Hyperbolic periodic points of a diffeomorphism are isolated

Question

Let $f$ be a diffeomorphism. Why are the hyperbolic periodic points isolated?

Answer 1

They aren't isolated. Here's a counterexample.

Let $f : S^1 \times S^1 \to S^1 \times S^1$ be the diffeomorphism induced under the universal covering map $\mathbb R \times \mathbb R \to S^1 \times S^1$ by the matrix $\begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix}$ acting on $\mathbb R \times \mathbb R$. Far from being isolated, the periodic points of $f$ form a countable dense subset of $S^1 \times S^1$.