When is it meaningful to talk about a countably infinite composition of automorphisms of a given object? I'm assuming such infinite compositions exist since (correct me if I'm wrong) but an infinite composition of the identity map would give the identity map.
This question came up in the following context:
Let $C$ be the canonical Cantor set and let $C_\alpha$ be a Cantor-type set, i.e. the same construction but removing the middle $\alpha$-interval at each iteration instead of the middle third. Prove "stuff"...
I wanted to prove the "stuff" by showing existence of an order preserving homeomorphism $f : [0, 1] \to [0, 1]$ with $f(C) = C_\alpha$.
The series of composite automorphisms $\{\varphi_k\}$ would certainly need some convergence properties. But I'm hoping for some examples or contexts in which such infinite compositions would arise (or not).
In particular, when does an infinite composition of automorphisms produce another automorphism?
-XXP
Those composition don't need to converge in some sense. Take for example the real numbers with addition as a group. Then the mappling $f:\mathbb{R}\to \mathbb{R}; \ x\mapsto \frac{x}2$ is an automorphism. As $f^k : \mathbb{R} \to \mathbb{R} ; \ x\mapsto \frac{x}{2^k}$ the composition converges to the constant 0 function.
Maybe you are looking for Lie-groups which are -saying heuristical (as I only know them in that way)- manifolds which are a group too and have a differential group operation. So for example $SO(2)$ is a Lie group, those are the rotations on $\mathbb{R}^2$. This group is diffeomorphic to $S^1$ which is the unit ball in $\mathbb{R}^2$. Furthermore the composition of two rotations is just adding the angles, and a rotation in $\mathbb{R}^2$ is uniquely determined by the angle of its rotation. So as $S^1$ is a closed subspace of a Banach-space the infinite composition of rotations should converge to a rotation when the infinite composition converges pointwise.