Julia Sets and Complex Singularities of Free Energies

Qiao, Jianyong

doi:10.1090/memo/1102

Julia Sets and Complex Singularities of Free Energies

About this Title

Jianyong Qiao

Publication: Memoirs of the American Mathematical Society
Publication Year: 2015; Volume 234, Number 1102
ISBNs: 978-1-4704-0982-1 (print); 978-1-4704-2029-1 (online)
DOI: http://dx.doi.org/10.1090/memo/1102
Published electronically: July 28, 2014
Keywords:Julia set, Fatou set, renormalization transformation, iterate

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Chapters

Introduction
Chapter 1. Complex dynamics and Potts models
Chapter 2. Dynamical complexity of renormalization transformations
Chapter 3. Connectivity of Julia sets
Chapter 4. Jordan domains and Fatou components
Chapter 5. Critical exponent of free energy

Abstract

We study a family of renormalization transformations of generalized diamond hierarchical Potts models through complex dynamical systems. We prove that the Julia set (unstable set) of a renormalization transformation, when it is treated as a complex dynamical system, is the set of complex singularities of the free energy in statistical mechanics. We give a sufficient and necessary condition for the Julia sets to be disconnected. Furthermore, we prove that all Fatou components (components of the stable sets) of this family of renormalization transformations are Jordan domains with at most one exception which is completely invariant. In view of the problem in physics about the distribution of these complex singularities, we prove here a new type of distribution: the set of these complex singularities in the real temperature domain could contain an interval. Finally, we study the boundary behavior of the first derivative and second derivative of the free energy on the Fatou component containing the infinity. We also give an explicit value of the second order critical exponent of the free energy for almost every boundary point.

