On topological entropy of commuting tree maps

Let T$T$ be a tree with s$s$ ends and f,g$f,g$ be continuous maps from T$T$ to T$T$ with f°g=g°f$f°g=g°f$. In this note we show that if there exists a positive integer m≥2$m\ge 2$ such that gcd(m,l)=1$gcd(m,l)=1$ for any 2≤l≤s$2\le l\le s$ and f,g$f,g$ share a periodic point which is a km$km$-periodic point of f$f$ for some positive integer k$k$, then the topological entropy of f°g$f°g$ is positive.