Julia Sets and Complex Singularities of Free Energies

102 Pages·2015·2.049 MB·English

by  Jianyong Qiao

MEMOIRS
of the
American Mathematical Society
Volume 234 • Number 1102 (second of 5 numbers) • March 2015
Julia Sets and Complex Singularities
of Free Energies
Jianyong Qiao
ISSN 0065-9266 (print) ISSN 1947-6221 (online)
American Mathematical Society
MEMOIRS
of the
American Mathematical Society
Volume 234 • Number 1102 (second of 5 numbers) • March 2015
Julia Sets and Complex Singularities
of Free Energies
Jianyong Qiao
ISSN 0065-9266 (print) ISSN 1947-6221 (online)
American Mathematical Society
Providence, Rhode Island
Library of Congress Cataloging-in-Publication Data
Qiao,Jianyong,1962–
Juliasetsandcomplexsingularitiesoffreeenergies/JianyongQiao.
pages cm. – (Memoirs of the AmericanMathematicalSociety, ISSN 0065-9266; volume 234,
number1102)
Includesbibliographicalreferences.
ISBN978-1-4704-0982-1(alk. paper)
1.Juliasets. 2.Fractals. I.Title.
QA614.86.Q53 2014
515(cid:2).39–dc23 2014041891
DOI:http://dx.doi.org/10.1090/memo/1102
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Contents
Introduction 1
Chapter 1. Complex dynamics and Potts models 5
1.1. Iterations of a rational map 5
1.2. Julia sets related to Potts models 6
Chapter 2. Dynamical complexity of renormalization transformations 11
2.1. Factorization of renormalization transformations 11
2.2. Classification of dynamical systems 12
2.3. Iteration of real maps 14
2.4. Iteration of the real map U 16
mnλ
2.5. Complex singularities associated with Feigenbaum phenomenon 25
Chapter 3. Connectivity of Julia sets 29
3.1. J(U ) with variant parameters 29
mnλ
3.2. Connectivity numbers of periodic domains of U 35
mnλ
3.3. The proof of Theorem 3.3 44
3.4. Disconnected Julia set J(U ) 45
2nλ
Chapter 4. Jordan domains and Fatou components 51
4.1. Local connectedness of J(U ) 51
mnλ
4.2. Jordan domains in F(U ) 58
mnλ
4.3. Jordan curve and J(U ) 65
mnλ
Chapter 5. Critical exponent of free energy 71
5.1. Free energy on Fatou set 71
5.2. Boundary behavior of free energy 76
5.3. Thermodynamical formalism 78
5.4. Critical exponent 82
Acknowledgements 85
Bibliography 87
iii

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 arenormalization transformation, whenit is treatedas a
complex dynamical system, is the set of complex singularities of the free energy in
statisticalmechanics. WegiveasufficientandnecessaryconditionfortheJuliasets
tobedisconnected. Furthermore,weprovethatallFatoucomponents(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
realtemperaturedomaincouldcontainaninterval. Finally, westudytheboundary
behaviorofthefirstderivativeandsecondderivativeofthefreeenergyontheFatou
componentcontainingtheinfinity. Wealsogiveanexplicitvalueofthesecondorder
critical exponent of the free energy for almost every boundary point.
ReceivedbytheeditorJune25,2011,and,inrevisedform,January23,2013.
ArticleelectronicallypublishedonJuly28,2014.
DOI:http://dx.doi.org/10.1090/memo/1102
2010MathematicsSubjectClassification. Primary37F10,37F45;Secondary: 82B20,82B28.
Key words and phrases. Julia set, Fatou set, renormalization transformation, iterate, phase
transition.
TheresearchwassupportedbytheNationalNaturalScienceFoundationofChina,theStateKey
DevelopmentProgramofBasicResearchofChina.
Affiliation at time of publication: School of Science and School of Computer Science, Beijing
UniversityofPostsandTelecommunications,Beijing,100876,People’sRepublicofChina;email:
[email protected].
(cid:3)c2014 American Mathematical Society
v

