 Core interval of an expanding tent map

Restrict $T_s(x)=1-s|x|$ to $[-1,1]$, with $1<s\le 2$. Its image is the closed intervalBecause $0\in A$ and ${\max }_{x\in A}|x|=1$, its next image is again $T_s(A)=A$. Every state in $[-1,1]$ therefore enters this invariant core after at most one iterate and remains there. This is the core used in interval exactness of a tent-map core.

The domain restriction matters. If the same formula defines a map on the whole real line, any $x<-1/(s-1)$ stays on the negative branch and satisfiesThus points outside the declared phase interval need not enter $A$, and $A$ is not a globally absorbing interval for the real-line extension.