Related papers: A dynamical property unique to the Lucas sequence
In some particular cases we give criteria for morphic sequences to be almost periodic (=uniformly recurrent). Namely, we deal with fixed points of non-erasing morphisms and with automatic sequences. In both cases a polynomial-time algorithm…
The theory of polynomial-like maps is of fundamental importance in holomorphic dynamics. We study dynamical properties of a larger class of maps. Our main result is that, under some natural conditions, a map of this class has a completely…
In this paper, plane polynomial systems having a singular point attracting all orbits in positive time are classified up to topological equivalence. This is done by assigning a combinatorial invariant to the system (a so-called "feasible…
Let ${\cal P}$ be the set of palindromes occurring in the Fibonacci sequence. In this note, we establish three structures of $\mathcal{P}$ and and discuss their properties: cylinder structure, chain structure and recursive structure. Using…
A simple method called symbolic representation for piecewise linear functions on the real line is introduced and used to compute the numbers of periodic points of all periods for some such functions. Since, for every positive integer m, the…
The degree sequence of the algebraic numbers in an algebraic linear recurrence sequence is shown to be virtually periodic. This is proved using the Skolem-Mahler-Lech theorem. It has applications to the degree sequence and the minimal…
We consider the Lorenz equations, a system of three dimensional ordinary differential equations modeling atmospheric convection. These equations are chaotic and hard to study even numerically, and so a simpler "geometric model" has been…
By counting the numbers of periodic points of all periods for some interval maps, we obtain infinitely many new congruence identities in number theory.
An existing dialogue between number theory and dynamical systems is advanced. A combinatorial device gives necessary and sufficient conditions for a sequence of non-negative integers to count the periodic points in a dynamical system. This…
We introduce the notion of almost realizability, an arithmetic generalization of realizability for integer sequences, which is the property of counting periodic points for some map. We characterize the intersection between the set of…
We prove an infinite family of lacunary recurrences for the Lucas numbers using combinatorial means.
We develop a new approach to recurrence and the existence of non-constant harmonic functions on infinite weighted graphs. The approach is based on the capacity of subsets of metric boundaries with respect to intrinsic metrics. The main tool…
We study Gibonacci sequences mod $m$, giving special attention to the Lucas numbers. It is known which $m$ have the property that the Fibonacci sequence contains all residues mod $m$. When $m$ has this property, we say that the Fibonacci…
A composition of birational maps given by Laurent polynomials need not be given by Laurent polynomials; however, sometimes---quite unexpectedly---it does. We suggest a unified treatment of this phenomenon, which covers a large class of…
We prove that every Peano continuum (a space that is a continuous image of $[0,1]$) admits a topologically mixing but not exact map. The constructed map has a dense set of periodic points.
It will be shown that the renormalization operator, acting on the space of smooth unimodal maps with critical exponent greater than 1, has periodic points of any combinatorial type.
We prove a special case of a dynamical analogue of the classical Mordell-Lang conjecture. In particular, let $\phi$ be a rational function with no superattracting periodic points other than exceptional points. If the coefficients of $\phi$…
We propose an evolutionary mechanism of phyllotaxis, regular arrangement of leaves on a plant stem. It is shown that the phyllotactic pattern with the Fibonacci sequence has a selective advantage, for it involves the least number of…
In this paper the existence and unicity of a stable periodic orbit is proven, for a class of piecewise affine differential equations in dimension 3 or more, provided their interaction structure is a negative feedback loop. It is also shown…
We study the uniqueness properties of classical semiconjugations for analytic self-maps of the disk, by characterizing them as canonical solutions to certain functional equations. As a corollary, we obtain a complete description of all…