# American Institute of Mathematical Sciences

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

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

 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.
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.

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##### 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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