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Bau-Sen Du

Publications and source records attributed to Bau-Sen Du.

At least 19 recordsLinked to original sources
arXiv2020

On the Chaos in Continuous Weakly Mixing Maps

Let $\mathcal X$ be an infinite locally compact separable metric space with metric $ρ$ and let $f : \mathcal X \longrightarrow \mathcal X$ be a continuous weakly mixing map. Let $β= \sup \big\{ ρ(x, y): \{x, y \} \subset \mathcal X \big\}$. In this note, we show (Theorem 4) that, for any countably infinite set $\{x_1, x_2, \cdots\}$ of points in $\mathcal X$ with compact orbit closures $\overline{O_f(x_i)}$'s, there exist an infinite set $\mathcal M$ of positive integers and countably infinitely many pairwise disjoint Cantor sets ${\mathcal S}^{(1)}, {\mathcal S}^{(2)}, \cdots$ of totally transitive points of $f$ such that (1) for any integers $\ell \ge 1$ and $n \ge 1$, $\ell!$ divides all sufficiently large integers in $\mathcal M$ and for any distinct points $a_1, a_2, \cdots, a_n$ in ${\mathbb S} = \bigcup_{j=1}^\infty \, {\mathcal S}^{(j)}$, the set $\{ F_n^m\big((a_1, a_2, \cdots, a_n)\big): m \in \mathcal M \}$ is dense in $\mathcal X \times \mathcal X \times \cdots \times \mathcal X$ ($n$ terms), where $F_n\big((a_1, a_2, \cdots, a_n)\big) = \big(f(a_1), f(a_2), \cdots, f(a_n)\big)$; (2) ${\mathbb S}$ is a dense $β$-scrambled set of $f^n$ for all $n \ge 1$; (3) for any $x$ in $\{x_1, x_2, \cdots\}$ and any $c$ in $\widehat {\mathbb S} = \bigcup_{i=0}^\infty \, f^i({\mathbb S})$, $\{ x, c \}$ is a ($β/2$)-scrambled set of $f$. Furthermore, if $f$ has a fixed point and $δ= \inf_{n \ge 1} \big\{ \sup\{ ρ(f^n(x), x): x \in \mathcal X \} \big\} \ge 0$, then the above Cantor sets ${\mathcal S}^{(1)}, {\mathcal S}^{(2)}, \cdots$ can be chosen to satisfy the additional property that $\widehat {\mathbb S} = \bigcup_{i=0}^\infty f^i({\mathbb S})$ is a dense {\it invariant} $δ$-scrambled set of $f^n$ for all $n \ge 1$. For continuous mixing maps on $\mathcal X$, we have a stronger result (Theorem 5). A notion of chaos is also introduced.

math.DS↗
arXiv2019

Continuous Transitive Maps on the Interval Revisited

In this note, continuous transitive maps $f$ on the interval $I$ are re-addressed, where $I$ denotes one of the intervals: $(-\infty, \infty)$, $(-\infty, a]$, $[b, \infty)$, $[a, b]$, where $a < b$ are real numbers. Such maps must have a fixed point, say $z$, in the interior of $I$. Some well-known properties of such maps are re-proved in a systematic way according to the following : (1) $f$ moves some point $c \ne z$ away from $z$, i.e., fo some point $c \ne z$, we have $f(c) \le c < z$ or $z < c \le f(c)$; (2) $f$ moves some point $\hat c \ne z$ towards but not "over" $z$, i.e., for some point $\hat c \ne z$, we have $\hat c < f(\hat c) < z$ or $z < f(\hat c) < \hat c$; and (3) $f$ moves all points $x \ne z$ to the other side of $z$, i.e., for all points $x \ne z$, we have $x < z \le f(x)$ and $f(x) \le x < z$. The proofs are arranged in such ways that they yield the same results. For example, Theorem 3 treats maps satisfying Condition (1) or Condition (2) while Theorem 4 treats separately maps satisfying Condition (1) and, Conditions (2) or (3). Characterizations of continuous bitransitive maps on an interval are re-addressed and a new chaotic property of continuous bitransitive maps is also introduced (Theorem 8, p.21). In this revision, we correct Corollary 9 and some errors in the proof of Theorem 8 and move the Appendix to a different paper [15] in which we generalize Theorem 8 for continuous weakly mixing (i.e., bitransitive) maps on intervals to continuous weakly mixing maps and continuous mixing maps on infinite separable locally compact metric spaces.

