Discrete Dynamical Systems Embedded in Cantor Sets

While the notion of chaos is well established for dynamical systems on manifolds, it is not so for dynamical systems over discrete spaces with $N$ variables, as binary neural networks and cellular automata. The main difficulty is the choice of a suitable topology to study the limit $N\to\infty$. By embedding the discrete phase space into a Cantor set we provided a natural setting to define topological entropy and Lyapunov exponents through the concept of error-profile. We made explicit calculations both numerical and analytic for well known discrete dynamical models.Comment: 36 pages, 13 figures: minor text amendments in places, time running top to bottom in figures, to appear in J. Math. Phy

Equidistribution Rates, Closed String Amplitudes, and the Riemann Hypothesis

We study asymptotic relations connecting unipotent averages of $Sp(2g,\mathbb{Z})$ automorphic forms to their integrals over the moduli space of principally polarized abelian varieties. We obtain reformulations of the Riemann hypothesis as a class of problems concerning the computation of the equidistribution convergence rate in those asymptotic relations. We discuss applications of our results to closed string amplitudes. Remarkably, the Riemann hypothesis can be rephrased in terms of ultraviolet relations occurring in perturbative closed string theory.Comment: 15 page

ERRATA: "GEOMETRIC GALOIS THEORY, NONLINEAR NUMBER FIELDS AND A GALOIS GROUP INTERPRETATION OF THE IDELE CLASS GROUP"

In this errata we correct several mistakes Secs. 1 and 3–6 that appear in the published version of our paper. The corrections have been implemented in the revised version [1]. In addition, in Sec. 2 we clarify an important point which was not adequately addressed in the published version; in Sec. 7 we point out an enhancement of the hyperbolization scheme included in [1]. The reader may also wish to consult [2]. </jats:p

QUATERNION DYNAMICS AND FRACTALS IN ℝ⁴

In this paper we study the Fatou–Julia theory for some quaternionic rational functions in the quaternion skew-field ℍ. We obtain new dynamically-defined fractals in ℝ4 as the corresponding Julia sets. We also define the quaternionic Mandelbrot set. </jats:p