= Solution
Use the <interval exactness of a tent-map core> from the preceding part. Choose two separated nondegenerate <closed intervals> $J_0,J_1$ inside $J=(1-s,1)$. There are integers $n_i$ with $T_s^{n_i}(J_i)=A$. Taking $N=\max(n_0,n_1)$ works for both, because $T_s(A)=A$:
$$
F(J_0)=F(J_1)=A,\qquad F=T_s^N.
$$
Within each $J_i$, the <continuous function> $F$ attains both endpoint levels $1-s$ and $1$. Choose a segment between those levels, reversing its orientation if necessary. Starting from a point at level $1-s$, take the first point at level one, and then the last point at level $1-s$ before it. Between these two points, continuity and the first-passage choice give $1-s<F(x)<1$, with all intermediate levels attained. The interior $K_i$ of that segment therefore satisfies $F(K_i)=J$. The intervals $K_0,K_1$ are disjoint and lie inside $J$.
Thus <interval exactness produces a horseshoe>:
$$
\boxed{T_s^N(K_0)=T_s^N(K_1)=J,\qquad K_0\cap K_1=\varnothing.}
$$
This is a <horseshoe for an interval map> for an iterate of $T_s$, proving <Glendinning chaos> for every $\sqrt2<s\leq2$. The proof does not require the same horseshoe iterate $N$ to work for every parameter.
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