[AH] Lars V. Ahlfors, Conformal invariants: topics in geometric function theory, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. McGraw-Hill Series in Higher Mathematics. MR 0357743
[BL] P. M. Bleher and M. Yu. Lyubich, Julia sets and complex singularities in hierarchical Ising models, Comm. Math. Phys. 141 (1991), no. 3, 453–474. MR 1134933
[BO] A. N. Berker $\&$ S. Ostlund, Renormalization group calculations of finite systems, J. Phys. C: 12(1979), 4961-4975.
[BOW] Rufus Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Mathematics, Vol. 470, Springer-Verlag, Berlin-New York, 1975. MR 0442989
[CA] C. Carathéodory, Über die Begrenzung einfach zusammenhängender Gebiete, Math. Ann. 73 (1913), no. 3, 323–370 (German). MR 1511737, 10.1007/BF01456699
[CE] Pierre Collet and Jean-Pierre Eckmann, Iterated maps on the interval as dynamical systems, Progress in Physics, vol. 1, Birkhäuser, Boston, Mass., 1980. MR 613981
[CG] Lennart Carleson and Theodore W. Gamelin, Complex dynamics, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993. MR 1230383
[CJS] G. Z. Cui, Y. P.Jiang $\&$ D. Sullivan, Dynamics of geometrically finite rational maps, manuscript, 1996.
[CJY] Lennart Carleson, Peter W. Jones, and Jean-Christophe Yoccoz, Julia and John, Bol. Soc. Brasil. Mat. (N.S.) 25 (1994), no. 1, 1–30. MR 1274760, 10.1007/BF01232933
[DDI] B. Derrida, L. de Seze, and C. Itzykson, Fractal structure of zeros in hierarchical models, J. Statist. Phys. 33 (1983), no. 3, 559–569. MR 732376, 10.1007/BF01018834
[DH] A. Douady $\&$ J. H. Hubbard, Etudes dynamique des polynômes complexes, Publications Mathematiques d $'$ Orsay, 1984.
[DIL] B. Derrida, C. Itzykson, and J. M. Luck, Oscillatory critical amplitudes in hierarchical models, Comm. Math. Phys. 94 (1984), no. 1, 115–132. MR 763765
[FA1] P. Fatou, Sur les solutions uniformes de certains équations fonctionnelle, C. R. Acad. Sci. Paris 143(1906), 546-548.
[FA2] P. Fatou, Sur les équations fonctionnelles, Bull. Soc. Math. France 47(1919), 161-271; 48(1920), 33-94, 208-314.
[FE1] Mitchell J. Feigenbaum, Quantitative universality for a class of nonlinear transformations, J. Statist. Phys. 19 (1978), no. 1, 25–52. MR 0501179
[FE2] Mitchell J. Feigenbaum, The transition to aperiodic behavior in turbulent systems, Comm. Math. Phys. 77 (1980), no. 1, 65–86. MR 588687
[FI] M. E. Fisher, The nature of critical points, Lectures in Theor. Phys. VIIc (edited by W. E. Brittin) 1-160, University of Colorado Press, Boulder, 1965.
[GA] R. G. Ghulghazaryan and N. S. Ananikian, Partition function zeros of the one-dimensional Potts model: the recursive method, J. Phys. A 36 (2003), no. 23, 6297–6312. MR 1986953, 10.1088/0305-4470/36/23/302
[GO] G. M. Golusin, Geometic Function Theory in Complex Variable, Moscow, 1966.
[GU] Z. Glumac and K. Uzelac, The partition function zeros in the one-dimensional $q$-state Potts model, J. Phys. A 27 (1994), no. 23, 7709–7717. MR 1312278
[HL] Bambi Hu and Bin Lin, Yang-Lee zeros, Julia sets, and their singularity spectra, Phys. Rev. A (3) 39 (1989), no. 9, 4789–4796. MR 994768, 10.1103/PhysRevA.39.4789
[HU] Kerson Huang, Statistical mechanics, 2nd ed., John Wiley & Sons, Inc., New York, 1987. MR 1042093
