Dynamics of a family of rational maps concerning renormalization transformation

Abstract

Considering a family of rational maps T_{n\lambda }concerning renormalization transformation, we give a perfect description about the dynamical properties of T_{n\lambda } and the topological properties of the Fatou components F (T_{n\lambda }). Furthermore, we discuss the continuity of the Hausdorff dimension HD(J (T_{n\lambda })) about real parameter λ.

Keywords

Completely invariant domain / quasi-circle / Hausdorff dimension / renormalization transformation

References

[1] Beardon A F. Iteration of Rational Functions. New York: Springer-Verlag, 1991

[2] Bleher P M, Lyubich M Y. Julia sets and complex singularities in hierarchical Ising models. Comm Math Phys, 1991, 141(3): 453–474

[3] Carleson L, Gamelin T W. Complex Dynamics. New York: Springer-Verlag, 1993

[4] Carleson L, Jones P W, Yoccoz J C. Julia and John. Bull Braz Math Soc, 1994, 25(1): 1–30

[5] Collet P, Eckmann J-P. Iterated Map on the Interval as Dynamical Systems. Boston: Birkhäuser, 2009

[6] Denker M, Urbanski M. Hausdorff measures on Julia sets of subexpanding rational maps. Israel J Math, 1991, 76(1-2): 193–214

[7] Derrida B, De Seze L, Itzykson C. Fractal structure of zeros in hierarchical models. J Stat Phys, 1983, 33(3): 559–569

[8] Derrida B, Itzykson C, Luck J M. Oscillatory critical amplitudes in hierarchical models. Comm Math Phys, 1984, 94(1): 115–132

[9] Fisher M E. The nature of critical points. In: Brittin W E. Lectures in Theoretical Physics, Vol VII C: Statistical Physics, Weak Interactions, Field Theory. New York: Gordon Breach, 1965, 1–159

[10] Gao J. On quasi-discs and Fatou components of a family of rational maps concerning renormalization transformation. Chaos Solitons Fractals, 2018, 114: 31–37

[11] Gao J, Qiao J, Zhai Y. Misiurewicz points on the Mandelbrot-like set concerning renormalization transformation. Sci China Math, 2014, 57(12): 2539–2548

[12] Lee T D, Yang C N. Statistical theory of equations of state and phase transitions. II. Lattice gas and Ising model. Phys Rev, 1952, 87(3): 410–419

[13] McMullen C T. Hausdorff dimension and conformal dynamics II: Geometrically finite rational maps. Comment Math Helv, 2000, 75(4): 535–593

[14] Milnor J. Dynamics in One Complex Variable. 3rd ed. Ann of Math Stud, Vol 160. Princeton: Princeton Univ Press, 2006

[15] Monroe J L. Julia sets associated with the Potts model on the Bethe lattice and other recursively solved systems. J Phys A, 2001, 34(33): 6405–6412

[16] Pommerenke C. Boundary Behaviour of Conformal Maps. Grundlehren Math Wiss, Vol 299. Berlin: Springer, 1992

[17] Qiao J. On the preimages of parabolic periodic points. Nonlinearity, 2000, 13(3): 813–818

[18] Qiao J. Julia sets and complex singularities in diamond-like hierarchical Potts models. Sci China Ser A: Math, 2005, 48(3): 388–412

[19] Qiao J, Li Y. On connectivity of Julia sets of Yang-Lee zeros. Comm Math Phys, 2001, 222(2): 319–326

[20] Qiao J, Yin Y, Gao J. Feigenbaum Julia sets of singularities of free energy. Ergodic Theory Dynam Systems, 2010, 30(5): 1573–1591

[21] Qiu W, Wang X, Yin Y. Dynamics of McMullen maps. Adv Math, 2012, 229(4): 2525–2577

[22] Qiao J. Julia Sets and Complex Singularities of Free Energies. Mem Amer Math Soc, Vol 234, No 1102. Providence: Amer Math Soc, 2015

[23] Rivera-Letelier J. On the continuity of Hausdorff dimension of Julia sets and similarity between the Mandelbrot set and Julia sets. Fund Math, 2001, 170(3): 287–317

[24] Ruelle D. Repellers for real analytic maps. Ergodic Theory Dynam Systems, 1982, 2(1): 99–107

[25] Sullivan D. Conformal Dynamical Systems. Berlin: Springer, 1983

[26] Wang X, Qiu W, Yin Y, Qiao J, Gao J. Connectivity of the Mandelbrot set for the family of renormalization transformations. Sci China Math, 2010, 53(3): 849–862

[27] Wilson K G. Renormalization group and critical phenomena. I. Renormalization group and the Kadanoff scaling picture. Phys Rev B, 1971, 4(9): 3174–3183

[28] Yang C N, Lee T D. Statistical theory of equations of state and phase transitions. I. Theory of condensation. Phys Rev, 1952, 87(3): 404–409

[29] Yang F, Zeng J. On the dynamics of a family of generated renormalization transformations. J Math Anal Appl, 2014, 413(1): 361–377

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Higher Education Press