scholarly journals On the Continuity of Hausdorff Dimension of Julia Sets Concerning Potts Models

2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Gang Liu ◽  
Junyang Gao
Keyword(s):  
Hausdorff Dimension ◽  
Statistical Mechanics ◽  
Julia Sets ◽  
Rational Maps ◽  
Two Dimensional ◽  
Potts Models

Considering the Julia sets of a family of rational maps concerning two-dimensional diamond hierarchical Potts models in statistical mechanics, we show the continuity of their Hausdorff dimension.

The Hausdorff dimension of Julia sets of hyperbolic meromorphic functions II

2000 ◽  
Vol 20 (3) ◽  
pp. 895-910 ◽  
Author(s):  
GWYNETH M. STALLARD
Keyword(s):  
Continuous Function ◽  
Hausdorff Dimension ◽  
Meromorphic Function ◽  
Statistical Mechanics ◽  
Rational Function ◽  
Meromorphic Functions ◽  
Julia Set ◽  
Julia Sets ◽  
Real Analytic ◽  
Analytic Maps

Ruelle (Repellers for real analytic maps. Ergod. Th. & Dynam. Sys.2 (1982), 99–108) used results from statistical mechanics to show that, when a rational function $f$ is hyperbolic, the Hausdorff dimension of the Julia set, $\dim J(f)$, depends real analytically on $f$. We give a proof of the fact that $\dim J(f)$ is a continuous function of $f$ that does not depend on results from statistical mechanics and we show that this result can be extended to a class of transcendental meromorphic functions. This enables us to show that, for each $d \in (0,1)$, there exists a transcendental meromorphic function $f$ with $\dim J(f) = d$.

Variation of Hausdorff dimension of Julia sets

1993 ◽  
Vol 13 (1) ◽  
pp. 167-174 ◽  
Author(s):  
T. J. Ransford
Keyword(s):  
Hausdorff Dimension ◽  
Connected Domain ◽  
Harmonic Functions ◽  
Julia Set ◽  
Julia Sets ◽  
Rational Maps ◽  
Analytic Family ◽  
Simply Connected Domain ◽  
Image Position ◽  
Simply Connected

AbstractLet (Rλ)λ∈D be an analytic family of rational maps of degree d ≥ 2, where D is a simply connected domain in ℂ, and each Rλ is hyperbolic. Then the Hausdorff dimension δ(λ) of the Julia set of Rλ satisfieswhere ℋ is a collection of harmonic functions u on D. We examine some consequences of this, and show how it can be used to obtain estimates for the Hausdorff dimension of some particular Julia sets.

Hausdorff Dimension of Julia Sets

2019 ◽  
pp. 153-192
Author(s):  
Xin-Hou Hua ◽  
Chung-Chun Yang
Keyword(s):  
Hausdorff Dimension ◽  
Julia Sets

Mixing Times of Critical Two-Dimensional Potts Models

2017 ◽  
Vol 71 (5) ◽  
pp. 994-1046 ◽  
Author(s):  
Reza Gheissari ◽  
Eyal Lubetzky
Keyword(s):  
Mixing Times ◽  
Two Dimensional ◽  
Potts Models

scholarly journals GENERALIZED YANG-BAXTER EQUATION

1993 ◽  
Vol 08 (24) ◽  
pp. 2299-2309 ◽  
Author(s):  
R. M. KASHAEV ◽  
YU. G. STROGANOV
Keyword(s):  
Single Layer ◽  
Lattice Models ◽  
Baxter Equation ◽  
Explicit Solutions ◽  
Generalized Equations ◽  
Two Dimensional ◽  
Layer Transfer ◽  
Transfer Matrices ◽  
Dimensional Lattice ◽  
Potts Models

A generalization of the Yang-Baxter equation is proposed. It enables us to construct integrable two-dimensional lattice models with commuting two-layer transfer matrices, while single-layer ones are not necessarily commutative. Explicit solutions to the generalized equations are found. They are related with Boltzmann weights of the sl (3) chiral Potts models.

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