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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

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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

K. M. Brucks and H. Bruin, "Topics From One-Dimensional Dynamics," London Mathematical Society Student Texts, 62, Cambridge University Press, Cambridge, 2004.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

H. F. Cui and Y. M. Ding, Renormalization and conjugacy of piecewise linear Lorenz maps,, preprint, ().   Google Scholar

[8]

Y. M. Ding and W. T. Fan, The asymptotic periodicity of Lorenz maps, Acta Math. Sci., 19 (1999), 114-120.  Google Scholar

[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.  Google Scholar

[10]

P. Glendinning, Topological conjugation of Lorenz maps by $\beta$-transformations, Math. Proc. Camb. Phil. Soc., 107 (1990), 401-413. doi: 10.1017/S0305004100068675.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

J. Guckenheimer and R. F. Williams, Structural stability of Lorenz attractors, IHES Publ. Math., 50 (1979), 59-72.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

M. I. Malkin, Rotation intervals and the dynamics of Lorenz type mappings, Selecta Mathematica Sovietica, 10 (1991), 265-275.  Google Scholar

[19]

M.Martens and W. de Melo, Universal models for Lorenz maps, Ergodic Theory Dynam. Systems, 21 (2001), 833-860. doi: 10.1017/S0143385701001420.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

M. R. Palmer, "On the Classification of Measure Preserving Transformations of Lebesgue Spaces," Ph. D. thesis, University of Warwick, 1979. Google Scholar

[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.  Google Scholar

[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  Google Scholar

[25]

C. Robinson, Nonsymmetric Lorenz attractors from a homoclinic bifurcation, SIAM J. Math. Anal., 32 (2000), 119-141. doi: 10.1137/S0036141098343598.  Google Scholar

[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.  Google Scholar

[27]

C. Sparrow, "The Lorenz Equations: Bifurcations, Chaos and Strange Attractors," Applied Mathematical Sciences, 41, Springer-Verlag, 1982.  Google Scholar

[28]

W. Tucker, The Lorenz attractor exists, C. R. Acad. Sci. Paris Sér. I Math., 328 (1999), 1197-1202.  Google Scholar

[29]

W. Tucker, A rigorous ODE solver and Smale's 14th problem, Found. Comput. Math., 2 (2002), 53-117.  Google Scholar

[30]

M. Viana, What's new on Lorenz strange attractors?, Math. Intelligencer, 22 (2000), 6-19. doi: 10.1007/BF03025276.  Google Scholar

[31]

R. F. Williams, The structure of Lorenz attractors, IHES Publ. Math., 50 (1979), 73-99.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

K. M. Brucks and H. Bruin, "Topics From One-Dimensional Dynamics," London Mathematical Society Student Texts, 62, Cambridge University Press, Cambridge, 2004.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

H. F. Cui and Y. M. Ding, Renormalization and conjugacy of piecewise linear Lorenz maps,, preprint, ().   Google Scholar

[8]

Y. M. Ding and W. T. Fan, The asymptotic periodicity of Lorenz maps, Acta Math. Sci., 19 (1999), 114-120.  Google Scholar

[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.  Google Scholar

[10]

P. Glendinning, Topological conjugation of Lorenz maps by $\beta$-transformations, Math. Proc. Camb. Phil. Soc., 107 (1990), 401-413. doi: 10.1017/S0305004100068675.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

J. Guckenheimer and R. F. Williams, Structural stability of Lorenz attractors, IHES Publ. Math., 50 (1979), 59-72.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

M. I. Malkin, Rotation intervals and the dynamics of Lorenz type mappings, Selecta Mathematica Sovietica, 10 (1991), 265-275.  Google Scholar

[19]

M.Martens and W. de Melo, Universal models for Lorenz maps, Ergodic Theory Dynam. Systems, 21 (2001), 833-860. doi: 10.1017/S0143385701001420.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

M. R. Palmer, "On the Classification of Measure Preserving Transformations of Lebesgue Spaces," Ph. D. thesis, University of Warwick, 1979. Google Scholar

[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.  Google Scholar

[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  Google Scholar

[25]

C. Robinson, Nonsymmetric Lorenz attractors from a homoclinic bifurcation, SIAM J. Math. Anal., 32 (2000), 119-141. doi: 10.1137/S0036141098343598.  Google Scholar

[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.  Google Scholar

[27]

C. Sparrow, "The Lorenz Equations: Bifurcations, Chaos and Strange Attractors," Applied Mathematical Sciences, 41, Springer-Verlag, 1982.  Google Scholar

[28]

W. Tucker, The Lorenz attractor exists, C. R. Acad. Sci. Paris Sér. I Math., 328 (1999), 1197-1202.  Google Scholar

[29]

W. Tucker, A rigorous ODE solver and Smale's 14th problem, Found. Comput. Math., 2 (2002), 53-117.  Google Scholar

[30]

M. Viana, What's new on Lorenz strange attractors?, Math. Intelligencer, 22 (2000), 6-19. doi: 10.1007/BF03025276.  Google Scholar

[31]

R. F. Williams, The structure of Lorenz attractors, IHES Publ. Math., 50 (1979), 73-99.  Google Scholar

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