On a set of numbers arising in the dynamics of unimodal maps

This article deals with the question of local connectivity of the Julia set of polynomials and rational maps. It essentially presents conjectures and questions.

We study the values taken by the Riemann zeta-function $\zeta$ on discrete sets. We show that infinite vertical arithmetic progressions are uniquely determined by the values of $\zeta$ taken on this set. Moreover, we prove a joint discrete universality theorem for $\zeta$ with respect to certain permutations of the set of positive integers. Finally, we study a generalization of the classical denseness theorems for $\zeta$.

This short note presents a peculiar generalization of the Riemann hypothesis, as the action of the permutation group on the elements of continued fractions. The problem is difficult to attack through traditional analytic techniques, and thus this note focuses on providing a numerical survey. These results indicate a broad class of previously unexamined functions may obey the Riemann hypothesis in general, and even share the non-trivial zeros in particular.

We give an historical account, including recent progress, on some problems of Erd\H os in number theory.

We describe in this survey several results relating Fractal Geometry, Dynamical Systems and Diophantine Approximations, including a description of recent results related to geometrical properties of the classical Markov and Lagrange spectra and generalizations in Dynamical Systems and Differential Geometry.

We propose a set of questions on the dynamics of H\'enon maps from the real, complex, algebraic and arithmetic points of view.

We discuss analogues of the prime number theorem for a hyperbolic rational map f of degree at least two on the Riemann sphere. More precisely, we provide counting estimates for the number of primitive periodic orbits of f ordered by their multiplier, and also obtain equidistribution of the associated holonomies; both estimates have power savings error terms. Our counting and equidistribution results will follow from a study of dynamical zeta functions that have been twisted b...

We argue that discrete dynamics has natural links to the theory of analytic functions. Most important, bifurcations and chaotic dynamical properties are related to intersections of algebraic varieties. This paves the way to identification of boundaries of Mandelbrot sets with discriminant varieties in moduli spaces, which are the central objects in the worlds of chaos and order (integrability) respectively. To understand and exploit this relation one needs first to develop th...

We straighten a result of [5] about arithmetic properties of the Laurent coefficients of the conformal isomorphism from the complement of the unit disk onto the complement of the Mandelbrot set. This confirms an empirical observation by Don Zagier, see [1]

In this paper we investigate the primeness of a class of entire functions and discuss the dynamics of a periodic member f of this class with respect to a transcendental entire function g that permutes with f. In particular we show that the Julia sets of f and g are identical.