Renormalization and \alpha-limit set for expanding Lorenz maps

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July 2011, 29(3): 979-999. doi: 10.3934/dcds.2011.29.979

Renormalization and $\alpha$-limit set for expanding Lorenz maps

Yiming Ding ^1,

1.	Wuhan Institute of Physics and Mathematics, The Chinese Academy of Sciences, Wuhan 430071, China

Received November 2009 Revised September 2010 Published November 2010

Abstract
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We show that there is a bijection between the renormalizations and proper completely invariant closed sets of expanding Lorenz map, which enables us to distinguish periodic and non-periodic renormalizations. Based on the properties of the periodic orbit with minimal period, the minimal completely invariant closed set is constructed. Topological characterizations of the renormalizations and $\alpha$-limit sets are obtained via consecutive renormalizations. Some properties of periodic renormalizations are collected in Appendix.

Keywords: Expanding Lorenz map, $\alpha$-limit set., renormalization.

Mathematics Subject Classification: Primary: 37E05, 37F25; Secondary: 54H2.

Citation: Yiming Ding. Renormalization and $\alpha$-limit set for expanding Lorenz maps. Discrete and Continuous Dynamical Systems, 2011, 29 (3) : 979-999. doi: 10.3934/dcds.2011.29.979

References:

[1]	V. S. Afraimovich, V. V. Bykov and L. P. Shil'nikov., On the appearance and structure of the Lorenz attractor, Dokl. Acad. Sci. USSR, 234 (1977), 336-339.
[2]	L. Alsedà and A. Falcò, On the topological dynamics and phase-locking renormalization of Lorenz-like maps, Ann. Inst. Fourier, Grenoble, 53 (2003), 859-883.
[3]	L. Alsedà, J. Llibre, M. Misiurewicz and C. Tresser, Periods and entropy for Lorenz-like maps, Ann. Inst. Fourier, Grenoble, 39 (1989), 929-952.
[4]	K. M. Brucks and H. Bruin, "Topics From One-Dimensional Dynamics," London Mathematical Society Student Texts, 62, Cambridge University Press, Cambridge, 2004.
[5]	Y. Choi, Attractors from one dimensional Lorenz-like maps, Discrete Contin. Dyn. Syst., 11 (2004), 715-730. doi: 10.3934/dcds.2004.11.715.
[6]	H. F. Cui and Y. M. Ding, The $\alpha$-limit sets of a unimodal map without homtervals, Topology Appl., 157 (2010), 22-28. doi: 10.1016/j.topol.2009.04.054.
[7]	H. F. Cui and Y. M. Ding, Renormalization and conjugacy of piecewise linear Lorenz maps, preprint, arXiv:0906.3131.
[8]	Y. M. Ding and W. T. Fan, The asymptotic periodicity of Lorenz maps, Acta Math. Sci., 19 (1999), 114-120.
[9]	L. Flatto and J. C. Lagarias, The lap-counting function for linear mod one transformations. I. Explicit formulas and renormalizability, Ergodic Theory Dynam. Systems, 16 (1996), 451-491. doi: 10.1017/S0143385700008920.
[10]	P. Glendinning, Topological conjugation of Lorenz maps by $\beta$-transformations, Math. Proc. Camb. Phil. Soc., 107 (1990), 401-413. doi: 10.1017/S0305004100068675.
[11]	P. Glendinning and T. Hall, Zeros of the kneading invariant and topological entropy for Lorenz maps, Nonlinearity, 9 (1996), 999-1014. doi: 10.1088/0951-7715/9/4/010.
