 Home

Codex Wiki

List of universities

University of Cambridge

Course of the University of Cambridge

Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2022

Paper 2

Set
$$
I_0=[x_0,x_1],\qquad I_1=[x_1,x_3],\qquad I_2=[x_3,x_2].
$$
Their covering graph is
$$
I_0\longrightarrow I_1,I_2,\qquad
I_1\longrightarrow I_0,I_1,I_2,\qquad
I_2\longrightarrow I_0,I_1,
$$
with <adjacency matrix of a directed graph>[adjacency matrix]
$$
A=\begin{pmatrix}0&1&1\\1&1&1\\1&1&0\end{pmatrix}.
$$
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 <Glendinning chaos>[chaotic]. Since
$$
\operatorname{tr}A=1,
\qquad
\operatorname{tr}(A^3)=13,
$$
<counting cycles in an interval covering graph> gives
$$
\frac{13-1}{3}=\boxed4
$$
distinct 3-cycles in the minimum case, attained by the corresponding <connect-the-dots interval map>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 2 / 33B / b / ii / Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 2 / 33B / b / ii / Solution

 Ancestors (12)

 Discussion (0)

 Articles by others on the same topic (0)

Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 2 / 33B / b / ii / Solution

 Ancestors (12)

 Discussion (0)  Subscribe (1)

 Articles by others on the same topic (0)

 Discussion (0)