Real Analysis Exchange

Properties of Topologically Transitive Maps on the Real Line

Anima Nagar

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Abstract

We prove that every topologically transitive map $f$ on the real line must satisfy the following properties:

(1) The set $C$ of critical points is unbounded.

(2)The set $f(C)$ of critical values is also unbounded.

(3)Apart from the empty set and the whole set, there can be at most one open invariant set.

(4)With a single possible exception, for every element $x$ the backward orbit $\{y\in {\mathbb R} : f^n(y) = x$ for some $n$ in ${\mathbb N}\}$ is dense in ${\mathbb R}$.

Article information

Source
Real Anal. Exchange, Volume 27, Number 1 (2001), 325-334.

Dates
First available in Project Euclid: 6 June 2008

Permanent link to this document
https://projecteuclid.org/euclid.rae/1212763972

Mathematical Reviews number (MathSciNet)
MR1887863

Zentralblatt MATH identifier
1015.37030

Subjects
Primary: 54H20: Topological dynamics [See also 28Dxx, 37Bxx]

Keywords
Topologically transitive maps critical points critical values invariant set

Citation

Nagar, Anima. Properties of Topologically Transitive Maps on the Real Line. Real Anal. Exchange 27 (2001), no. 1, 325--334. https://projecteuclid.org/euclid.rae/1212763972


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References

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