Fixed Point Equivalences of Axiom of Choice

Question

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.

Answer 1

An example is Alexander Abian's result in his paper:

A. Abian, A fixed point theorem equivalent to the axiom of choice, Arch. math. Logik 25 (1985), 173-174

The result says:

Theorem: The Axiom of Choice is equivalent to the statement that "Every noncontractive mapping from a pre-inductive partially ordered set into itself has a fixed point".

For definitions note that:

Definition 1: We call a partially ordered set $(P, \leq)$ pre-inductive if and only if every well ordered subset of $P$ has an upper bound in $P$.

Definition 2: We call a mapping $f$ from $(P,\leq)$ into $(P,\leq)$ noncontractive if and only if $x\leq f(x)$ for every $x \in P$.

Remark: Note that every pre-inductive partially ordered set is nonempty.

Answer 2

There are several results about fixed points, related to the axiom of choice. Some of them:

1) Fixed-Point Brouwer Theorem (1912) is a consequence of that axiom.

2) the Banach fixed-point theorem is another result of it.

3) the theorem of Schauder fixed point is a generalization of the Brouwer Theorem, and therefore, it is another consequence of the Axiom.

4) the "fixed point theorem to expanding relations" is equivalent to the axiom cited in conjuntista model 'ZF + ¬Axiom of Regularity'.

5) "fixed point theorem to inflationary relations" is equivalent to the axiom cited in conjuntista model 'ZF + ¬Axiom of Regularity'.

note: article 2 of the above results: "Book ----> Trends in Logic Volume 42 2015 Freedom and Enforcement in Action A Study in Action Formal Theory authors:

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6) the fixed point theorem of Bourbaki-Kneser is equivalent to the axiom cited. In order theory, this theorem is a version of Zorn's Lemma.

7) the fixed point theorem of Knaster-Tarski is a consequence of Zorn's Lemma.