[JI1] Yunping Jiang, Renormalization and geometry in one-dimensional and complex dynamics, Advanced Series in Nonlinear Dynamics, vol. 10, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. MR 1442953
[JI2] Yunping Jiang, Infinitely renormalizable quadratic polynomials, Trans. Amer. Math. Soc. 352 (2000), no. 11, 5077–5091. MR 1675198, 10.1090/S0002-9947-00-02514-9
[JI3] Yun Ping Jiang, Markov partitions and Feigenbaum-like mappings, Comm. Math. Phys. 171 (1995), no. 2, 351–363. MR 1344730
[JU] G. Julia, Mémoire sur l $'$ itératin de fonctions rationnelles, J. Math. Pure Appl. 8(1918), 47-245.
[KA] L. P. Kadanoff, Notes on Migdal $'$ s recursion formulate, Ann. Phys. 100(1976), 359-394.
[KI] Seung-Yeon Kim, Yang-Lee zeros of the antiferromagnetic Ising model, Phys. Rev. Lett. 93 (2004), no. 13, 130604, 4. MR 2119237, 10.1103/PhysRevLett.93.130604
[LI] T. C. Lubensky, J. Isaacsom, Field theory for the statistics of branched polymers, gelation, and vulcanization, Phys. Rev. Lett. 41(1978), 829-832.
[LM] Mikhail Lyubich and John Milnor, The Fibonacci unimodal map, J. Amer. Math. Soc. 6 (1993), no. 2, 425–457. MR 1182670, 10.1090/S0894-0347-1993-1182670-0
[LY] T. D. Lee and C. N. Yang, Statistical theory of equations of state and phase transitions. II. Lattice gas and Ising model, Physical Rev. (2) 87 (1952), 410–419. MR 0053029
[LYU] Mikhail Lyubich, Dynamics of quadratic polynomials. I, II, Acta Math. 178 (1997), no. 2, 185–247, 247–297. MR 1459261, 10.1007/BF02392694
[MAK] N. G. Makarov, On the distortion of boundary sets under conformal mappings, Proc. London Math. Soc. (3) 51 (1985), no. 2, 369–384. MR 794117, 10.1112/plms/s3-51.2.369
[MAN] Ricardo Mañé, On a theorem of Fatou, Bol. Soc. Brasil. Mat. (N.S.) 24 (1993), no. 1, 1–11. MR 1224298, 10.1007/BF01231694
[MC] Curtis T. McMullen, Complex dynamics and renormalization, Annals of Mathematics Studies, vol. 135, Princeton University Press, Princeton, NJ, 1994. MR 1312365
[MI] John Milnor, Dynamics in one complex variable, Friedr. Vieweg & Sohn, Braunschweig, 1999. Introductory lectures. MR 1721240
[MO1] James L. Monroe, Potts models with period doubling cascades, chaos, etc, J. Phys. A 29 (1996), no. 17, 5421–5427. MR 1419030, 10.1088/0305-4470/29/17/016
[MO2] James L. Monroe, Julia sets associated with the Potts model on the Bethe lattice and other recursively solved systems, J. Phys. A 34 (2001), no. 33, 6405–6412. MR 1862966, 10.1088/0305-4470/34/33/305
[MS] Welington de Melo and Sebastian van Strien, One-dimensional dynamics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 25, Springer-Verlag, Berlin, 1993. MR 1239171
[OS] A. H. Osbaldestin, $1/s$-expansion for generalized dimensions in a hierarchical $s$-state Potts models, J. Phys. A 28 (1995), no. 20, 5951–5962. MR 1364770
[PI] Kevin M. Pilgrim, Rational maps whose Fatou components are Jordan domains, Ergodic Theory Dynam. Systems 16 (1996), no. 6, 1323–1343. MR 1424402, 10.1017/S0143385700010051
[PO] Ch. Pommerenke, Boundary behaviour of conformal maps, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 299, Springer-Verlag, Berlin, 1992. MR 1217706
[PR] H.-O. Peitgen and P. H. Richter, The beauty of fractals, Springer-Verlag, Berlin, 1986. Images of complex dynamical systems. MR 852695
