Related papers: Square patterns in dynamical orbits

In this paper, we classify, up to three possible exceptions, all monic, post-critically finite quadratic polynomials $f(x)\in \mathbb{Z}[x]$ with an iterate reducible module every prime, but all of whose iterates are irreducible over…

We study a random polynomial of degree $n$ over the finite field $\mathbb{F}_q$, where the coefficients are independent and identically distributed and uniformly chosen from the squares in $\mathbb{F}_q$. Our main result demonstrates that…

We give necessary and sufficient conditions for a sequence to be exactly realizable as the sequence of numbers of periodic points in a dynamical system. Using these conditions, we show that no non-constant polynomial is realizable, and give…

We evaluate the variance of coefficients of the characteristic polynomial for binary quantum graphs using a dynamical approach. This is the first example where a spectral statistic can be evaluated in terms of periodic orbits for a system…

Let k be an algebraically closed field of characteristic 0, let X=P^1\times A^N and let f be a rational endomorphism of X given by (x,y)--->(g(x), A(x)y), where g is a rational function, while A is an N-by-N matrix with entries in k(x). We…

Let $f_1,\dots,f_m$ be polynomials in $n$ variables with coefficients in a finite field $\mathbb{F}_q$. We estimate the number of points $\underline{x}$ in $\mathbb{F}_q^n$ such that each value $f_i(\underline{x})$ is a nonzero square in…

Let $q$ be a prime power and $\phi$ a rational function with coefficients in a finite field $\mathbb{F}_q$. For $n \geq 1$, each element of $\mathbb{P}^1(\F_{q^n})$ is either periodic or strictly preperiodic under iteration of $\phi$.…

Given a prime $p\ge5$ and an integer $s\ge1$, we show that there exists an integer $M$ such that for any quadratic polynomial $f$ with coefficients in the ring of integers modulo $p^s$, such that $f$ is not a square, if a sequence…

Let $f:(\mathbb{C}^n,0)\mapsto(\mathbb{C}^n,0)$ be a germ of an $n$-dimensional holomorphic map. Assume that the origin is an isolated fixed point of each iterate of $f$. Then $\{\mathcal{N}_q(f)\}_{q=1}^{\infty}$, the sequence of the…

Mutually orthogonal frequency squares (MOFS) of type $F(m\lambda;\lambda)$ generalize the structure of mutually orthogonal Latin squares: rather than each of $m$ symbols appearing exactly once in each row and in each column of each square,…

We prove that for a generic family of circle diffeomorphisms every parameter value that corresponds to an irrational rotation number is approximated by parameter values for which the diffeomorphisms have arbitrarily large finite numbers of…

Classes of polynomial differential equations of degree n are considered. An explicit upper bound on the size of the coefficients are given which implies that each equation in the class has exactly n complex periodic solutions. In most of…

It is often assumed that a warped galaxy can be modeled by a set of rings. This paper verifies numerically the validity of this assumption by the study of periodic orbits populating a heavy self-gravitating warped disk. The phase space…

Let $f(x) \in \mathbb{F}_p[x]$, and define the orbit of $x\in \mathbb{F}_p$ under the iteration of $f$ to be the set \[ \mathcal{O}(x):=\{x,f(x),(f\circ f)(x),(f\circ f\circ f)(x),\dots\}. \] An orbit is a $k$-cycle if it is periodic of…

Let f(x) be a continuous function from a compact real interval into itself with a periodic orbit of minimal period m, where m is not an integral power of 2. Then, by Sharkovsky's theorem, for every positive integer n with m \prec n in the…

A Latin square of order $n$ is an $n \times n$ matrix of $n$ symbols, such that each symbol occurs exactly once in each row and column. For an odd prime power $q$ let $\mathbb{F}_q$ denote the finite field of order $q$. A quadratic Latin…

For an $S$-valued function $f$ of $m \geq 1$ variables we consider the dynamical process in which the output $f(\overline{v})$ replaces exactly one entry of the input $\overline{v} \in S^m$ at each step. This can be viewed as a special case…

The surface corresponding to the moduli space of quadratic endomorphisms of $\mathbb{P}^1$ with a marked periodic point of order $n$ is studied. It is shown that the surface is rational over $\mathbb{Q}$ when $n\le 5$ and is of general type…

We construct a countable family of multi-dimensional continued fraction algorithms, built out of five specific multidimensional continued fractions, and find a wide class of cubic irrational real numbers a so that either (a, a^2) or (a,…

In this paper, we prove existence and orbital stability results of periodic standing waves for the cubic-quintic nonlinear Schr\"odinger equation. We use the implicit function theorem to construct a smooth curve of explicit periodic waves…