March  2017, 37(3): 1651-1678. doi: 10.3934/dcds.2017068

Renormalizable Expanding Baker Maps: Coexistence of strange attractors

1. 

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

2. 

Centro de Matemática da Universidade do Porto, Rua do Campo Alegre 687,4169-007 Porto, Portugal

Received  June 2016 Revised  November 2016 Published  December 2016

We introduce the concept of Expanding Baker Maps and renormalizable Expanding Baker Maps in a two-dimensional scenario. For a one-parameter family of Expanding Baker Maps we prove the existence of an interval of parameters for which the respective transformation is renormalizable. Moreover, we show the existence of intervals of parameters for which coexistence of strange attractors takes place.

Citation: Antonio Pumariño, José Ángel Rodríguez, Enrique Vigil. Renormalizable Expanding Baker Maps: Coexistence of strange attractors. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1651-1678. doi: 10.3934/dcds.2017068
References:
[1]

V. BaladiD. RockmoreN. Tongring and C. Tresser, Renormalization on the n-dimensional torus, Nonlinearity, 5 (1992), 1111-1136.  doi: 10.1088/0951-7715/5/5/005.  Google Scholar

[2]

M. Benedicks and L. Carleson, On iterations of $ 1-ax^2 $ on $(-1, 1)$, Ann. Math., 122 (1985), 1-25.  doi: 10.2307/1971367.  Google Scholar

[3]

M. Benedicks and L. Carleson, The dynamics of the Hénon map, Ann. Math., 133 (1991), 73-169.  doi: 10.2307/2944326.  Google Scholar

[4]

K. M. Brucks and H. Bruin, Topics from One-Dimensional Dynamics Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511617171.  Google Scholar

[5]

E. Colli, Infinitely many coexisting strange attractors, Ann. Inst. H. Poincaré, 15 (1998), 539-579.  doi: 10.1016/S0294-1449(98)80001-2.  Google Scholar

[6]

J. M. GambaudoS. van Strein and C. Tresser, Henon-like maps with strange attractors: There exist $ C^{∞} $ Kupka-Smale diffeomorphisms on $ S^2 $ with neither sinks nor sources, Nonlinearity, 2 (1989), 287-304.  doi: 10.1088/0951-7715/2/2/005.  Google Scholar

[7]

M. Lyubich and M. Martens, Renormalization of Hénon Maps: A survey, Dynamics, Games and Science. I, Springer Proc. Math., 1 (2011), 597-618.  doi: 10.1007/978-3-642-11456-4_37.  Google Scholar

[8]

W. de Melo, Renormalization in one-dimensional dynamics, J. Difference Equ. Appl., 17 (2011), 1185-1197.  doi: 10.1080/10236190902998016.  Google Scholar

[9]

W. de Melo and S. van Strien, One-Dimensional Dynamics Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-642-78043-1.  Google Scholar

[10]

L. Mora and M. Viana, Abundance of strange attractors, Acta Mathematica, 171 (1993), 1-71.  doi: 10.1007/BF02392766.  Google Scholar

[11]

S. Newhouse, Diffeomorphisms with infinitely many sinks, Topology, 13 (1974), 9-18.  doi: 10.1016/0040-9383(74)90034-2.  Google Scholar

[12]

J. Palis and J. C. Yoccoz, Homoclinic tangencies for hyperbolic sets of large Hausdorff dimension, Acta Math., 172 (1994), 91-136.  doi: 10.1007/BF02392792.  Google Scholar

[13]

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

[14]

A. Pumariño and J. A. Rodríguez, Coexistence and persistence of infinitely many strange attractors, Ergod. Theory Dyn. Syst., 21 (2001), 1511-1523.  doi: 10.1017/S0143385701001730.  Google Scholar

[15]

A. PumariñoJ. A. RodríguezJ. C. Tatjer and E. Vigil, Piecewise linear bidimensional maps as models of return maps for 3D-diffeomorphisms, Progress and Challenges in Dynamical Systems, Springer, 54 (2013), 351-366.  doi: 10.1007/978-3-642-38830-9_22.  Google Scholar

[16]

A. PumariñoJ. A. RodríguezJ. C. Tatjer and E. Vigil, Expanding Baker Maps as models for the dynamics emerging from 3D-homoclinic bifurcations, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 523-541.  doi: 10.3934/dcdsb.2014.19.523.  Google Scholar

[17]

A. PumariñoJ. A. RodríguezJ. 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

[18]

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

[19]

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

[20]

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

[21]

J. A. Yorke and K. T. Alligood, Cascades of period doubling bifurcations: A prerequisite for horseshoes, Bull. A.M.S., 9 (1983), 319-322.  doi: 10.1090/S0273-0979-1983-15191-1.  Google Scholar

show all references

References:
[1]

V. BaladiD. RockmoreN. Tongring and C. Tresser, Renormalization on the n-dimensional torus, Nonlinearity, 5 (1992), 1111-1136.  doi: 10.1088/0951-7715/5/5/005.  Google Scholar

[2]

M. Benedicks and L. Carleson, On iterations of $ 1-ax^2 $ on $(-1, 1)$, Ann. Math., 122 (1985), 1-25.  doi: 10.2307/1971367.  Google Scholar

[3]

M. Benedicks and L. Carleson, The dynamics of the Hénon map, Ann. Math., 133 (1991), 73-169.  doi: 10.2307/2944326.  Google Scholar

[4]

K. M. Brucks and H. Bruin, Topics from One-Dimensional Dynamics Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511617171.  Google Scholar

[5]

E. Colli, Infinitely many coexisting strange attractors, Ann. Inst. H. Poincaré, 15 (1998), 539-579.  doi: 10.1016/S0294-1449(98)80001-2.  Google Scholar