[PT] Kevin Pilgrim and Tan Lei, Rational maps with disconnected Julia set, Astérisque 261 (2000), xiv, 349–384 (English, with English and French summaries). Géométrie complexe et systèmes dynamiques (Orsay, 1995). MR 1755447
[QG] Qiao Jianyong and Gao Junyang, Jordan domain and Fatou set concerning diamond-like hierarchical Potts models, Nonlinearity 20 (2007), no. 1, 119–131. MR 2285108, 10.1088/0951-7715/20/1/008
[QI1] Jianyong Qiao, Julia sets and complex singularities in diamond-like hierarchical Potts models, Sci. China Ser. A 48 (2005), no. 3, 388–412. MR 2158293, 10.1360/04ys0180
[QI2] Jianyong Qiao, On the preimages of parabolic periodic points, Nonlinearity 13 (2000), no. 3, 813–818. MR 1759001, 10.1088/0951-7715/13/3/316
[QI3] J. Y. Qiao, On Fatou sets concerning Yang-Lee theory, Science in China, Ser. A 35(2005), 191-205.
[QI4] Jianyong Qiao, On Julia sets concerning phase transitions, Sci. China Ser. A 46 (2003), no. 3, 415–431. MR 2010149
[QI5] J. Y. Qiao, Complex Dynamics of Renormalization Transformations (in Chinese), Science Press, Beijing, 2010.
[QL] Jianyong Qiao and Yuhua Li, On connectivity of Julia sets of Yang-Lee zeros, Comm. Math. Phys. 222 (2001), no. 2, 319–326. MR 1859602, 10.1007/s002200100507
[QYG] Jianyong Qiao, Yongcheng Yin, and Junyang Gao, Feigenbaum Julia sets of singularities of free energy, Ergodic Theory Dynam. Systems 30 (2010), no. 5, 1573–1591. MR 2718910, 10.1017/S0143385709000522
[RO] P. Roesch, On local connectivity for the Julia set of rational maps: Newton’s famous example, Ann. of Math. (2) 168 (2008), no. 1, 127–174. MR 2415400, 10.4007/annals.2008.168.127
[RU1] David Ruelle, Thermodynamic formalism, Encyclopedia of Mathematics and its Applications, vol. 5, Addison-Wesley Publishing Co., Reading, Mass., 1978. The mathematical structures of classical equilibrium statistical mechanics; With a foreword by Giovanni Gallavotti and Gian-Carlo Rota. MR 511655
[SINA] Ja. G. Sinaĭ, Gibbs measures in ergodic theory, Uspehi Mat. Nauk 27 (1972), no. 4(166), 21–64 (Russian). MR 0399421
[SING] David Singer, Stable orbits and bifurcation of maps of the interval, SIAM J. Appl. Math. 35 (1978), no. 2, 260–267. MR 0494306
[SU] Dennis Sullivan, Quasiconformal homeomorphisms and dynamics. I. Solution of the Fatou-Julia problem on wandering domains, Ann. of Math. (2) 122 (1985), no. 3, 401–418. MR 819553, 10.2307/1971308
[TY] Lei Tan and Yongcheng Yin, Local connectivity of the Julia set for geometrically finite rational maps, Sci. China Ser. A 39 (1996), no. 1, 39–47. MR 1397233
[WA] P. Walter, An Introduction to Ergodic Theory, Springer-Verlag, Berlin, 1982.
[WH] Gordon Thomas Whyburn, Analytic Topology, American Mathematical Society Colloquium Publications, v. 28, American Mathematical Society, New York, 1942. MR 0007095
[WI1] K. G. Wilson, Renormalization group and critical phenomena I., Phys. Rev. B 4(1971), 3174-3183.
[WI2] K. G. Wilson, Renormalization group and critical phenomena II., Phys. Rev. B 4(1971), 3184-3205.
[YA] Z. R. Yang, Family of diamond-type hierarchical lattices, Phys. Rev. B (3) 38 (1988), no. 1, 728–731. MR 949174, 10.1103/PhysRevB.38.728
[YL] C. N. Yang and T. D. Lee, Statistical theory of equations of state and phase transitions. I. Theory of condensation, Physical Rev. (2) 87 (1952), 404–409. MR 0053028

