Smooth Circle Covering with a Physical Measure on a Hyperbolic Repelling Fixed Point

We construct an example of a smooth ($C^\infty$) circle covering map topologically conjugate to the doubling map, such that it has a physical measure supported on a hyperbolic repelling fixed point.

By relaxing the smooth condition at a single point, we also construct an example where the basin of the physical measure has full measure.

A key technical step is a realization method of independent interest, which gives a canonical way to construct a full branch map given its induced map.