On the growth rate inequality for periodic points in the two sphere

Let f:S^2\to S^2 be a continuous map such that deg f = d, |d|>1. Suppose f has two attracting fixed points denoted N and S and let A=S^2\setminus \{N,S\}. Assume that if a loop \gamma\subset f^{-1}(A) is homotopically trivial in A, then f(\gamma) is also homotopically trivial in A. Then,... Show more