References

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[BO]A. N. Berker $\&$ S. Ostlund, Renormalization group calculations of finite systems, J. Phys. C: 12(1979), 4961-4975.
[BOW] R. Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Mathematics, Vol. 470, Springer-Verlag, Berlin, 1975. MR 0442989 (56 \#1364)
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[CG] L. Carleson and T. Gamelin, Complex dynamics, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993. MR 1230383 (94h:30033)
[CJS]G. Z. Cui, Y. P.Jiang $\&$ D. Sullivan, Dynamics of geometrically finite rational maps, manuscript, 1996.
[CJY] L. Carleson, P. W. Jones, and J.-C. Yoccoz, Julia and John, Bol. Soc. Brasil. Mat. (N.S.) 25 (1994), no. 1, 1–30. MR 1274760 (95d:30040), http://dx.doi.org/10.1007/BF01232933
[DDI] B. Derrida, L. de Seze, and C. Itzykson, Fractal structure of zeros in hierarchical models, J. Statist. Phys. 33 (1983), no. 3, 559–569. MR 732376 (84m:82112), http://dx.doi.org/10.1007/BF01018834
[DH]A. Douady $\&$ J. H. Hubbard, Etudes dynamique des polynômes complexes, Publications Mathematiques d $'$ Orsay, 1984.
[DIL] B. Derrida, C. Itzykson, and J. M. Luck, Oscillatory critical amplitudes in hierarchical models, Comm. Math. Phys. 94 (1984), no. 1, 115–132. MR 763765 (86e:82048)
[FA1]P. Fatou, Sur les solutions uniformes de certains équations fonctionnelle, C. R. Acad. Sci. Paris 143(1906), 546-548.
[FA2]P. Fatou, Sur les équations fonctionnelles, Bull. Soc. Math. France 47(1919), 161-271; 48(1920), 33-94, 208-314.
[FE1] M. J. Feigenbaum, Quantitative universality for a class of nonlinear transformations, J. Statist. Phys. 19 (1978), no. 1, 25–52. MR 0501179 (58 \#18601)
[FE2] M. J. Feigenbaum, The transition to aperiodic behavior in turbulent systems, Comm. Math. Phys. 77 (1980), no. 1, 65–86. MR 588687 (83e:58053)
[FI]M. E. Fisher, The nature of critical points, Lectures in Theor. Phys. VIIc (edited by W. E. Brittin) 1-160, University of Colorado Press, Boulder, 1965.
[GA] R. G. Ghulghazaryan and N. S. Ananikian, Partition function zeros of the one-dimensional Potts model: the recursive method, J. Phys. A 36 (2003), no. 23, 6297–6312. MR 1986953 (2004g:82036), http://dx.doi.org/10.1088/0305-4470/36/23/302
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[JI1] Y. P. Jiang, Renormalization and geometry in one-dimensional and complex dynamics, Advanced Series in Nonlinear Dynamics, vol. 10, World Scientific Publishing Co. Inc., River Edge, NJ, 1996. MR 1442953 (98e:58070)
[JI2] Y. P. Jiang, Infinitely renormalizable quadratic polynomials, Trans. Amer. Math. Soc. 352 (2000), no. 11, 5077–5091. MR 1675198 (2001b:37071), http://dx.doi.org/10.1090/S0002-9947-00-02514-9
[JI3] Y. P. Jiang, Markov partitions and Feigenbaum-like mappings, Comm. Math. Phys. 171 (1995), no. 2, 351–363. MR 1344730 (96g:58053)
[JU] G. Julia, Mémoire sur l $'$ itératin de fonctions rationnelles, J. Math. Pure Appl. 8(1918), 47-245.
[KA] L. P. Kadanoff, Notes on Migdal $'$ s recursion formulate, Ann. Phys. 100(1976), 359-394.
[KI] S.-Y. Kim, Yang-Lee zeros of the antiferromagnetic Ising model, Phys. Rev. Lett. 93 (2004), no. 13, 130604, 4. MR 2119237 (2005i:82029), http://dx.doi.org/10.1103/PhysRevLett.93.130604
[LI] T. C. Lubensky, J. Isaacsom, Field theory for the statistics of branched polymers, gelation, and vulcanization, Phys. Rev. Lett. 41(1978), 829-832.
[LM] M. Lyubich and J. Milnor, The Fibonacci unimodal map, J. Amer. Math. Soc. 6 (1993), no. 2, 425–457. MR 1182670 (93h:58080), http://dx.doi.org/10.2307/2152804
[LY] T. D. Lee and C. N. Yang, Statistical theory of equations of state and phase transitions. II. Lattice gas and Ising model, Physical Rev. (2) 87 (1952), 410–419. MR 0053029 (14,711c)
[LYU] M. Lyubich, Dynamics of quadratic polynomials. I, II, Acta Math. 178 (1997), no. 2, 185–247, 247–297. MR 1459261 (98e:58145), http://dx.doi.org/10.1007/BF02392694
[MAK] N. G. Makarov, On the distortion of boundary sets under conformal mappings, Proc. London Math. Soc. (3) 51 (1985), no. 2, 369–384. MR 794117 (87d:30012), http://dx.doi.org/10.1112/plms/s3-51.2.369
[MAN] R. Mañé, On a theorem of Fatou, Bol. Soc. Brasil. Mat. (N.S.) 24 (1993), no. 1, 1–11. MR 1224298 (94g:58188), http://dx.doi.org/10.1007/BF01231694
[MC] C. McMullen, Complex dynamics and renormalization, Annals of Mathematics Studies, vol. 135, Princeton University Press, Princeton, NJ, 1994. MR 1312365 (96b:58097)
[MI] J. Milnor, Dynamics in one complex variable, Friedr. Vieweg & Sohn, Braunschweig, 1999. Introductory lectures. MR 1721240 (2002i:37057)
[MO1] J. L. Monroe, Potts models with period doubling cascades, chaos, etc, J. Phys. A 29 (1996), no. 17, 5421–5427. MR 1419030 (97i:82027), http://dx.doi.org/10.1088/0305-4470/29/17/016
