I am working on the following exercise (Problem 6-b, pg. 6-22) from Milnor's Dynamical Systems notes:
Problem 6-b Let $f$ be any self map such that each iterate has only finitely many fixed points. Using the inequalities $$\#Per_k(f)\leq \#Fix(f^{\circ k})\leq \sum_{j=1}^k\#Per_j(f)$$ show that $$\limsup_{k\to \infty}\frac{\log^+\#Fix(f^{\circ k})}{k} = \limsup_{k\to \infty}\frac{\log^+\#Per_k(f)}{k}$$ (Here $\log^+(x)$ is defined to be log(x) for x ≥ 1 , but is defined to be zero for 0 ≤ x ≤ 1 . $\#Fix(f^{\circ k})$ and $\#Per_k(f)$ denote the number of fixed points of $f^{\circ k}$ and the number of points of period exactly $k$ respectively.)
My attempt: One direction of this equality is immediate. For the other, we have by subadditivity that $\log \#Fix(f^{\circ k})\leq \log \#Per_k(f) + \log \sum_{j=1}^{k-1}\#Per_j(f).$ If I could show that $\frac{\log \sum_{j=1}^{k-1}\#Per_j(f)}{k}\to 0$ we would be done. But why is that true? I imagine this follows easily from some well known "real-analysis property of he logarithm. And why is it important to work with $\log^+$ as opposed to the usual logarithm?