[6]

J. M. GambaudoS. van Strein and C. Tresser, Henon-like maps with strange attractors: There exist $ C^{∞} $ Kupka-Smale diffeomorphisms on $ S^2 $ with neither sinks nor sources, Nonlinearity, 2 (1989), 287-304.  doi: 10.1088/0951-7715/2/2/005.  Google Scholar

[7]

M. Lyubich and M. Martens, Renormalization of Hénon Maps: A survey, Dynamics, Games and Science. I, Springer Proc. Math., 1 (2011), 597-618.  doi: 10.1007/978-3-642-11456-4_37.  Google Scholar

[8]

W. de Melo, Renormalization in one-dimensional dynamics, J. Difference Equ. Appl., 17 (2011), 1185-1197.  doi: 10.1080/10236190902998016.  Google Scholar

[9]

W. de Melo and S. van Strien, One-Dimensional Dynamics Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-642-78043-1.  Google Scholar

[10]

L. Mora and M. Viana, Abundance of strange attractors, Acta Mathematica, 171 (1993), 1-71.  doi: 10.1007/BF02392766.  Google Scholar

[11]

S. Newhouse, Diffeomorphisms with infinitely many sinks, Topology, 13 (1974), 9-18.  doi: 10.1016/0040-9383(74)90034-2.  Google Scholar

[12]

J. Palis and J. C. Yoccoz, Homoclinic tangencies for hyperbolic sets of large Hausdorff dimension, Acta Math., 172 (1994), 91-136.  doi: 10.1007/BF02392792.  Google Scholar

[13]

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

[14]

A. Pumariño and J. A. Rodríguez, Coexistence and persistence of infinitely many strange attractors, Ergod. Theory Dyn. Syst., 21 (2001), 1511-1523.  doi: 10.1017/S0143385701001730.  Google Scholar

[15]

A. PumariñoJ. A. RodríguezJ. C. Tatjer and E. Vigil, Piecewise linear bidimensional maps as models of return maps for 3D-diffeomorphisms, Progress and Challenges in Dynamical Systems, Springer, 54 (2013), 351-366.  doi: 10.1007/978-3-642-38830-9_22.  Google Scholar

[16]

A. PumariñoJ. A. RodríguezJ. C. Tatjer and E. Vigil, Expanding Baker Maps as models for the dynamics emerging from 3D-homoclinic bifurcations, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 523-541.  doi: 10.3934/dcdsb.2014.19.523.  Google Scholar

[17]

A. PumariñoJ. A. RodríguezJ. 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

[18]

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

[19]

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

[20]

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

[21]

J. A. Yorke and K. T. Alligood, Cascades of period doubling bifurcations: A prerequisite for horseshoes, Bull. A.M.S., 9 (1983), 319-322.  doi: 10.1090/S0273-0979-1983-15191-1.  Google Scholar

Figure 1.  (a) A good fold. (b) A not good fold.
Figure 2.  Sequence of good folds.
Figure 3.  An example of EBM.
Figure 4.  Dynamics of $ \Lambda_t .$
Figure 5.  (a) Convex attractor for $\Lambda_t, \ t=0.95.$ (b) Non-simply-connected attractor for $\Lambda_t, \ t=0.765.$
Figure 6.  (a) 8-pieces attractor for $\Lambda_t, \ t=0.74 .$ (b) A magnification of the encircled piece of the previous attractor rotated by an angle $ \pi $.
Figure 7.  (a) Two 32-pieces attractors for $\Lambda_t, \ t=0.715 .$ (b) A magnification of the encircled piece of the previous attractors.
Figure 8.  The $\Lambda_t^8-$invariant domain. Encircled in a dashed blue line, the set $ \widetilde{\Delta}_0 .$
Figure 9.  Dynamic of $ \Gamma_{1, t} .$
Figure 10.  Renormalized attractor for $ t=0.74 .$ (a) Attractor for $\Lambda_t$ (magnification). (b) Attractor for $\Gamma_{1, t}.$
Figure 11.  The smoothness domains for a map in $ \mathbb{F} .$
Figure 12.  The sets $ \mathcal{T}, $ $ \mathcal{T}_{b, 1} $ and $ \mathcal{T}_{b, 2} $.
Figure 13.  The sets $T_{b, 2}, $ $ \mathcal{R}_1 $ and $ \mathcal{R}_{a, b} .$
Figure 14.  The set of parameters $ \mathcal{P}_0 .$ In dashed blue, the curve of parameters for which $ \Gamma_{1, t} $ displays a strictly invariant pentagonal domain.
Figure 15.  The iterates of $ \Delta_0.$ Encircled in a dashed blue line, the set $ \Delta_4 .$
Figure 16.  The iterates of $ \Pi_0 .$ Encircled a dashed green line, the set $ \Pi_4 .$
Figure 17.  The 4-pieces attractors for $ \Gamma_{1, t} $ and the iterates of $ \Delta_0 $ (pale gray or green, in the electronic version) and $ \Pi_0 $ (dark gray or blue, in the electronic version).
Figure 18.  Twice renormalized attractors for $t=0.715 .$ (a) Attractor for $ \Gamma_{2, 1, t} .$ (b) Attractor for $ \Gamma_{2, 2, t} .$
Figure 19.  Three-times renormalized attractors for $\Lambda_t .$ (a) Attractor for $ \Gamma_{3, 1}, \ t=0.71 .$ (b) Attractor for $ \Gamma_{3, 2}, \ t=0.7096 .$ (c) Attractor for $ \Gamma_{3, 3}, \ t=0.7093 .$ (d) Attractor for $ \Gamma_{3, 4}, \ t=0.7087 .$
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