Real Analysis Exchange

Orbits of Darboux-Like Real Functions

T. K. Subrahmonian Moothathu

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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.

Article information

Real Anal. Exchange, Volume 33, Number 1 (2007), 145-152.

First available in Project Euclid: 28 April 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 26A15: Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) {For properties determined by Fourier coefficients, see 42A16; for those determined by approximation properties, see 41A25, 41A27} 26A18: Iteration [See also 37Bxx, 37Cxx, 37Exx, 39B12, 47H10, 54H25] 54H20: Topological dynamics [See also 28Dxx, 37Bxx]

Darboux-like function orbit topological transitivity real sequence continuum hypothesis


Moothathu, T. K. Subrahmonian. Orbits of Darboux-Like Real Functions. Real Anal. Exchange 33 (2007), no. 1, 145--152.

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