Axiom of Choice has many known equivalences. Also there are many known fixed point theorems (unproved statements) which provide useful information about existence of fixed points for particular operators over spaces with special properties.
As fixed point theorems often deal with infinite iterations of an operator to build a fixed point and such a process may need Axiom of Choice, it seems reasonable to expect that there are some equivalences of Axiom of Choice in terms of "Fixed Point Statements".i.e. (statements about existence of a fixed point for a particular operator over a particular space).
Question: What are examples of fixed point statements that are known to be equivalent to Axiom of Choice or imply AC (within ZF)? Open conjectures about equivalent forms of Axiom of Choice in terms of fixed point statements are also welcome.