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.

If $\sqrt{2}<s\le 2$, every nondegenerate subinterval of $A=[1-s,1]$ eventually maps onto $A$. Away from the critical point, lengths multiply by $s$; one fold loses at most a factor of two. If an interval and its image both contain zero, its second image is $A$. Otherwise every two steps increase length by at least $s^2/2>1$, which cannot continue inside bounded $A$. Applying the intermediate value theorem to an interval that covers itself then gives a periodic point in every subinterval.

// Target: dynamical-systems.bigb