math.DS↗
arXiv2017

An Interesting Application of the Intermediate Value Theorem: A Simple Proof of Sharkovsky's Theorem and the Towers of Periodic Points

This note is intended primarily for college calculus students right after the introduction of the Intermediate Value Theorem, to show them how the Intermediate Value Theorem is used repeatedly and straightforwardly to prove the celebrated Sharkovsky's theorem on the periods of coexistent periodic orbits of continuous maps on an interval. Furthermore, if the maps have a periodic orbit P of odd period > 1, then we find more periodic points, in the appendix, which constitute what we call the towers of periodic points associated with P. These towers of periodic points have infinitely many layers. In this note, no knowledge of Dynamical Systems Theory is required. In this revision, we add the result (Proposition 1 on page 9) that the periodic orbits of continuous unimodal maps on the interval $[0, 1]$ of least periods $\ge 2$ are nested in the sense that if $P$ and $Q$ are periodic orbits of a continuous unimodal map on $[0, 1]$ of least periods $\ge 2$ and if $\max P < \max Q$, then $[\min P, \max P] \subset [\min Q, \max Q]$.

math.HO↗
arXiv2015

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).

math.DS↗
arXiv2015

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).

math.DS↗
arXiv2014

On the number of parameters $c$ for which the point $x=0$ is a superstable periodic point of $f_c(x) = 1 - cx^2$

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 $f_c(x) = 1 - cx^2$.

math.DS↗
arXiv2012

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.

math.DS↗
arXiv2011

An example of unbounded chaos

Let $ϕ(x) = |1 - \frac 1x|$ for all $x > 0$. Then we extend $ϕ(x)$ 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 $[0, \infty]$, $ϕ(x)$ is topologically mixing, has dense irrational periodic points, and has topological entropy $\log λ$, where $λ$ is the unique positive zero of the polynomial $x^3 - 2x -1$; (2) $ϕ(x)$ 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 $\limsup_{n \to \infty} |ϕ^n(x) - ϕ^n(y)| = \infty$ and $\liminf_{n \to \infty} |ϕ^n(x) - ϕ^n(y)| = 0$. This demonstrates the true nature of chaos for $ϕ(x)$.

math.DS↗
arXiv2009

On the one-sided and two-sided similarities or weak similarities of permutations

Let $n \ge 3$ be an integer. Let $P_n = \{1, 2, 3, ..., n-1, n \}$ and let $S_n$ be the symmetric group of permutations on $P_n$. Motivated by the theory of discrete dynamical systems on the interval, we associate each permutation $\si_n$ in $S_n$ a (zero-one) Petrie matrix $M_{\si_n,n-1}$ in $GL(n-1,{\mathbb{R}})$ (which is generally not the same as the usual permutation matrix). Then, for any two permutations $\si_n$ and $ρ_n$ in $S_n$, the notions of right, left and two-sided similarities (and weak similarities respectively) of $\si_n$ and $ρ_n$ are introduced using the similarities (and the characteristic polynomials respectively) of the correspnding Petrie matrices of some extended permutations related to $\si_n$ and $ρ_n$ and examples are presented. As a by-product, we obtain ways to construct countably infinitely many pairs of Petrie matrices which are similar.

math.RA↗
arXiv2007

Bifurcation of the ACT map

In this paper, we study the Arneodo-Coullet-Tresser map $ F(x,y,z)=(ax-b(y-z), bx+a(y-z), cx-dx^k+e z)$ where $a,b,c,d,e$ are real with $bd\neq 0$ and $k>1$ is an integer. We obtain stability regions for fixed points of $F$ and symmetric period-2 points while $c$ and $e$ vary as parameters. Varying $a$ and $e$ as parameters, we show that there is a hyperbolic invariant set on which $F$ is conjugate to the full shift on two or three symbols. We also show that chaotic behaviors of $F$ while $c$ and $d$ vary as parameters and $F$ is near an anti-integrable limit. Some numerical results indicates $F$ has Hopf bifurcation, strange attractors, and nested structure of invariant tori.

math.DS↗
arXiv2007

A Simple Method Which Generates Infinitely Many Congruence Identities

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 number of periodic points of minimal period m must be divisible by m, we obtain infinitely many congruence identities.

math.NT↗
arXiv2007

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.

math.DS↗
arXiv2006

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.

math.DS↗