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

We propose an extension of the one dimensional (doubling) renormalization operator to the case of maps on the cylinder. The kind of maps considered are commonly referred as quasi-periodic forced one dimensional maps. We prove that the fixed point of the one dimensional renormalization operator extends to a fixed point of the quasi-periodic forced renormalization operator. We also prove that the operator is differentiable around the fixed point and we study its derivative. The...

In this article we study the local structure of the Fibonacci Partition Function by relating it to a cocycle over an irrational rotation.

We collect experimental evidence for several propositions, including the following: (1) For each Riemann zero $\rho$ (trivial or nontrivial) and each zeta fixed point $\psi$ there is a nearly logarithmic spiral $s_{\rho, \psi}$ with center $\psi$ containing $\rho$. (2) $s_{\rho, \psi}$ interpolates a subset $B_{\rho, \psi}$ of the backward zeta orbit of $\rho$ comprising a set of zeros of all iterates of zeta. (3) If zeta is viewed as a function on sets, $\zeta(B_{\rho, \psi}...

We show that the full features of the dynamics towards the Feigenbaum attractor, present in all low-dimensional maps with a unimodal leading component, form a hierarchical construction with modular organization that leads to a clear-cut emergent property. This well-known nonlinear model system combines a simple and precise definition, an intricate nested hierarchical dynamical structure, and emergence of a power-law dynamical property absent in the exponential-law that govern...

We investigate the arithmetic properties of the coefficients for the normalized conformal mapping of the exterior of the Multibrot set. In this paper, an estimate of the prime factor of the denominator of these coefficients is given. Bielefeld, Fisher and Haeseler presented Zagier's observation for the growth of denominator of the coefficients for the Mandelbrot set and Yamashita verified it in his master thesis. Our study takes into consideration both Zagier's observation an...

In this paper we are concerned with quasi-periodic forced one dimensional maps. We consider a two parametric family of quasi-periodically forced maps such that the one dimensional map (before forcing) is unimodal and it has a full cascade of period doubling bifurcations. Between one period doubling and the next one it is known that there exist a parameter value where the $2^n$-periodic orbit is superatracting. In a previous work we proposed an extension of the one-dimensional...

In this paper we describe the dynamics of certain rational maps of the form $k \cdot (x+x^{-1})$ over finite fields of odd characteristic.

We demonstrate that the dynamics towards and within the Feigenbaum attractor combine to form a q-deformed statistical-mechanical construction. The rate at which ensemble trajectories converge to the attractor (and to the repellor) is described by a q-entropy obtained from a partition function generated by summing distances between neighboring positions of the attractor. The values of the q-indices involved are given by the unimodal map universal constants, while the thermodyn...

Many interesting questions in arithmetic dynamics revolve, in one way or another, around the (local and/or global) reducibility behavior of iterates of a polynomial. We show that for very general families of integer polynomials $f$ (and, more generally, rational functions over number fields), the set of stable primes, i.e., primes modulo which all iterates of $f$ are irreducible, is a density zero set. Compared to previous results, our families cover a much wider ground, and ...

This review article brings forth some recent results in the theory of the Riemann zeta-function $qzeta(s)$.