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

TL;DR

The paper constructs smooth circle covering maps with physical measures supported on hyperbolic repelling fixed points, including cases with full measure basins, using a novel realization method.

Contribution

It introduces a new construction of smooth circle maps with physical measures at repelling points and a realization technique for full branch maps.

Findings

Constructed smooth circle maps with physical measures on repelling fixed points.

Achieved full measure basins by relaxing smoothness at a single point.

Developed a canonical realization method for full branch maps.

Abstract

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.