Frontiers of Mathematics in China >
Dynamics of a family of rational maps concerning renormalization transformation
Received date: 03 Mar 2020
Accepted date: 21 Jul 2020
Published date: 15 Aug 2020
Copyright
Considering a family of rational maps concerning renormalization transformation, we give a perfect description about the dynamical properties of and the topological properties of the Fatou components F (). Furthermore, we discuss the continuity of the Hausdorff dimension HD(J ()) about real parameter λ.
Yuhan ZHANG , Junyang GAO , Jianyong QIAO , Qinghua WANG . Dynamics of a family of rational maps concerning renormalization transformation[J]. Frontiers of Mathematics in China, 2020 , 15(4) : 807 -833 . DOI: 10.1007/s11464-020-0854-9
| 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
|
/
| 〈 |
|
〉 |