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Any Jordan curve in the complex plane can be approximated arbitrarily well in the Hausdorff topology by Julia sets of polynomials. Finite collections of disjoint Jordan domains can be approximated by the basins of attraction of rational…
Criticality in statistical physics naturally emerges at isolated points in the phase diagram. Jamming of spheres is not an exception: varying density, it is the critical point that separates the unjammed phase where spheres do not overlap…
Schrijver graphs are vertex-color-critical subgraphs of Kneser graphs having the same chromatic number. They also share the value of their fractional chromatic number but Schrijver graphs are not critical for that. Here we present an…
We study the dynamics of complex polynomials. We obtain results on Poincare return maps defined on certain neighborhoods of a point with bounded orbit under a polynomial. We introduce a generalization of the Yoccoz tau-function, the Yoccoz…
We show that near any given minimizing sequence of paths for the mountain pass lemma, there exists a critical point whose polarization is also a critical point. This is motivated by the fact that if any polarization of a critical point is…
In this article, we combine complex-analytic and arithmetic tools to study the preperiodic points of one-dimensional complex dynamical systems. We show that for any fixed complex numbers a and b, and any integer d at least 2, the set of…
An equicritical stratum is the locus of univariate monic squarefree complex polynomials where the critical points have prescribed multiplicities. Tracking the positions of both roots and critical points, there is a natural ``monodromy map''…
In this paper, we introduce the concept of central periodic points of a linear system as points which lies on orbits starting and ending at the central subgroup of the system. We show that this set is bounded if and only if the central…
We study the large-$N$ limit of adjoint fermion one-matrix models. We find one-cut solutions of the loop equations for the correlators of these models and show that they exhibit third order phase transitions associated with $m$-th order…
We show that if a polynomial filled Julia set has empty interior, then it is computable.
In recent years, there has been significant progress in the understanding of the dynamics of transcendental entire functions with bounded postsingular set. In particular, for certain classes of such functions, a complete description of…
In a new type of percolation phase transition, which was observed in a set of non-equilibrium models, each new connection between vertices is chosen from a number of possibilities by an Achlioptas-like algorithm. This causes preferential…
We consider the problem of sensitivity of threshold risk, defined as the probability of a function of a random variable falling below a specified threshold level $\delta >0.$ We demonstrate that for polynomial and rational functions of that…
We introduce the notion of almost realizability, an arithmetic generalization of realizability for integer sequences, which is the property of counting periodic points for some map. We characterize the intersection between the set of…
We consider a model of a random copolymer at a selective interface which undergoes a localization/delocalization transition. In spite of the several rigorous results available for this model, the theoretical characterization of the phase…
Every expanding Thurston map $f$ without periodic critical points is known to have an iterate $f^n$ which is the topological mating of two polynomials. This has been examined by Kameyama and Meyer; the latter who has offered an explicit…
We determine the exact Borel class of the points whose iterates under $\exp(z)+a$ tend to infinity. We also prove that the sets of non-escaping Julia points for many of these functions are topologically equivalent.
The quark number susceptibility near the QCD critical end-point (CEP), the tricritical point (TCP) and the O(4) critical line at finite temperature and quark chemical potential is investigated. Based on the universality argument and…
Sufficient conditions are given for the computation of accessing arcs and arcs that links boundary components of multiply connected domains. The existence of a not-computably-accessible but computable point on a computably compact arc is…
We consider fixed points of the Feigenbaum (periodic-doubling) operator whose orders tend to infinity. It is known that the hyperbolic dimension of their Julia sets go to 2. We prove that the Lebesgue measure of these Julia sets tend to…