[12]	P. Glendingning and C. Sparrow, Prime and renormalizable kneading invariants and the dynamics of expanding Lorenz maps, Physica D, 62 (1993), 22-50. doi: 10.1016/0167-2789(93)90270-B.
[13]	J. Guckenheimer and R. F. Williams, Structural stability of Lorenz attractors, IHES Publ. Math., 50 (1979), 59-72.
[14]	J. H. Hubbard and C. Sparrow, The classification of topologically expansive Lorenz maps, Comm. Pure Appl. Math., 43 (1990), 431-443. doi: 10.1002/cpa.3160430402.
[15]	G. Keller and P. Matthias, Topological and measurable dynamics of Lorenz maps, in "Ergodic Theory, Analysis and Efficient Simulation of Dynamical Systems," Springer, Berlin, (2001), 333-361.
[16]	S. Luzzatto, I. Melbourne and F. Paccaut, The Lorenz attractor is mixing, Commun. Math. Phys., 260 (2005), 393-401. doi: 10.1007/s00220-005-1411-9.
[17]	S. Luzzatto and W. Tucker, Non-uniformly expanding dynamics in maps with singularities and criticalities, Inst. Hautes Études Sci. Publ. Math. No., 89 (1999), 179-226.
[18]	M. I. Malkin, Rotation intervals and the dynamics of Lorenz type mappings, Selecta Mathematica Sovietica, 10 (1991), 265-275.
[19]	M.Martens and W. de Melo, Universal models for Lorenz maps, Ergodic Theory Dynam. Systems, 21 (2001), 833-860. doi: 10.1017/S0143385701001420.
[20]	C. A. Morales, M. J. Pacifico and B. San Martin, Expanding Lorenz attractors through resonant double homoclinic loops, SIAM J. Math. Anal., 36 (2005), 1836-1861. doi: 10.1137/S0036141002415785.
[21]	C. A. Morales, M. J. Pacifico and B. San Martin, Contracting Lorenz attractors through resonant double homoclinic loops, SIAM J. Math. Anal., 38 (2006), 309-332. doi: 10.1137/S0036141004443907.
[22]	M. R. Palmer, "On the Classification of Measure Preserving Transformations of Lebesgue Spaces," Ph. D. thesis, University of Warwick, 1979.
[23]	W. Parry, Symbolic dynamics and transformations of the unit interval, Trans. Amer. Math. Soc., 122 (1966), 368-378. doi: 10.1090/S0002-9947-1966-0197683-5.
[24]	W. Parry, The Lorenz attractor and a related population model, in "Ergodic Theory (Proc. Conf., Math. Forschungsinst., Oberwolfach, 1978)," Lecture Notes in Math., 729, Springer, Berlin, (1979), 169-187
[25]	C. Robinson, Nonsymmetric Lorenz attractors from a homoclinic bifurcation, SIAM J. Math. Anal., 32 (2000), 119-141. doi: 10.1137/S0036141098343598.
[26]	L. Silva and R. Sousa, Topological invariants and renormalization of Lorenz maps, Phys. D, 162 (2002), 233-243. doi: 10.1016/S0167-2789(01)00369-4.
[27]	C. Sparrow, "The Lorenz Equations: Bifurcations, Chaos and Strange Attractors," Applied Mathematical Sciences, 41, Springer-Verlag, 1982.
[28]	W. Tucker, The Lorenz attractor exists, C. R. Acad. Sci. Paris Sér. I Math., 328 (1999), 1197-1202.
[29]	W. Tucker, A rigorous ODE solver and Smale's 14th problem, Found. Comput. Math., 2 (2002), 53-117.
[30]	M. Viana, What's new on Lorenz strange attractors?, Math. Intelligencer, 22 (2000), 6-19. doi: 10.1007/BF03025276.
[31]	R. F. Williams, The structure of Lorenz attractors, IHES Publ. Math., 50 (1979), 73-99.

