Related papers: Dynamic Structures of 2-adic Fibonacci Polynomials
This paper investigates the dynamical properties of Dickson polynomials over finite fields, focusing on the periodicity and structural behavior of their iterated sequences. We introduce and analyze the sequence $[D_n(x, \alpha) \mod (x^q -…
In this paper we generalize to bivariate polynomials of Fibonacci and Lucas, properties obtained for Chebyshev polynomials. We prove that the coordinates of the bivariate polynomials over appropriate basis are families of integers…
In the paper we describe basin of attraction and the Siegel discs of the $p$-adic dynamical system $f(x)=x^{2n+1}+ax^{n+1}$ over complex $p$-adic field.
We discuss the construction of oscillator-like systems associated with orthogonal polynomials on the example of the Fibonacci oscillator. In addition, we consider the dimension of the corresponding lie algebras.
We begin by considering faithful matrix representations of elementary abelian groups in prime characteristic. The representations considered are seen to be determined up to change of bases by a single number. Studying this number leads to a…
We study dynamical properties of the Fibonacci trace map - a polynomial map that is related to numerous problems in geometry, algebra, analysis, mathematical physics, and number theory. Persistent homoclinic tangencies, stochastic sea of…
We show several properties related to the structure of the family of classes of two-dimensional periodic continued fractions. This approach to the study of the family of classes of nonequivalent two dimexsional periodic continued fractions…
We give a simplified presentation of some results about recurrences of certain sequences of binomial sums in terms of (generalized) Fibonacci and Lucas polynomials.
We derive some Fibonacci and Lucas identities which contain inverse binomial coefficients. Extension of the results to the general Horadam sequence is possible, in some cases.
We discuss selected topics of current research interest in the theory of dynamical systems, with emphasis on dimension theory, multifractal analysis, and quantitative recurrence. The topics include the quantitative versus the qualitative…
Recent results for rotations expressed as polynomials of spin matrices are derived here by elementary differential equation methods. Structural features of the results are then examined in the framework of biorthogonal systems, to obtain an…
We give a direct and intuitive proof (via sliding some columns up and down) of the following interesting fact: if we write out the Chebyshev polynomials in a chart and take the sums of coefficients along certain diagonals, we obtain the…
This paper continues our investigation of the dynamics of polynomial diffeomorphisms of C^2. We introduce a dynamical property of polynomial diffeomorphisms that generalizes hyperbolicity in the way that semi-hyperbolicity generalizes…
We describe results on the dynamics of polynomial diffeomorphisms of ${\bf C^2}$ and draw connections with the dynamics of polynomial maps of ${\bf C}$ and the dynamics of polynomial diffeomorphisms of ${\bf R^2}$ such as the H\'enon…
In this paper I consider all possible properties from commutative algebra for polynomial composites and monoid domains. The aim is full characterization of these structures. I start with the examination of group, ring, modules properties,…
A portrait is a combinatorial model for a discrete dynamical system on a finite set. We study the geometry of portrait moduli spaces, whose points correspond to equivalence classes of point configurations on the affine line for which there…
For each prime number $p$, the dynamical behavior of the square mapping on the ring $\mathbb{Z}_p$ of $p$-adic integers is studied. For $p=2$, there are only attracting fixed points with their attracting basins. For $p\geq 3$, there are a…
We derive some q-analogs of Euler-Cassini-type identities and of recurrence formulas for powers of Fibonacci polynomials.
We investigate algebraic and arithmetic properties of a class of sequences of sparse polynomials that have binomial coefficients both as exponents and as coefficients. In addition to divisibility and irreducibility results we also consider…
Properties of 2-adic valuation sequences for general quadratic polynomials with integer coefficients are determined directly from the coefficients. These properties include boundedness or unboundedness, periodicity, and valuations at…