相关论文: A dynamical property unique to the Lucas sequence

For a continuous map on a topological graph containing a unique loop S it is possible to define the degree and, for a map of degree 1, rotation numbers. It is known that the set of rotation numbers of points in S is a compact interval and…

We study absolutely periodic points and trajectories of Hamiltonian systems. Our main result is a necessary and sufficient for a Hamiltonian system to have the following property: if there exists one absolutely periodic trajectory then all…

The classical Gauss Map is a piecewise continuous map from the unit interval to itself. From this map we retrieve the continued fraction expansion of irrational numbers and its dynamical properties give information about some arithmetic and…

We discuss in detail the dynamics of maps $z\mapsto ae^z+be^{-z}$ for which both critical orbits are strictly preperiodic. The points which converge to $\infty$ under iteration contain a set $R$ consisting of uncountably many curves called…

This paper describes new, simple, recursive methods of construction for orientable sequences, i.e. periodic binary sequences in which any n-tuple occurs at most once in a period in either direction. As has been previously described, such…

Let $(F_n)_{n\ge0}$ and $(L_n)_{n\ge0}$ denote the sequences of Fibonacci and Lucas numbers respectively. This paper determines all Lucas numbers that can be represented as base $b$ mixed concatenations of a Fibonacci number and a Lucas…

We consider real sequences $(f_n)$ that satisfy a linear recurrence with constant coefficients. We show that the density of the positivity set of such a sequence always exists. In the special case where the sequence has no positive…

A continuous one-dimensional map with period three includes all periods. This raises the following question: Can we obtain any types of periodic orbits solely by learning three data points? In this paper, we report the answer to be yes.…

This article presents the exact solution of fixed points functions for the cycle of period four of the quadratic recurrence equations. The solution is demonstrated for the quadratic map and the logistic map. These recurrence equations,…

Let f be a transcendental map, and let U be an attracting or parabolic basin, or a doubly parabolic Baker domain. Assume U is simply connected. Then, we prove that periodic points are dense in the boundary of U, under certain hypothesis on…

The goal of this paper is twofold: (1) extend theory on certain statistics in the Fibonacci and Lucas sequences modulo $m$ to the Lucas sequences $U := \left(U_n(p,q)\right)_{n \geq 0}$ and $V := \left(V_n(p,q)\right)_{n \geq 0}$, and (2)…

This paper uses the framework of reverse mathematics to investigate the strength of two recurrence theorems of topological dynamics. It establishes that one of these theorems, the existence of an almost periodic point, lies strictly between…

Given a quadratic polynomial with rational coefficients, we investigate the existence of consecutive squares in the orbit of a rational point under the iteration of the polynomial. We display three different constructions of $1$-parameter…

The famous Bernoulli shift (or dyadic transformation) is perhaps the simplest deterministic dynamical system exhibiting chaotic dynamics. It is a piecewise linear time-discrete map on the unit interval with a uniform slope larger than one,…

We study for a dynamical system $f:X\longrightarrow X$ some of the principal topological recurrence-kind properties with respect to the induced maps $\overline{f}:\mathcal{K}(X)\longrightarrow\mathcal{K}(X)$, on the hyperspace of non-empty…

This paper first discusses the size and orientation of hat supertiles. Fibonacci and Lucas sequences, as well as a third integer sequence linearly related to the Lucas sequence are involved. The result is then generalized to any aperiodic…

Given an algebraically closed field $K$, a dynamical sequence over $K$ is a $K$-valued sequence of the form $a(n):= f(\phi^n(x_0))$, where $\phi\colon X\to X$ and $f\colon X\to\mathbb{A}^1$ are rational maps defined over $K$, and $x_0\in X$…

We consider dynamical systems given by interval maps with a finite number of turning points (including critical points, discontinuities) possibly of different critical orders from two sides. If such a map $f$ is continuous and piecewise…

Solutions to the random Fibonacci recurrence x_{n+1}=x_{n} + or - Bx_{n-1} decrease (increase) exponentially, x_{n} = exp(lambda n), for sufficiently small (large) B. In the limits B --> 0 and B --> infinity, we expand the Lyapunov exponent…

Let f be an orientation-preserving homeomorphism of the plane such that f-Id is contracting. Under these hypotheses, we establish the existence, for every periodic orbit, of a fixed point which has nonzero linking number with this periodic…