E. Vigil, Expanding Baker Maps: Coexistence of strange attractors, Discrete and Continuous Dynamical Systems -Series A, 37 (2017), 1651-1678. doi: 10.3934/dcds.2017068. Google Scholar [13] 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 [14] A. Pumariño and J. C. Tatjer, Attractors for return maps near homoclinic tangencies of three-dimensional dissipative diffeomorphism, Discrete and Continuous Dynamical Systems -Series B, 8 (2007), 971-1005. doi: 10.3934/dcdsb.2007.8.971. Google Scholar [15] S. Sternberg, On the structure of local homeomorphisms of euclidean n-space, American Journal of Mathematics, 80 (1958), 623-631. doi: 10.2307/2372774. Google Scholar [16] 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

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##### References:
 [1] M. Benedicks and L. Carleson, On iterations of $1-ax^2\;on\;(-1, 1)$, Annals of Mathematics, 122 (1985), 1-25. doi: 10.2307/1971367. Google Scholar [2] M. Benedicks and L. Carleson, The dynamics of the Hénon map, Annals of Mathematics, 133 (1991), 73-169. doi: 10.2307/2944326. Google Scholar [3] K. M. Brucks and H. Bruin, Topics from One-Dimensional Dynamics Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511617171. Google Scholar [4] 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 [5] W. de Melo and S. van Strien, One-Dimensional Dynamics Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-642-78043-1. Google Scholar [6] W. de Melo, Renormalization in one-dimensional dynamics, Journal of Difference Equations and Applications, 17 (2011), 1185-1197. doi: 10.1080/10236190902998016. Google Scholar [7] L. Mora and M. Viana, Abundance of strange attractors, Acta Mathematica, 171 (1993), 1-71. doi: 10.1007/BF02392766. Google Scholar [8] A. Pumariño and J. A. Rodríguez, Coexistence and Persistence of Strange Attractors Lecture Notes in Mathematics, 1658, Springer-Verlag, Berlin, 1997. doi: 10.1007/BFb0093337. Google Scholar [9] A. Pumariño, J. A. Rodríguez, J. C. Tatjer and E. Vigil, Piecewise linear bidimensional maps as models of return maps for 3D-diffeomorphisms, in Progress and Challenges in Dynamical Systems, Springer Proc. Math. Stat., 54, Springer, Heidelberg, (2013), 351-366. doi: 10.1007/978-3-642-38830-9_22. Google Scholar [10] A. Pumariño, J. A. Rodríguez, J. C. Tatjer and E. Vigil, Expanding Baker Maps as models for the dynamics emerging from 3D-homoclinic bifurcations, Discrete and Continuous Dynamical Systems -Series B, 19 (2014), 523-541. doi: 10.3934/dcdsb.2014.19.523. Google Scholar [11] 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 [12] A. Pumariño, J. A. Rodríguez and E. Vigil, Expanding Baker Maps: Coexistence of strange attractors, Discrete and Continuous Dynamical Systems -Series A, 37 (2017), 1651-1678. doi: 10.3934/dcds.2017068. Google Scholar [13] 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 [14] A. Pumariño and J. C. Tatjer, Attractors for return maps near homoclinic tangencies of three-dimensional dissipative diffeomorphism, Discrete and Continuous Dynamical Systems -Series B, 8 (2007), 971-1005. doi: 10.3934/dcdsb.2007.8.971. Google Scholar [15] S. Sternberg, On the structure of local homeomorphisms of euclidean n-space, American Journal of Mathematics, 80 (1958), 623-631. doi: 10.2307/2372774. Google Scholar [16] 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
Dynamics of $\Gamma_{\mu} .$
The smoothness domains for a map in $\mathbb{F} .$
The set $\mathcal{R}_1 .$
The set of parameters $\mathcal{P}_1 .$
The set $\mathcal{R}_{a, b} .$
The set of parameters $\mathcal{P}_2 .$
The set $\mathcal{R}_{a, b} .$
The sets $\Delta_{a, b}$ and $\Pi_{a, b} .$
The iterates of $\Delta_0:$ encircled in blue, the triangle $\Delta_0 ;$ encircled in a dashed blue line, the set $\Delta_4 .$
The set of parameters $\mathcal{P}_{\Delta} .$
The sets of parameters $\mathcal{P}_{\Delta}^{\prime}$ (dark grey) and $\mathcal{P}_{\Delta}$ (pale grey).
The iterates of $\Pi_0 :$ encircled in green, the triangle $\Pi_0 ;$ encircled in a dashed green line, the set $\Pi_4 .$
The set of parameters $\mathcal{P}_{\Pi} .$
(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 $H_{\Pi}(\mathcal{P}_3) .$
The relative position between $\gamma_0$ and $\eta_n.$
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