Buried points in complex dynamical systems and glassy transitions

Abstract

<p indent=0mm>A new interdisciplinary research direction is addressed across the fields of statistical mechanics and complex dynamical systems together with specific research topics, such as the connection between buried points and glassy transitions. First, pertinent results on complex dynamical systems are delivered. Let<italic> f</italic> be a complex analytical mapping from the Riemann sphere (or the complex plane) to itself, i.e., <italic>f</italic> is rational or entire. The stable set <italic>F</italic>(<italic>f</italic>) of this dynamical system is called the Fatou set of <italic>f</italic>, and the unstable set <italic>J</italic>(<italic>f</italic>) the Julia set. Every component of <italic>F</italic>(<italic>f</italic>) is called a Fatou component. Obviously, the boundary of each Fatou component belongs to <italic>J</italic>(<italic>f</italic>). If a point in <italic>J</italic>(<italic>f</italic>) does not belong to the boundary of any Fatou component, then it is a buried point of <italic>f</italic>, which is often used to describe the topological complexity of the Julia set. In this quest, Makienko’s conjecture is undoubtedly one of the most important problems which is stated as follows: Let <italic>R</italic> be a rational mapping and <italic>F</italic>(<italic>R</italic>) contain infinitely many components. If every Fatou component is not completely invariant, then <italic>R</italic> has buried points. The known result is the following: If <italic>J</italic>(<italic>R</italic>) is disconnected, or connected and locally connected, then Makienko’s conjecture is true; if <italic>J</italic>(<italic>R</italic>) is disconnected and <italic>R</italic> has a buried point, then it must have buried components. Concerning entire functions, we have: If <italic>F</italic>(<italic>f</italic>) is not empty, then <italic>f</italic> has a buried point if and only if <italic>F</italic>(<italic>f</italic>) is disconnected. Following the above results, interesting topological properties on the sets of buried points are deduced. However, many open problems remain unsolved in regard to these special points. In the second part, we explained the topological complexity of the set of Yang-Lee zeros. In 1952, Yang C.N. and Lee T.D. established an analytic theory in statistical mechanics, summarized as the circle theorem: For statistical models like 2-dimensional lattice gas, the zeros of the partition function condense to the unit circle. Henceforth, the importance of the distribution of Yang-Lee zeros in the complex plane is emphasized for general models. In 1983, researchers including B. Derrida, L. De Seze and C. Itzykson found that the distribution of Yang-Lee zeros can be cast into a problem on Julia sets of renormalization transformations. In the third part, we describe the connection between buried points and glass transitions. Concerning the Potts model on a generalized diamond hierarchical lattice, we present a new type of distribution of Julia sets of renormalization transformations, which is called the real Feigenbaum-type distribution of buried points. Physically, this means that the limiting set of Yang-Lee zeros could contain an interval of the real axis, all of which are singularities of the free energy function. This important observation indicates that the exploration of renormalization transformation dynamics is essential to the understanding of the phenomenon of glassy transitions. Finally, we discussed the theoretical value and practical significance of the above research topics and directions. It is well known that <italic>Science</italic> published 125 most challenging scientific questions in 2005 to celebrate the Journal’s 125th anniversary, among which Question 47 is “What is the essence of the glassy substance?”. There are many theoretical models for glassy transitions, but up to now, a complete theory remains illusive. Perhaps, the connection between complex dynamics and glassy transition established above could shed new light on this old problem with the help of powerful tools from mathematics, as ever before.