show all references

References:

[1]	V. S. Afraimovich, V. V. Bykov and L. P. Shil'nikov., On the appearance and structure of the Lorenz attractor, Dokl. Acad. Sci. USSR, 234 (1977), 336-339.
[2]	L. Alsedà and A. Falcò, On the topological dynamics and phase-locking renormalization of Lorenz-like maps, Ann. Inst. Fourier, Grenoble, 53 (2003), 859-883.
[3]	L. Alsedà, J. Llibre, M. Misiurewicz and C. Tresser, Periods and entropy for Lorenz-like maps, Ann. Inst. Fourier, Grenoble, 39 (1989), 929-952.
[4]	K. M. Brucks and H. Bruin, "Topics From One-Dimensional Dynamics," London Mathematical Society Student Texts, 62, Cambridge University Press, Cambridge, 2004.
[5]	Y. Choi, Attractors from one dimensional Lorenz-like maps, Discrete Contin. Dyn. Syst., 11 (2004), 715-730. doi: 10.3934/dcds.2004.11.715.
[6]	H. F. Cui and Y. M. Ding, The $\alpha$-limit sets of a unimodal map without homtervals, Topology Appl., 157 (2010), 22-28. doi: 10.1016/j.topol.2009.04.054.
[7]	H. F. Cui and Y. M. Ding, Renormalization and conjugacy of piecewise linear Lorenz maps, preprint, arXiv:0906.3131.
[8]	Y. M. Ding and W. T. Fan, The asymptotic periodicity of Lorenz maps, Acta Math. Sci., 19 (1999), 114-120.
[9]	L. Flatto and J. C. Lagarias, The lap-counting function for linear mod one transformations. I. Explicit formulas and renormalizability, Ergodic Theory Dynam. Systems, 16 (1996), 451-491. doi: 10.1017/S0143385700008920.
[10]	P. Glendinning, Topological conjugation of Lorenz maps by $\beta$-transformations, Math. Proc. Camb. Phil. Soc., 107 (1990), 401-413. doi: 10.1017/S0305004100068675.
[11]	P. Glendinning and T. Hall, Zeros of the kneading invariant and topological entropy for Lorenz maps, Nonlinearity, 9 (1996), 999-1014. doi: 10.1088/0951-7715/9/4/010.
[12]	P. Glendingning and C. Sparrow, Prime and renormalizable kneading invariants and the dynamics of expanding Lorenz maps, Physica D, 62 (1993), 22-50. doi: 10.1016/0167-2789(93)90270-B.
[13]	J. Guckenheimer and R. F. Williams, Structural stability of Lorenz attractors, IHES Publ. Math., 50 (1979), 59-72.
[14]	J. H. Hubbard and C. Sparrow, The classification of topologically expansive Lorenz maps, Comm. Pure Appl. Math., 43 (1990), 431-443. doi: 10.1002/cpa.3160430402.
[15]	G. Keller and P. Matthias, Topological and measurable dynamics of Lorenz maps, in "Ergodic Theory, Analysis and Efficient Simulation of Dynamical Systems," Springer, Berlin, (2001), 333-361.
[16]	S. Luzzatto, I. Melbourne and F. Paccaut, The Lorenz attractor is mixing, Commun. Math. Phys., 260 (2005), 393-401. doi: 10.1007/s00220-005-1411-9.
[17]	S. Luzzatto and W. Tucker, Non-uniformly expanding dynamics in maps with singularities and criticalities, Inst. Hautes Études Sci. Publ. Math. No., 89 (1999), 179-226.
[18]	M. I. Malkin, Rotation intervals and the dynamics of Lorenz type mappings, Selecta Mathematica Sovietica, 10 (1991), 265-275.
[19]	M.Martens and W. de Melo, Universal models for Lorenz maps, Ergodic Theory Dynam. Systems, 21 (2001), 833-860. doi: 10.1017/S0143385701001420.
[20]	C. A. Morales, M. J. Pacifico and B. San Martin, Expanding Lorenz attractors through resonant double homoclinic loops, SIAM J. Math. Anal., 36 (2005), 1836-1861. doi: 10.1137/S0036141002415785.
[21]	C. A. Morales, M. J. Pacifico and B. San Martin, Contracting Lorenz attractors through resonant double homoclinic loops, SIAM J. Math. Anal., 38 (2006), 309-332. doi: 10.1137/S0036141004443907.
[22]	M. R. Palmer, "On the Classification of Measure Preserving Transformations of Lebesgue Spaces," Ph. D. thesis, University of Warwick, 1979.
[23]	W. Parry, Symbolic dynamics and transformations of the unit interval, Trans. Amer. Math. Soc., 122 (1966), 368-378. doi: 10.1090/S0002-9947-1966-0197683-5.
[24]	W. Parry, The Lorenz attractor and a related population model, in "Ergodic Theory (Proc. Conf., Math. Forschungsinst., Oberwolfach, 1978)," Lecture Notes in Math., 729, Springer, Berlin, (1979), 169-187
[25]	C. Robinson, Nonsymmetric Lorenz attractors from a homoclinic bifurcation, SIAM J. Math. Anal., 32 (2000), 119-141. doi: 10.1137/S0036141098343598.
[26]	L. Silva and R. Sousa, Topological invariants and renormalization of Lorenz maps, Phys. D, 162 (2002), 233-243. doi: 10.1016/S0167-2789(01)00369-4.
[27]	C. Sparrow, "The Lorenz Equations: Bifurcations, Chaos and Strange Attractors," Applied Mathematical Sciences, 41, Springer-Verlag, 1982.
[28]	W. Tucker, The Lorenz attractor exists, C. R. Acad. Sci. Paris Sér. I Math., 328 (1999), 1197-1202.
[29]	W. Tucker, A rigorous ODE solver and Smale's 14th problem, Found. Comput. Math., 2 (2002), 53-117.
[30]	M. Viana, What's new on Lorenz strange attractors?, Math. Intelligencer, 22 (2000), 6-19. doi: 10.1007/BF03025276.
[31]	R. F. Williams, The structure of Lorenz attractors, IHES Publ. Math., 50 (1979), 73-99.

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