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 Incoming links: Connect-the-dots interval map

Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 2 / 33B / b / ii / Solution Created 2026-09-24 Updated 2026-09-25

I_{0} = [x_{0}, x_{1}], I_{1} = [x_{1}, x_{3}], I_{2} = [x_{3}, x_{2}] .

Their covering graph is

I_{0} ⟶ I_{1}, I_{2}, I_{1} ⟶ I_{0}, I_{1}, I_{2}, I_{2} ⟶ I_{0}, I_{1},

with adjacency matrix

A = 011111110 .

The closed walks

I_{1} \to I_{1} \to I_{1}

and

I_{1} \to I_{0} \to I_{1}

give a horseshoe from two closed covering walks for

F^{2}

, so

F

must be chaotic. Since

tr A = 1, tr (A^{3}) = 13,

counting cycles in an interval covering graph gives

\frac{13 - 1}{3} = 4

distinct 3-cycles in the minimum case, attained by the corresponding connect-the-dots interval map.

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