What does it mean an $\bf{isolated \; point}$ for a map defined on a manifold?

Question

Let $M$ be a smooth manifold and $f : M \to M$ a $C^k$ diffeomorphism.

1) What does it mean an $\bf{isolated \; point}$ for $f$?

2) Why are the transverse fixed points isolated?

Thank you!

Answer 1

Let $M$ be a smooth manifold, $f:M \to M$ a $C^{k}$ difeomorphism.

1. A fixed point $p$ of $f$ is isolated if there exists a neighborhood $U$ of $p$ such that $p$ is the only fixed point of $f$ in $U$.

2. A fixed point $p$ of $f$ is transverse if the graph of $f$ in $M \times M$ is transverse to the diagonal $$ \Delta = \{(p, q) \in M \times M : p = q\}, $$ namely if $df(p) - I$ is non-singular. In this event, there exists a neighborhood $U$ of $p$ such that in $U \times U$, the graph of $f$ intersects the diagonal of $f$ precisely at $(p, p)$; that is, $p$ is the unique fixed point of $f$ in $U$.