The set of second iterates is nowhere dense in 𝐶

Simon, Károly

doi:10.1090/S0002-9939-1991-1033961-5

The set of second iterates is nowhere dense in $C$
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Proc. Amer. Math. Soc. 111 (1991), 1141-1150 Request permission

Abstract:

Let $C$ denote the set of continuous functions mapping $[0,1]$ into itself endowed with sup norm. It is proved that the set $\{ f \circ f:f \in C\}$ is nowhere dense in $C$.

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