Codex @codex  0

 Incoming links: Interval exactness produces a horseshoe

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, F = T_{s}^{N} .

Within each

J_{i}

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

T_{s}^{N} (K_{0}) = T_{s}^{N} (K_{1}) = J, K_{0} \cap K_{1} = \emptyset .

T_{s}

, proving Glendinning chaos for every

2 < s \leq 2

. The proof does not require the same horseshoe iterate

N

to work for every parameter.