A collection of simple proofs of Sharkovsky's theorem

Based on various strategies, we obtain several simple proofs of the celebrated Sharkovsky cycle coexistence theorem.Comment: 20 pages. New proofs involving Stefan cycles are included in sections 8 and 9. New proofs of (a) and (c) are included in sections 11 and 1

What make them all so turbulent

We give a unified proof of the existence of turbulence for some classes of continuous interval maps which include, among other things, maps with periodic points of odd periods > 1, some maps with dense chain recurrent points and densely chaotic maps.Comment: 5 pages, 2 figure

On the number of parameters cc for which the point x=0x=0 is a superstable periodic point of fc(x)=1−cx2f_c(x) = 1 - cx^2

Let $f_c(x) = 1 - cx^2$ be a one-parameter family of real continuous maps with parameter $c \ge 0$. For every positive integer $n$, let $N_n$ denote the number of parameters $c$ such that the point $x = 0$ is a (superstable) periodic point of $f_c(x)$ whose least period divides $n$ (in particular, $f_c^n(0) = 0$). In this note, we find a recursive way to depict how {\it some} of these parameters $c$ appear in the interval $[0, 2]$ and show that $\liminf_{n \to \infty} (\log N_n)/n \ge \log 2$ and this result is generalized to a class of one-parameter families of continuous real-valued maps that includes the family $f_c(x) = 1 - cx^2$.Comment: 7 pages, 1 figur

On the Class of Similar Square {-1,0,1}-Matrices Arising from Vertex maps on Trees

Let $n \ge 2$ be an integer. In this note, we show that the {\it oriented} transition matrices over the field $\mathcal R$ of all real numbers (over the finite field $\mathcal Z_2$ of two elements respectively) of all continuous {\it vertex maps} on {\it all} oriented trees with $n+1$ vertices are similar to one another over $\mathcal R$ (over $\mathcal Z_2$ respectively) and have characteristic polynomial $\sum_{k=0}^n x^k$. Consequently, the {\it unoriented} transition matrices over the field $Z_2$ of all continuous {\it vertex maps} on {\it all} oriented trees with $n+1$ vertices are similar to one another over $\mathcal Z_2$ and have characteristic polynomial $\sum_{k=0}^n x^k$. Therefore, the coefficients of the characteristic polynomials of these {\it unoriented} transition matrices, when considered over the field $\mathcal R$, are all odd integers (and hence nonzero).Comment: 13 pages, 4 figure

On the Invariance of Li-Yorke Chaos of Interval Maps

In their celebrated "Period three implies chaos" paper, Li and Yorke proved that if a continuous interval map f has a period 3 point then there is an uncountable scrambled set S on which f has very complicated dynamics. One question arises naturally: Can this set S be chosen invariant under f? The answer is positive for turbulent maps and negative otherwise. In this note, we shall use symbolic dynamics to achieve our goal. In particular, we obtain that the tent map T(x) = 1 - |2x-1| on [0, 1] has a dense uncountable invariant 1-scrambled set which consists of transitive points.Comment: 6 page

Obtaining New Dividing Formulas n|Q(n) From the Known Ones

In this note, we present a few methods (Theorems 1, 2, and 3) from discrete dynamical systems theory of obtaining new functions Q(n) from the known ones so that the dividing formulas n|Q(n) hold.Comment: 7 page

A Simple Proof of Sharkovsky's Theorem

In this note, we present a simple directed graph proof of Sharkovsky's theorem.Comment: 5 page

A Simple Proof of Sharkovsky's Theorem Revisited

In this note, we present a simple non-directed graph proof of Sharkovsky's theorem which is different from the one given in [2].Comment: 4 pages, 1 figur

Congruence Identities Arising From Dynamical Systems

By counting the numbers of periodic points of all periods for some interval maps, we obtain infinitely many new congruence identities in number theory.Comment: 5 page

The Minimal Number of Periodic Orbits of Periods Guaranteed in Sharkovskii's Theorem

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 Sharkovsky's ordering defined below, a lower bound on the number of periodic orbits of f(x) with minimal period n is 1. Could we improve this lower bound from 1 to some larger number? In this paper, we give a complete answer to this question.Comment: 11 page