Topological Entropy — Smale Horseshoe Map

The Smale horseshoe map stretches, folds, and contracts a square. Iterating reveals the invariant Cantor set. Topological entropy h = log 2 — exactly one bit of information is generated per iteration.

Iterations n

3

Stretch λ

3.0

Contraction μ

0.25

Topological entropy:
h = log(2) ≈ 0.693

Periodic orbits of period n:
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Strips shown:
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Cantor set dimension:
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