Introduction
The theory of complex dynamical systems is the study of a dynamical system
generated by an non-invertible analytic map R : S → S of a Riemann surface
S. An important example is a rational map of the Riemann sphere C. In this
case the dynamical system is the semigroup of iterations Rj (j = 1,2,...). The
basic problem is to understand the phase portrait of such a system, that is, the
typical behavior of orbits {Rj(z)}∞ , as well as the character of change of the
j=0
phase portrait under the deformation of R.
Itiswellknownthatthetheoryofcomplexdynamicalsystemswasfirststudied
atthebeginningofthelastcenturybyG.Julia([JU])andP.Fatou([FA1],[FA2]).
During 1919 to 1921, by applying the theory of normal families to the theory of
iterationsofanalyticmaps,theyestablishedthefoundationofthetheoryofcomplex
dynamicalsystemswhichiscalledtheFatou-Juliatheorynow. However,thistheory
passedthroughafifty-yearepochofstagnationbefore1980,whenitenteredaperiod
of great development by introduceing modern techniques into the study. It focuses
in itself ideas and methods of very diverse areas of mathematics. This kind of
dynamical systems provides an understanding of the nature of chaos, the fractal
property, and structural stability. Thus it has became one of the main sources for
the formulation of problems in the nonlinear theory.
For a complex dynamical system, the Julia set is an unstable set, while the
Fatou set is a stable set. It is well known that a typical Julia set is fractal, the
dynamicalsystemontheJuliasetischaotic. In1983,Derrida,DeSeze,andItzykso
([DDI]) found a connection between the phase transition in statistical mechanics
and Julia sets in complex dynamical systems. In 1952, Yang and Lee proved the
celebrated Yang-Lee theorem ([LY], [YL]) in statistical mechanics. The theorem
deals with the analytic continuation of the free energy on the complex plane, here
the free energy means the logarithm of the partition function. They studied the
distribution of zerosof the partition functionwhich is considered as afunctionof a
complex magnetic field (Yang-Lee zeros). They proved the famous circle theorem
which states that zeros of the partition function of an Ising ferromagnet lie on the
unit circle in an externally applied complex magnetic field plane. Hence complex
singularitiesofthefreeenergylieontheunitcircleaswell. Afterthispioneerwork,
Fisher ([FI]) in 1964 initiated the study of zeros of the partition function in the
complex temperature plane (Fisher zeros). These methods were then extended to
othertypesofinteractionsandfoundawiderangeofapplications(see[BO],[GA],
[GU], [KI], [LI], [MO1], [MO2]).
Animportantproblemstatedin[YL]istostudythelimitdistributionofzeros
ofthegrandpartitionfunction. Thereasonisthatthefreeenergycanbeexpressed
asalogarithmicpotentialoverthisdistribution. Since1952,numerousarticleshave
dealt with various properties of complex singularities of ferromagnetic models (see
1
2 INTRODUCTION
[GU], [KI], [LI], [MO1], [MO2]). However, properties of complex singularities of
antiferromagneticmodelsaremuchlesswellunderstoodthanthoseofferromagnetic
models (see [KI]). It was generally assumed for a long time that zeros of the grand
partition function lie on a smooth curve. But in 1983, it was realized that the
picture of the distribution of this kind of zeros is not so simple. Derrida, De Seze
and Itzykso ([DDI]) found fractal patterns in so-called hierarchical lattices. It has
been shown for many examples that these singularities are located on the Julia set
associatedwitharenormalizationtransformation(see[DDI],[MO2],[PR]).Some
interesting relationships between critical exponents, critical amplitudes and the
shape of a Julia set have been found ([DIL]). In [BL], Bleher and Lyubich studied
Julia sets and complex singularities in diamond-like hierarchical Ising models. For
a general model, they reformulated the following problem: How are singularities of
the free energy continued to the complex space and what is their global structure
in the complex space?
In this article, we deal with a λ-state Potts model on a generalized diamond
hierarchical lattice which is a natural generalization of a diamond-like hierarchical
Ising models studied in many papers in the past thirty years (see [BL], [DDI],
[DIL], [PR], [QI5], [QL], [QYG], [YA]). A λ-state Potts model (for integer or
non-integer values of λ) plays an important role in the general theory of phase
transitions and critical phenomena ([GU], [HU], [LI], [OS]). In this article, it is
proved that the limit distribution of complex singularities of the free energy of a
