1
$\begingroup$

I started working with a map $f:\mathbb{R} \to \mathbb{R}$ such that it is continuous except on a finite set. I started looking for a definition of topological transitivity and topological mixing in this context. So, I am asking if there is a good reference where I can find such definition.

I am aware the works of Kolyada et al. where some of these aspects are discussed, however the considered maps are always continuous. Also, I know some references where topological transitivity for piecewise continuous maps on the unit interval is discussed, for example Glendinning's work "Topological conjugation of Lorenz maps by β-transformations".

Thanks in advance for your help.

Improve this question
$\endgroup$
1
  • 2
    $\begingroup$ Assuming that $f$ has 1-sided limits at every point, there exists a countable subset of $\mathbf{R}\cup\{\pm\infty\}$ that is stable under taking $x\mapsto f(x^-)$ and $x\mapsto f(x^+)$. You can split real elements in this subset into a pair $\{x^-,x^+\}$ to obtain a compact space $X$ projecting onto $\mathbf{R}\cup\{\pm\infty\}$. Under a mild additional assumption (namely that $f$ is monotonic locally at every 1-sided neighb. of every point), $f$ lifts to a unique continuous self-map of $X$. Then the various topological dynamical considerations for $f$ should be understood in this space $X$. $\endgroup$
    – YCor
    Commented Jan 20, 2018 at 21:51

0

Reset to default

You must log in to answer this question.