Open Access
2007/2008 Orbits of Darboux-Like Real Functions
T. K. Subrahmonian Moothathu
Real Anal. Exchange 33(1): 145-152 (2007/2008).


We show that, with respect to the dynamics of iteration, Darboux-like functions from $\mathbb{R}$ to $\mathbb{R}$ can exhibit some strange properties which are impossible for continuous functions. To be precise, we show that (i) there is an extendable function from $\mathbb{R}$ to $\mathbb{R}$ which is `universal for orbits' in the sense that it possesses every orbit of every function from $\mathbb{R}$ to $\mathbb{R}$ up to an arbitrary small translation, and which has orbits asymptotic to any real sequence, (ii) there is a function $f\:mathbb{R}\to \mathbb{R}$ such that for every $n\in \mathbb{N}$, $f^n$ is almost continuous and the graph of $f^n$ is dense in $\mathbb{R}^2$, in spite of the fact that all $f$-orbits are finite. To prove (i) we assume the Continuum Hypothesis.


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T. K. Subrahmonian Moothathu. "Orbits of Darboux-Like Real Functions." Real Anal. Exchange 33 (1) 145 - 152, 2007/2008.


Published: 2007/2008
First available in Project Euclid: 28 April 2008

zbMATH: 1132.54023
MathSciNet: MR2402869

Primary: 26A15 , 26A18 , 54H20‎

Keywords: Continuum hypothesis , Darboux-like function , Orbit , real sequence , topological transitivity

Rights: Copyright © 2007 Michigan State University Press

Vol.33 • No. 1 • 2007/2008
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