Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 4 / 32A / c

Table of contents
- i c
  - Solution i
- ii c
  - Solution ii

i

Solution

For the stated map, the graph cycles give the period-two orbit

1/3 \mapsto 5/6 \mapsto 1/3,

and the period-four orbit

2/9 \mapsto 13/18 \mapsto 5/9 \mapsto 8/9 \mapsto 2/9.

ii

Solution

The two period-five orbits are

1/9 \to 11/18 \to 7/9 \to 4/9 \to 17/18 \to 1/9,

2/15 \to 19/30 \to 11/15 \to 8/15 \to 14/15 \to 2/15.

Direct use of the two affine branches verifies every arrow.

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