General concepts of graphs

A little general abstract combinatorial nonsense delivered in this note is a presentation of some old and basic concepts, central to discrete mathematics, in terms of new words. The treatment is from a structural and systematic point of view. This note consists essentially of definitions and summaries.Comment: 20 page

More simple proofs of Sharkovsky's theorem

Recently we have obtained two simple proofs of Sharkovsky's theorem, one with directed graphs [7] and the other without [8]. In this note, we present yet more simple proofs of Sharkovsky's theorem.Comment: 5 page

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

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

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

The lives of period-3 orbits for some quadratic polynomials

In this note, we consider the following two families of quadratic polynomials $S_{a,c}(x) = a - cx^2$ and $T_{a,c}(x) = a - c(1 + x^2)$ and show that their respective period-3 orbits live very different lives.Comment: 5 pages, 2 figure

An example of unbounded chaos

Let $\phi(x) = |1 - \frac 1x|$ for all $x > 0$. Then we extend $\phi(x)$ in the usual way to become a continuous map from the compact topological (but not metric) space $[0, \infty]$ onto itself which also maps the set of irrational points in $(0, \infty)$ onto itself. In this note, we show that (1) on $[0, \infty]$, $\phi(x)$ is topologically mixing, has dense irrational periodic points, and has topological entropy $\log \lambda$, where $\lambda$ is the unique positive zero of the polynomial $x^3 - 2x -1$; (2) $\phi(x)$ has bounded uncountable {\it invariant} 2-scrambled sets of irrational points in $(0, 3)$; (3) for any countably infinite set $X$ of points (rational or irrational) in $(0, \infty)$, there exists a dense unbounded uncountable {\it invariant} $\infty$-scrambled set $Y$ of irrational transitive points in $(0, \infty)$ such that, for any $x \in X$ and any $y \in Y$, we have $\limsup_{n \to \infty} |\phi^n(x) - \phi^n(y)| = \infty$ and $\liminf_{n \to \infty} |\phi^n(x) - \phi^n(y)| = 0$. This demonstrates the true nature of chaos for $\phi(x)$.Comment: 10 pages, 2 figure

A Simple Proof of Sharkovsky's Theorem Rerevisited

Based on various strategies and a new general doubling operator, we obtain several simple proofs of the celebrated Sharkovsky's cycle coexistence theorem. A simple non-directed graph proof which is especially suitable for a calculus course right after the introduction of Intermediate Value Theorem is also given (in section 3).Comment: 28 pages, 5 figures, In this revision, we replace a detailed proof of (a), (b) and (c) in section 3 and a detailed proof of Sharkovsky's theorem in section 11. arXiv admin note: substantial text overlap with arXiv:math/070359