Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 16 / 3 / b / Solution

Let $H:X\to Y$ be a homeomorphism with $H\circ f=g\circ H$. This is topological conjugacy. Choose a compatible metric $e$ on $Y$ and pull it back to $X$:Because $H$ is a homeomorphism, $d_H$ is a compatible metric. The conjugacy identity implies $H{f}^k=g^{k}H$ for every $k\ge 0$, so the corresponding Bowen metrics satisfyThe bijection $H$ therefore transports separated sets in either direction without changing their separation, and $s_n(d_H,ϵ)=s_n(e,ϵ)$. Their exponential growth rates, and then their limits as $ϵ\downarrow 0$, agree. Part (a) lets us replace $d_H$ by any compatible metric on $X$. Hence