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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 |
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.
References:
[1] |
V. Baladi, D. Rockmore, N. Tongring and C. Tresser,
Renormalization on the n-dimensional torus, Nonlinearity, 5 (1992), 1111-1136.
doi: 10.1088/0951-7715/5/5/005. |
[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. |
[3] |
M. Benedicks and L. Carleson,
The dynamics of the Hénon map, Ann. Math., 133 (1991), 73-169.
doi: 10.2307/2944326. |
[4] |
K. M. Brucks and H. Bruin,
Topics from One-Dimensional Dynamics Cambridge University Press, Cambridge, 2004.
doi: 10.1017/CBO9780511617171. |
[5] |
E. Colli,
Infinitely many coexisting strange attractors, Ann. Inst. H. Poincaré, 15 (1998), 539-579.
doi: 10.1016/S0294-1449(98)80001-2. |
[6] |
J. M. Gambaudo, S. 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. |
[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. |
[8] |
W. de Melo,
Renormalization in one-dimensional dynamics, J. Difference Equ. Appl., 17 (2011), 1185-1197.
doi: 10.1080/10236190902998016. |
[9] |
W. de Melo and S. van Strien,
One-Dimensional Dynamics Springer-Verlag, Berlin, 1993.
doi: 10.1007/978-3-642-78043-1. |
[10] |
L. Mora and M. Viana,
Abundance of strange attractors, Acta Mathematica, 171 (1993), 1-71.
doi: 10.1007/BF02392766. |
[11] |
S. Newhouse,
Diffeomorphisms with infinitely many sinks, Topology, 13 (1974), 9-18.
doi: 10.1016/0040-9383(74)90034-2. |
[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. |
[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. |
[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. |
[15] |
A. Pumariño, J. A. Rodríguez, J. 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. |
[16] |
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 Contin. Dyn. Syst. Ser. B, 19 (2014), 523-541.
doi: 10.3934/dcdsb.2014.19.523. |
[17] |
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. |
[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. |
[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. |
[20] |
J. C. Tatjer,
Three-dimensional dissipative diffeomorphisms with homoclinic tangencies, Ergodic Theory and Dynamical Systems, 21 (2001), 249-302.
doi: 10.1017/S0143385701001146. |
[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. |
show all references
References:
[1] |
V. Baladi, D. Rockmore, N. Tongring and C. Tresser,
Renormalization on the n-dimensional torus, Nonlinearity, 5 (1992), 1111-1136.
doi: 10.1088/0951-7715/5/5/005. |
[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. |
[3] |
M. Benedicks and L. Carleson,
The dynamics of the Hénon map, Ann. Math., 133 (1991), 73-169.
doi: 10.2307/2944326. |
[4] |
K. M. Brucks and H. Bruin,
Topics from One-Dimensional Dynamics Cambridge University Press, Cambridge, 2004.
doi: 10.1017/CBO9780511617171. |
[5] |
E. Colli,
Infinitely many coexisting strange attractors, Ann. Inst. H. Poincaré, 15 (1998), 539-579.
doi: 10.1016/S0294-1449(98)80001-2. |
[6] |
J. M. Gambaudo, S. 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. |
[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. |
[8] |
W. de Melo,
Renormalization in one-dimensional dynamics, J. Difference Equ. Appl., 17 (2011), 1185-1197.
doi: 10.1080/10236190902998016. |
[9] |
W. de Melo and S. van Strien,
One-Dimensional Dynamics Springer-Verlag, Berlin, 1993.
doi: 10.1007/978-3-642-78043-1. |
[10] |
L. Mora and M. Viana,
Abundance of strange attractors, Acta Mathematica, 171 (1993), 1-71.
doi: 10.1007/BF02392766. |
[11] |
S. Newhouse,
Diffeomorphisms with infinitely many sinks, Topology, 13 (1974), 9-18.
doi: 10.1016/0040-9383(74)90034-2. |
[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. |
[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. |
[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. |
[15] |
A. Pumariño, J. A. Rodríguez, J. 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. |
[16] |
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 Contin. Dyn. Syst. Ser. B, 19 (2014), 523-541.
doi: 10.3934/dcdsb.2014.19.523. |
[17] |
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. |
[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. |
[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. |
[20] |
J. C. Tatjer,
Three-dimensional dissipative diffeomorphisms with homoclinic tangencies, Ergodic Theory and Dynamical Systems, 21 (2001), 249-302.
doi: 10.1017/S0143385701001146. |
[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. |















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