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].
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arXiv subjects
Bau-Sen Du
Publications and source records attributed to Bau-Sen Du.
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.
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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.
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On the nature of chaos
Based on newly discovered properties of the shift map (Theorem 1), we believe that chaos should involve not only nearby points can diverge apart but also faraway points can get close to each other. Therefore, we propose to call a continuous map $f$ from an infinite compact metric space $(X, d)$ to itself chaotic if there exists a positive number $δ$ such that for any point $x$ and any nonempty open set $V$ (not necessarily an open neighborhood of $x$) in $X$ there is a point $y$ in $V$ such that $\limsup_{n \to \infty} d(f^n(x), f^n(y)) \ge δ$ and $\liminf_{n \to \infty} d(f^n(x), f^n(y)) = 0$.
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