Related papers: Around the boundary of complex dynamics
Analogy between an approximate version of Feigenbaum renormalization group analysis in complex domain and the phase transition theory of Yang-Lee (based on consideration of formally complexified thermodynamic values) is discussed. It is…
The effects of a boundary on the circuit complexity are studied in two dimensional theories. The analysis is performed in the holographic realization of a conformal field theory with a boundary by employing different proposals for the dual…
This text reviews certain notions in metric geometry that may have further applications to problems in complex geometry and holomorphic dynamics in several variables. The discussion contains a few unrecorded results and formulates a number…
A stochastic dynamics framework for the study of complex systems is presented.
Complex dynamics deals with the iteration of holomorphic functions. As is well- known, the first functions to be studied which gave non-trivial dynamics were quadratic polynomials, which produced beautiful computer generated pictures of…
We give a concrete description for the boundary of the central quadratic hyperbolic component. The connectedness of the Julia sets of the boundary maps are also considered.
These notes are the outgrowth of a series of lectures given at MSRI in January 1995 at the beginning of the special semester in complex dynamics and hyperbolic geometry. In these notes, the primary aim is to motivate the study of complex…
Hyperbolic numbers are a variation of complex numbers, but their dynamics is quite different. The hyperbolic Mandelbrot set for quadratic functions over hyperbolic numbers is simply a filled square, and the filled Julia set for hyperbolic…
First, for the family P_{n,c}(z) = z^n + c, we show that the geometric limit of the Mandelbrot sets M_n(P) as n tends to infinity exists and is the closed unit disk, and that the geometric limit of the Julia sets J(P_{n,c}) as n tends to…
Bieberbach constructed in 1933 domains in $ \bf {C}^2$ which were biholomorphic to $ \bf {C}^2$ but not dense. The existence of such domains was unexpected. The special domains Bieberbach considered are basins of attraction of a cubic…
This article summarises a Web-book on "Complexity" that was developed to introduce undergraduate students to interesting complex systems in the biological, physical and social sciences, and the common tools, principles and concepts used for…
Let $ E $ be a non-empty compact subset of the Riemann sphere and $T$ be a rational map of degree at least two. We study the associated \emph{orbital set}, that is, the backwards orbit of $E$ under $T$, and study the relationship between…
It is shown that the boundary of the Mandelbrot set $M$ has Hausdorff dimension two and that for a generic $c \in \bM$, the Julia set of $z \mapsto z^2+c$ also has Hausdorff dimension two. The proof is based on the study of the bifurcation…
We consider limit sets of some conformal iterated function systems, and introduce classes of subsets of the limit set, with the property that the classes are closed under countable intersections and all sets in the classes have large…
We study a linear problem that arises in the study of dynamic boundaries, in particular in free boundary problems in connection with fluid dynamics. The equations are also very natural and of interest on their own.
We relate periodic and recurrent points in dendritic Julia sets. This generalizes well-known results for interval dynamics.
We study rational functions satisfying summability conditions - a family of weak conditions on the expansion along the critical orbits. Assuming their appropriate versions, we derive many nice properties: There exists a unique, ergodic, and…
We treat the boundary problem for complex varieties (with isolated singularities) of dimension greater than one, which are contained in a suitable class of strictly pseudoconvex, unbounded domains of C^n.
This article concerns a new geometric characterization of the Julia set. By using Ahlfors-Shimizu's characteristic, we establish some growth results which indicates the characterization of the Julia set. The main technique is to estimate…
This paper focuses on polynomial dynamical systems over finite fields. These systems appear in a variety of contexts, in computer science, engineering, and computational biology, for instance as models of intracellular biochemical networks.…