Dynamical stability of the typical continuous function

Year, pages: 2005, 503 - 514

We consider the typical behavior of two maps. The first is the set valued function $Λ $ taking $f$ in $C(I,I)$ to its collection of $ω$@-limit points $Λ (f)=\bigcup_{x\in I}ω (x,f)$, and the second is the map $Ω $ taking $f$ in $ C(I,I)$ to its collection of $ω$@-limit sets $Ω (f)=\{ω (x,f): x\in I\}$. After reviewing results which characterize those functions $f$ in $C(I,I)$ at which each of our maps $Λ $ and $Ω $ is continuous, we show that both $Λ $ and $Ω $ are continuous on a residual subset of $C(I,I)$. We go on to investigate the relationship between the continuity of $Λ $ and $Ω $ at some function $f$ in $C(I,I)$ with the chaotic nature of that function.

Steele, T. 2005. Dynamical stability of the typical continuous function. In Mathematica Slovaca, vol. 55, no.5, pp. 503-514. 0139-9918.

Steele, T. (2005). Dynamical stability of the typical continuous function. Mathematica Slovaca, 55(5), 503-514. 0139-9918.