Continuous Transitive Maps on the Interval Revisited

Abstract

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.

49 pages

References in corpus (2)

• General definitions of chaos for continuous and discrete-time processes
• On the Chaos in Continuous Weakly Mixing Maps

Cited by in corpus (2)

• On the Chaos in Continuous Weakly Mixing Maps
• Transitivity and Mixing Properties of Set-Valued Dynamical Systems