Topological conjugacy of iterated random orientation-preserving
homeomorphisms of the circle
Suppose we have a parameter-dependent orientation-preserving circle
homeomorphism, together with a probability measure $\nu$ on the parameter
space. This naturally generates a "random dynamical system" on the circle,
where at each time step a parameter is randomly chosen with distribution
$\nu$ (independently of all previous time steps) and the associated
homeomorphism is applied. Given two parameter-dependent
orientation-preserving homeomorphisms defined over the same parameter space
(with the same measure), one can define a notion of "topological conjugacy"
between the random dynamical systems that they generate. Under sufficient
"regularity" assumptions, we will classify such parameter-dependent
homeomorphisms up to topological conjugacy.