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Random Mapping Statistics
 IN ADVANCES IN CRYPTOLOGY
, 1990
"... Random mappings from a finite set into itself are either a heuristic or an exact model for a variety of applications in random number generation, computational number theory, cryptography, and the analysis of algorithms at large. This paper introduces a general framework in which the analysis of ..."
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Cited by 110 (6 self)
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Random mappings from a finite set into itself are either a heuristic or an exact model for a variety of applications in random number generation, computational number theory, cryptography, and the analysis of algorithms at large. This paper introduces a general framework in which the analysis of about twenty characteristic parameters of random mappings is carried out: These parameters are studied systematically through the use of generating functions and singularity analysis. In particular, an open problem of Knuth is solved, namely that of finding the expected diameter of a random mapping. The same approach is applicable to a larger class of discrete combinatorial models and possibilities of automated analysis using symbolic manipulation systems ("computer algebra") are also briefly discussed.
Bifurcations in the space of exponential maps
 Preprint 3 (2004), Institute for Mathematical Sciences, SUNY StonyBrook. ArXiv:math.DS/0311480
"... Abstract. This article investigates the parameter space of the exponential family z ↦→ exp(z)+κ. We prove that the boundary (in C) of every hyperbolic component is a Jordan arc, as conjectured by Eremenko and Lyubich as well as Baker and Rippon, and that ∞ is not accessible through any nonhyperbolic ..."
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Cited by 11 (6 self)
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Abstract. This article investigates the parameter space of the exponential family z ↦→ exp(z)+κ. We prove that the boundary (in C) of every hyperbolic component is a Jordan arc, as conjectured by Eremenko and Lyubich as well as Baker and Rippon, and that ∞ is not accessible through any nonhyperbolic (“queer”) stable component. The main part of the argument consists of demonstrating a general “Squeezing Lemma”, which controls the structure of parameter space near infinity. We also prove a second conjecture of Eremenko
Classification of escaping exponential maps
"... Abstract. We give a complete classification of the set of parameters κ for which the singular value of Eκ: z ↦ → exp(z) + κ escapes to ∞ under iteration. In particular, we show that every pathconnected component of this set is a curve to infinity. 1. ..."
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Abstract. We give a complete classification of the set of parameters κ for which the singular value of Eκ: z ↦ → exp(z) + κ escapes to ∞ under iteration. In particular, we show that every pathconnected component of this set is a curve to infinity. 1.
Eremenko. Entire functions of slow growth whose Julia set coincides with the plane. Ergodic Theory Dynam
 Systems
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On the Convergence of Hyperbolic Components in Families of Finite Type
 MATHEMATICA GOTTINGIENSIS
, 1995
"... We study approximations of analytic families G(; \Delta) of entire functions by analytic families F n (; \Delta), for example polynomials, where 2 C . In order to control the dynamics of these functions the families are assumed to be of constant finite type. In this setting we prove the convergenc ..."
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Cited by 3 (2 self)
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We study approximations of analytic families G(; \Delta) of entire functions by analytic families F n (; \Delta), for example polynomials, where 2 C . In order to control the dynamics of these functions the families are assumed to be of constant finite type. In this setting we prove the convergence of the hyperbolic components as kernels in the sense of Carath'eodory, which is a stronger notion than the usual pointwise convergence. We give an example that in general the hyperbolic components do not converge in the Hausdorff metric as the Julia sets do.
Structural instability of exponential functions
 Department of Mathematical Sciences, University of Liverpool
, 1994
"... transactions of the american mathematical society ..."
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Growth in Complex Exponential Dynamics
 Ed.), University of West Bohemia Press, Plzen, Czech republic
, 1998
"... Both the computer drawing of the complement of the Mandelbrotlike set of a oneparameter dependent complex exponential family of maps and the computer drawing of the Julia sets of the maps of this family, grow with the maximal number of iterations we choose. Some graphic examples of this growth, ..."
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Both the computer drawing of the complement of the Mandelbrotlike set of a oneparameter dependent complex exponential family of maps and the computer drawing of the Julia sets of the maps of this family, grow with the maximal number of iterations we choose. Some graphic examples of this growth, which evoke the image of a garden, are shown here. 1. INTRODUCTION During the period 191820 the field of complex analytic dynamics showed a vigorous growth when Julia [1] and Fatou [2] became interested in the behavior of complex functions under iteration. Sometimes the results of iteration were quite tame or stable; at other times these iterations behaved in a dramatically different fashion what we now call chaotic behavior (see a historical overview of complex dynamics in [3]). In 1980, Mandelbrot [4] used computer graphics to explore complex dynamics. His discovery of the Mandelbrot set [4, 5] prompted many mathematicians as Douady and Hubbard [6], Peitgen and Richter [7], Branner...
HYPERBOLIC ENTIRE FUNCTIONS WITH BOUNDED FATOU COMPONENTS
"... Abstract. We show that an invariant Fatou component of a hyperbolic transcendental entire function is a Jordan domain (in fact, a quasidisc) if and only if it contains only finitely many critical points and no asymptotic curves. We use this theorem to prove criteria for the boundedness of Fatou com ..."
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Abstract. We show that an invariant Fatou component of a hyperbolic transcendental entire function is a Jordan domain (in fact, a quasidisc) if and only if it contains only finitely many critical points and no asymptotic curves. We use this theorem to prove criteria for the boundedness of Fatou components and local connectivity of Julia sets for hyperbolic entire functions, and give examples that demonstrate that our results are optimal. A particularly strong dichotomy is obtained in the case of a function with precisely two critical values. 1.