generalized diamond hierarchical Potts model is exactly the Julia set of a renor-
malizationtransformationwiththreeparameters(Theorem1.1). Themainsubject
of this article is the structure of this family of Julia sets. In view of the problem
concerningthedistributionofcomplexsingularitiesproposedin[YL]and[BL],we
give a complete description about the connectivity and the local connectivity of
these Julia sets (Theorem 3.1-3.3, Theorem 4.1). One of significant results is that
the Julia set of the renormalization transformation for some parameters contains a
small Feigenbaum Julia set which intersects with the positive real axis in a closed
interval (Theorem 2.2). This is an interesting phenomenon which has never been
foundbefore. Sincethepositiverealaxiscorrespondstotherealworld, itmaylead
to new problems in the research of statistical physics. In order to deal with the
free energy on the Riemann sphere, we study the regularity of boundaries of all
components of the Fatou set of the renormalization transformation (Theorem 4.2
and Theorem 4.3). These results will help in the study of the boundary behavior
ofthefreeenergy. Finally, anexplicitvalueofthesecondordercriticalexponentof
the freeenergy for almost all points on the boundary of the immediately attractive
basin of infinity is given (Theorem 5.4).
In this article, we shall use U to denote the above renormalization trans-
mnλ
formation, where m,n ∈ N and λ ∈ R are three parameters. In Chapter 1, we
introduce basic notations and fundamental results in complex dynamical systems.
We also give a definition of a generalized diamond hierarchical Potts model. By a
classical theorem in the theory of complex dynamical systems we can deduce that
the set of complex singularities of a generalized diamond hierarchical Potts model
is the Julia set of the renormalization transformation U (Theorem1.1).
mnλ
Chapter 2 is devoted to study the dynamical complexity of renormalization
transformations U with variant parameters m, n and λ. Firstly, we give a
mnλ
marvellous factorization of U . It is very helpful to us for dealing with the
mnλ
INTRODUCTION 3
dynamics of U in this article. Furthermore, we give a classification about the
mnλ
complexity of the dynamics of the renormalization transformation U (Theorem
mnλ
2.1). Afterexploring locationsof real fixedpoints andthe post-critical set of U
mnλ
deeply, we find a very interesting phenomenon about the distribution of complex
singularities. WeprovethattheJuliasetJ(U )couldcontainasmallFeigenbaum
2nλ
Julia set which meets the positive real axis at a closed interval. This leads to a
mystical distribution of complex singularities: the set of complex singularities in
the real temperature domain could contain an interval (Theorem 2.2). This is a
very interesting phenomenon.
In Chapter 3, we deal with the connectivity of the Julia set J(U ) of the
mnλ
renormalizationtransformationU . Firstly,inthischapterweprovethatJ(U )
mnλ mnλ
is connected when m = n or m and n are both odd (Theorem 3.1 and Theorem
3.2). Furthermore, we give a sufficient and necessary condition for the Julia set
J(U ) to be a disconnected set (Theorem 3.3).
mnλ
In Chapter 4, we deal with topological properties of the Fatou components of
U . The main result in this chapter is that all components of the Fatou set of
mnλ
U areJordandomains withatmost oneexceptionwhichisacompletelyinvari-
mnλ
ant domain (Theorem 4.2). In order to prove this result, we need a result about
the local connectivity of the Julia set J(U ) which tells us that all components
mnλ
of J(U ) are locally connected (Theorem 4.1). When the absolute value of λ is
mnλ
large enough, we show that the Julia set J(U ) is actually a quasicircle. In this
mnλ
case the Fatou set F(U ) consists of two Jordan domains (Theorem 4.3).
mnλ
Chapter 5 is devoted to dealing with the critical exponent of the free energy
of a generalized diamond hierarchical Potts model. Considering the immediate at-
tractive basin A (∞) which corresponds to the ”high temperature” domain, we
mnλ
show thatthe derivative f(cid:3) of the freeenergy f is analyticon A (∞) and
mnλ mnλ mnλ
the boundary ∂A (∞) is the natural boundary of f(cid:3) for some parameters m,
mnλ mnλ
n and λ (Theorem 5.2). Noting that the second derivatives f(cid:3)(cid:3) is not contin-
mnλ
uous up to the boundary ∂A (∞) (Theorem 5.3), we give an explicit value of
mnλ
the second order critical exponent α(2) of f for almost every point τ on the
τ mnλ
boundary of A (∞) (Theorem 5.4). The main method used for the proof of this
mnλ
result is the thermodynamical formalism following Bowen, Ruelle and Sinai (see
[BOW],[RU1],[SINA]).