= Solution
The image of the full interval under the <tent map> is $T_s([-1,1])=[1-s,1]$. Since $s>1$, this image contains the <critical point> zero and the right endpoint one; its next image is again $[1-s,1]$. Moreover, any interval $B$ with $T_s(B)=B$ lies inside the image of $[-1,1]$. Thus the <core interval of an expanding tent map> is
$$
\boxed{A=[1-s,1]=[T_s^2(0),T_s(0)].}
$$
Every point outside $A$ enters $A$ after one iterate and then stays there. For $s=2$ there are no points of the domain outside $A$.
To prove <interval exactness of a tent-map core>, first take a nondegenerate <closed interval> $L\subseteq A$ and put $L_n=T_s^n(L)$. Each $L_n$ is an interval. If $L_n$ does not straddle zero, the corresponding branch is affine with slope of magnitude $s$, so $|L_{n+1}|=s|L_n|$. If it straddles zero, its image has right endpoint one and
$$
|L_{n+1}|=s\max\{|\inf L_n|,|\sup L_n|\}\geq\frac{s}{2}|L_n|.
$$
If both $L_n$ and $L_{n+1}$ contain zero, then $L_{n+1}$ contains $[0,1]$, whence $L_{n+2}=A$. Otherwise at most one of these two steps folds across the <critical point>. Consequently, unless the full core has already been reached,
$$
|L_{n+2}|\geq\frac{s^2}{2}|L_n|.
$$
Since $s^2/2>1$, indefinite failure to reach $A$ would give intervals of unbounded length inside an interval of length $s$. This is impossible, proving $\boxed{T_s^k(L)=A\text{ for some }k}$. For an interval with open endpoints, apply the argument to any nondegenerate closed subinterval inside it; its image already covers $A$, while the image of the original interval cannot leave $A$.
Now let $U$ be any nonempty relatively open subset of $A$ and choose a closed nondegenerate $J=[u,v]\subset U$. Some $F=T_s^k$ maps $J$ onto $A$, so there are points $p,q\in J$ with $F(p)=1-s$ and $F(q)=1$. Therefore $F(p)-p\leq0$ and $F(q)-q\geq0$. The <intermediate value theorem> yields $z\in J$ with $F(z)=z$. This is a <periodic point> of $T_s$, of period dividing $k$. Every such $U$ contains one, so \b[the <periodic points> are dense in the whole core $A$].
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