Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-26T01:02:20.369Z Has data issue: false hasContentIssue false
  • English
  • Français

Feigenbaum Julia sets of singularities of free energy

Published online by Cambridge University Press:  13 October 2009

JIANYONG QIAO ,
YONGCHENG YIN  and
JUNYANG GAO
JIANYONG QIAO
Affiliation:
Center of Mathematics, China University of Mining and Technology (Beijing), Beijing 100083, China (email: [email protected], [email protected])
YONGCHENG YIN
Affiliation:
School of Mathematical Sciences, Fudan University, Shanghai, 200433, China (email: [email protected])
JUNYANG GAO
Affiliation:
Center of Mathematics, China University of Mining and Technology (Beijing), Beijing 100083, China (email: [email protected], [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

Considering the Potts model on diamond-like hierarchical lattices, in this paper, we show that the Julia sets of the singularities of the free energy may be a Feigenbaum Julia set. Furthermore, we prove that these Julia sets are locally connected if they are connected.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 30 , Issue 5 , October 2010 , pp. 1573 - 1591
Copyright
Copyright © Cambridge University Press 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Barata, J. C. A. and Goldbaum, P. S.. On the distribution and gap structure of Lee–Yang zeros for the Ising model, periodic and aperiodic couplings. J. Stat. Phys. 103 (2001), 857–891.CrossRefGoogle Scholar
[2]Beardon, A. F.. Iteration of Rational Functions. Springer, Berlin, 1991.CrossRefGoogle Scholar
[3]Bleher, P. M. and Lyubich, M.. Julia sets and complex singularities in hierarchical Ising models. Comm. Math. Phys. 141 (1991), 453–474.CrossRefGoogle Scholar
[4]Carleson, L., Jones, P., Yoccoz, J. C., Julia, and John, . Bol. Soc. Bras. Mat. 25 (1994), 1–30.CrossRefGoogle Scholar
[5]Carleson, L. and Gamelin, T. W.. Complex Dynamics. Springer, Berlin, 1991.Google Scholar
[6]Derrida, B., De Seze, L. and Itzykson, C.. Fractal structure of zeros in hierarchical models. J. Stat. Phys. 33 (1983), 559–569.CrossRefGoogle Scholar
[7]Derrida, B., Itzykson, C. and Luck, J. K.. Oscillatory critical amplitudes in hierarchical models. Comm. Math. Phys. 94 (1984), 115–132.CrossRefGoogle Scholar
[8]Douady, A. and Hubbard, J. H.. Étude dynamique des polynômes complexes, I-II. Publications Mathématiques d’Orsay, 1984.Google Scholar
[9]Erzan, A.. Hierarchical q-state Potts models with periodic and aperiodic renormalization group trajectories. Phys. Lett. A 93 (1983), 140–237.CrossRefGoogle Scholar
[10]Fisher, M. E.. The Nature of Critical Points (Lectures in Theoretical Physics, VIIC). Ed. Brittin, W. B.. Gordon and Breach, New York, 1965, pp. 1–157.Google Scholar
[11]Großmann, S.. Analytic Properties of Thermodynamic Functions and Phase Transitions (Festkörperprobleme IX). Ed. Madelung, O.. Vieweg, Brounschweig Wiesbaden, 1969, pp. 207–245.Google Scholar
[12]Guckenheimer, J.. On the bifurcation of maps of the interval. Invent. Math. 39 (1977), 165.CrossRefGoogle Scholar
[13]Jiang, Y.. Renormalization and Geometry in One-Dimensional and Complex Dynamics. World Scientific, Singapore, 1996.CrossRefGoogle Scholar
[14]Mañé, R.. On a theorem of Fatou. Bol. Soc. Bras. Mat. 24 (1993), 1–11.CrossRefGoogle Scholar
[15]de Melo, W. and Van Strien, S.. One Dimensional Dymamics. Springer, Berlin, 1993.CrossRefGoogle Scholar
[16]Milnor, J.. Dynamics in One Complex Variable, 2nd edn. Vieweg, Brounschweig Wiesbaden, 2000.CrossRefGoogle Scholar
[17]Näkki, R. and Väsälä, J.. John disks. Expo. Math. (1991), 3–43.Google Scholar
[18]Pilgrim, K. and Tan, L.. Rational map with disconnected Julia set. Astérisque 261 (2000), 349–384.Google Scholar
[19]Pommerenke, Ch.. Boundary Behaviour of Conformal Maps. Springer, Berlin, 1992.CrossRefGoogle Scholar
[20]Qiao, J.. Julia sets and complex singularities in diamond-like hierarchical Potts models. Sci. China. Ser. A 48(3) (2005), 388–412.CrossRefGoogle Scholar
[21]Qiao, J. and Li, Y.. On connectivity of Julia sets of Yang–Lee zeros. Comm. Math. Phys. 222 (2001), 319–326.CrossRefGoogle Scholar
[22]Qiao, J. and Gao, J.. Jordan domain and Fatou set concerning diamond-like hierarchical models. Nonlinearity 40 (2007), 119–131.Google Scholar
[23]Roesch, P.. On local connectivity for the Julia set of rational maps: Newton’s famous example. Ann. of Math. (2) 165 (2007), 1–48.Google Scholar
[24]Shishikura, M. and Tan, L.. An alternative proof of Mañé’s theorem on non-expanding Julia sets. The Mandelbrot Set, Theme and Variations (London Mathematical Society Lecture Note Series, 274). Ed. Tan, L.. Cambridge University Press, Cambridge, 2000, pp. 265–279.CrossRefGoogle Scholar
[25]Singer, D.. Stable orbits and bifurcation of maps of the interval. SIAM J. Appl. Math. 35(2) (1978), 260–267.CrossRefGoogle Scholar
[26]Tan, L. and Yin, Y.. Local connectivity of the Julia set for geometrically finite rational maps. Sci. China Ser. A 39 (1996), 39–47.Google Scholar
[27]Whyburn, G.. Analytic Topology (American Mathematical Society Colloquium Publications, 28). American Mathematical Society, Providence, RI, 1942.CrossRefGoogle Scholar
[28]Yang, C. N. and Lee, T. D.. Statistical theory of equations of state and phase transitions. I. Theory of condensation. Phys. Rev. 87 (1952), 404–419.CrossRefGoogle Scholar