[MO2] J. L. Monroe, Julia sets associated with the Potts model on the Bethe lattice and other recursively solved systems, J. Phys. A 34 (2001), no. 33, 6405–6412. MR 1862966 (2002i:82014), http://dx.doi.org/10.1088/0305-4470/34/33/305
[MS] W. de Melo and S. van Strien, One-dimensional dynamics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 25, Springer-Verlag, Berlin, 1993. MR 1239171 (95a:58035)
[OS] A. H. Osbaldestin, $1/s$ -expansion for generalized dimensions in a hierarchical $s$ -state Potts models, J. Phys. A 28 (1995), no. 20, 5951–5962. MR 1364770 (96m:82004)
[PI] K. M. Pilgrim, Rational maps whose Fatou components are Jordan domains, Ergodic Theory Dynam. Systems 16 (1996), no. 6, 1323–1343. MR 1424402 (97k:58141), http://dx.doi.org/10.1017/S0143385700010051
[PO] Ch. Pommerenke, Boundary behaviour of conformal maps, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 299, Springer-Verlag, Berlin, 1992. MR 1217706 (95b:30008)
[PR] H.-O. Peitgen and P. H. Richter, The beauty of fractals, Springer-Verlag, Berlin, 1986. Images of complex dynamical systems. MR 852695 (88e:00019)
[PT] K. Pilgrim and L. Tan, Rational maps with disconnected Julia set, Astérisque 261 (2000), xiv, 349–384 (English, with English and French summaries). Géométrie complexe et systèmes dynamiques (Orsay, 1995). MR 1755447 (2001e:37056)
[QG] J. Y. Qiao and J. Y. Gao, Jordan domain and Fatou set concerning diamond-like hierarchical Potts models, Nonlinearity 20 (2007), no. 1, 119–131. MR 2285108 (2007k:37057), http://dx.doi.org/10.1088/0951-7715/20/1/008
[QI1] J. Y. Qiao, Julia sets and complex singularities in diamond-like hierarchical Potts models, Sci. China Ser. A 48 (2005), no. 3, 388–412. MR 2158293 (2006f:82023), http://dx.doi.org/10.1360/04ys0180
[QI2] J. Y. Qiao, On the preimages of parabolic periodic points, Nonlinearity 13 (2000), no. 3, 813–818. MR 1759001 (2001d:37056), http://dx.doi.org/10.1088/0951-7715/13/3/316
[QI3]J. Y. Qiao, On Fatou sets concerning Yang-Lee theory, Science in China, Ser. A 35(2005), 191-205.
[QI4] J. Y. Qiao, On Julia sets concerning phase transitions, Sci. China Ser. A 46 (2003), no. 3, 415–431. MR 2010149 (2004h:82030)
[QI5]J. Y. Qiao, Complex Dynamics of Renormalization Transformations (in Chinese), Science Press, Beijing, 2010.
[QL] J. Y. Qiao and Y. H. Li, On connectivity of Julia sets of Yang-Lee zeros, Comm. Math. Phys. 222 (2001), no. 2, 319–326. MR 1859602 (2002k:82020), http://dx.doi.org/10.1007/s002200100507
[QYG] J. Y. Qiao, Y. C. Yin, and J. Y. Gao, Feigenbaum Julia sets of singularities of free energy, Ergodic Theory Dynam. Systems 30 (2010), no. 5, 1573–1591. MR 2718910 (2011m:37072), http://dx.doi.org/10.1017/S0143385709000522
[RO] P. Roesch, On local connectivity for the Julia set of rational maps: Newton's famous example, Ann. of Math. (2) 168 (2008), no. 1, 127–174. MR 2415400 (2009c:37043), http://dx.doi.org/10.4007/annals.2008.168.127
[RU1] D. Ruelle, Thermodynamic formalism, Encyclopedia of Mathematics and its Applications, vol. 5, Addison-Wesley Publishing Co., Reading, Mass., 1978. The mathematical structures of classical equilibrium statistical mechanics; With a foreword by Giovanni Gallavotti and Gian-Carlo Rota. MR 511655 (80g:82017)
[SINA] Ya. G. Sina\i, Gibbs measures in ergodic theory, Uspehi Mat. Nauk 27 (1972), no. 4(166), 21–64 (Russian). MR 0399421 (53 \#3265)
[SING] D. Singer, Stable orbits and bifurcation of maps of the interval, SIAM J. Appl. Math. 35 (1978), no. 2, 260–267. MR 0494306 (58 \#13206)
[SU] D. Sullivan, Quasiconformal homeomorphisms and dynamics. I. Solution of the Fatou-Julia problem on wandering domains, Ann. of Math. (2) 122 (1985), no. 3, 401–418. MR 819553 (87i:58103), http://dx.doi.org/10.2307/1971308
[TY] L. Tan and Y. C. Yin, Local connectivity of the Julia set for geometrically finite rational maps, Sci. China Ser. A 39 (1996), no. 1, 39–47. MR 1397233 (97g:58142)
[WA]P. Walter, An Introduction to Ergodic Theory, Springer-Verlag, Berlin, 1982.
[WH] G. Whyburn, Analytic Topology, American Mathematical Society Colloquium Publications, v. 28, American Mathematical Society, New York, 1942. MR 0007095 (4,86b)
[WI1]K. G. Wilson, Renormalization group and critical phenomena I., Phys. Rev. B 4(1971), 3174-3183.
[WI2]K. G. Wilson, Renormalization group and critical phenomena II., Phys. Rev. B 4(1971), 3184-3205.
[YA] Z. R. Yang, Family of diamond-type hierarchical lattices, Phys. Rev. B (3) 38 (1988), no. 1, 728–731. MR 949174 (89d:82148), http://dx.doi.org/10.1103/PhysRevB.38.728
[YL] C. N. Yang and T. D. Lee, Statistical theory of equations of state and phase transitions. I. Theory of condensation, Physical Rev. (2) 87 (1952), 404–409. MR 0053028 (14,711b)

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Julia Sets and Complex Singularities of Free Energies