Why are we interested in auto-homeomorphisms? I am studying extensions of auto-homeomorphisms over compactifications and have a question:
A homeomorphism between two different spaces tells us that the spaces are topologically indistinguishable. Then, why would we study auto-homeomrphisms which are homeomorphisms between one and the same space? What else about the space can the autohomeomorphism tell us?
Thank you for your insights!
 A: You may view this as extending the concept of symmetries (which would by auto-isometries instead of auto-homeomorphism) to a more general situation. E.g., a cube has a larger symmetry group than a general parallelepiped, and in fact the subgroup structure of the symmetry group of the cube compared to that of the tetrahedron can tell us how to obtain a tetrahedron by picking every other vertex. If you loosen the concept from isometries to homeomorphism you can get similar classifying insights for topological spaces (instead of metric spaces)
A: You always have the identity as an automorphism, of course. And the inverse of an automorphism is also one, as is the composition of two. So they naturally form a group, and in some cases we can put a natural topology on that group and have a nice new topological group.
If there are many automorphisms, lots of points topologically "behave the same", a space $X$ such that for every $x,y \in X$ there is an automorphism of $X$ that maps $x$ to $y$ is called homogeneous. It implies that the automorphism group is "rich" or "large". Many natural spaces in topology, like $\Bbb R, \Bbb Q$, the irrationals, the Cantor set etc. are homogeneous and have interesting automorphism groups. Showing a space to be non-homogeneous (like the unit disk) can be a challenging problem.
A: It will appear in many places naturally.
Example 1: If one point can be continuously moved to another point by automorphism.Then these two points have some common properties.
Example 2: Computing $\operatorname{Aut}(\Bbb C \cup \{\infty\})$ or $\operatorname{Aut}(\text{unit disk})$ is an interesting problem.
