A kind of dynamic model of prime numbers

Reductions Modulo Primes of Systems of Polynomial Equations and Algebraic Dynamical Systems

We give bounds for the number and the size of the primes $p$ such that a reduction modulo $p$ of a system of multivariate polynomials over the integers with a finite number $T$ of complex zeros, does not have exactly $T$ zeros over the algebraic closure of the field with $p$ elements. We apply these bounds to study the periodic points and the intersection of orbits of algebraic dynamical systems over finite fields. In particular, we establish some links between these problems and the uniform dynamical Mordell-Lang conjecture.

On a dynamical approach to some prime number sequences

In this paper we show how the cross-disciplinary transfer of techniques from Dynamical Systems Theory to Number Theory can be a fruitful avenue for research. We illustrate this idea by exploring from a nonlinear and symbolic dynamics viewpoint certain patterns emerging in some residue sequences generated from the prime number sequence. We show that the sequence formed by the residues of the primes modulo $k$ are maximally chaotic and, while lacking forbidden patterns, display a non-trivial spectrum of Renyi entropies which suggest that every block of size $m>1$, while admissible, occurs with d…

Primes, knots and periodic orbits

This is a text written for the Ennio De Giorgi Colloquio volume. It covers analogies between algebraic number theory and knot theory, analogies between analytic number theory and certain dynamical systems, and a report on our construction of dynamical systems for arithmetic schemes which realize some of these analogies.

Vojta's conjecture, heights associated with subschemes, and primitive prime divisors in arithmetic dynamics

Assuming Vojta's conjecture, we give a sufficient condition for the limit \[ \lim_{n \to \infty} \frac{h_{Y}(f^{n}(x))}{h_{H}(f^{n}(x))} \] is equal to zero, where $f \colon X \longrightarrow X$ is a surjective self-morphism on a smooth projective variety $X$, $h_{H}$ is an ample height function on $X$, and $h_{Y}$ is a global height function associated with a closed subscheme $Y \subset X$ of codimension at least two. Based on this, we propose a conjecture on a sufficient condition for the limit to be zero. We point out that our conjecture implies Dynamical Mordell-Lang conjecture for endomor…

On the Number of Periodic Points of Quadratic Dynamical Systems Modulo a Prime

In 2004 Vasiga and Shallit studied the number of periodic points of two particular discrete quadratic maps modulo prime numbers. They found the asymptotic behaviour of the sum of the number of periodic points for all primes less than some bound, assuming the Extended Riemann Hypothesis. Later that same year Chou and Shparlinski proved this asymptotic result without assuming any unproven hypotheses. Inspired by this we perform experiments and find a striking pattern in the behaviour of the sum of the number of periodic points for quadratic maps other than the two particular ones studied previou…

Prime Ends Dynamics in Parametrised Families of Rotational Attractors

We provide several new examples in dynamics on the $2$-sphere, with the emphasis on better understanding the induced boundary dynamics of invariant domains in parametrized families. First, motivated by a topological version of the Poincaré-Bendixson Theorem obtained recently by Koropecki and Passeggi, we show the existence of homeomorphisms of $\mathbb{S}^2$ with Lakes of Wada rotational attractors, with an arbitrarily large number of complementary domains, and with or without fixed points, that are arbitrarily close to the identity. This answers a question of Le Roux. Second, from reduced Arn…

Mean Li-Yorke chaos along polynomials of several variables and prime numbers

In this paper, for any given polynomial, by analyzing the limiting behavior of ergodic averages along polynomials of several variables and prime numbers, we prove that for a topology dynamical system, positive entropy implies mean Li-Yoke chaos along non-constant polynomials of several variables and prime numbers.

Prime orbit theorems for expanding Thurston maps

We obtain an analogue of the prime number theorem for a class of branched covering maps on the $2$-sphere called expanding Thurston maps $f$, which are topological models of some rational maps without any smoothness or holomorphicity assumption. More precisely, by studying dynamical zeta functions and, more generally, dynamical Dirichlet series for $f$, we show that the number of primitive periodic orbits of $f$, ordered by a weight on each point induced by a non-constant (eventually) positive real-valued Hölder continuous function $ϕ$ on $S^2$ satisfying some additional regularity conditions,…

幂零 Cantor 作用的素谱与动力学

Prime spectrum and dynamics for nilpotent Cantor actions

本文研究了幂零 Cantor 作用的稳定性，证明了具有有限素谱的作用必稳定，并构造了不可数个具有无限素谱且不稳定的 Heisenberg 群作用。