We study properties of the metric space of pluriregular sets and of contractions on that space induced by finite families of proper polynomial mappings of several complex variables. In particular, we show that closed balls in the space of pluriregular sets do not have to be compact and we give a simple proof of applicability of the so-called chaos game in the case of composite Julia sets. Part of the construction of those sets also leads to a computationally viable approximation by simpler sets based on Monte-Carlo simulation.
We investigate the metric space of pluriregular sets as well as the contractions on that space induced by infinite compact families of proper polynomial mappings of several complex variables. The topological semigroups generated by such families, with composition as the semigroup operation, lead to the constructions of a variety of Julia-type pluriregular sets. The generating families can also be viewed as infinite iterated function systems with compact attractors. We show that such attractors can be approximated both deterministically and probabilistically in a manner of the classic chaos game.
We define two norms on R-3 as follows. For (a, b,c) is an element of R-3, we let parallel to (a, b, c)parallel to (R) = sup {/ax(2) + bx + c/ : x is an element of [-1, 1]} and parallel to (a, b, c)parallel to (C) = sup "/az(2) +bz +c/ : z is an element of
It is shown that iteration of analytic set-valued functions can be used to generate composite Julia sets in C-N. Then it is shown that the composite Julia sets generated by a finite family of regular polynomial mappings of degree at least 2 in C-N, depend
A natural metric is introduced on the family of all polynomially convex compact L-regular sets in C-n, thus turning this family into a complete metric space. An application in complex dynamics is described.
The recent passing of Professor Jozef Siciak inevitably brings about reflections on his legacy, not just in terms of mathematical results he had obtained or inspired, but also in terms of shaping the way mathematics is being developed internally and as a part of science in general. Based on several decades of close personal contacts, the author attempts to outline Professor Siciak's views concerning these matters.
It is shown that both plurisubharmonicity and convexity of functions can be characterized in terms of Brownian motions with individually scaled components. Applications related to multivariable Jensen's inequality are also given.
Using a construction similar to an iterated function system, but withfunctions changing at each step of iteration, we provide a natural example of a continuous one-parameter family of holomorphic functions of infinitely many variables. This family is parametrized by the compact space of positive integer sequences of prescribed growth and hence it can also be viewed as a parametric description of a trivial analytic multifunction.
The metric space of pluriregular sets was introduced over two decades ago but to this day most of its topological properties remain a mystery. The purpose of this short survey is to present the current state of knowledge concerning this space.
With the aim of understanding the mathematical structure of the fluctuation–dissipation theorem in non-equilibrium statistical physics and then constructing a mathematical principle in the modeling problem for time series analysis, we have developed the theory of KM2O-Langevin equations for discrete time stochastic processes. In this paper, as a new method for model analysis in the theory of KM2O-Langevin equations, we show that block frames provide a natural mathematical language for dealing with minimum norm expansions of multi-dimensional stochastic processes which do not necessarily satisfy stationarity and non-degeneracy conditions.