Continuity of the stable transfer under non-uniform contraction

Determine whether replacing uniform fiber contraction by non-uniform or tempered fiber contraction still ensures that the stable series defines a continuous transfer function.

Background

The paper’s cohomological reduction theorem relies on uniform exponential contraction within the fibers. Under that hypothesis, the stable-difference series converges uniformly and produces a continuous transfer function that removes the fiber dependence of a Hölder potential up to a coboundary.

The authors explicitly identify the extension to non-uniform or tempered contraction as unresolved. The problem is to determine whether weaker contraction assumptions still provide sufficient convergence and regularity of the stable series to define a continuous transfer, thereby extending the reduction framework beyond uniformly contracting extensions.