$C^1$ Circle Covering with a Physical Measure on a Hyperbolic Repelling Fixed Point

We construct an example of a $C^1$ 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 $C^1$ 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 lemma of independent interest, which gives a canonical way to construct a full branch map given its induced map.

💡 Research Summary

The paper investigates expanding circle covering maps that are topologically conjugate to the doubling map and demonstrates that a hyperbolic repelling fixed point can support a physical (SRB) measure. The authors work within the class D of orientation‑preserving circle coverings with two full branches, each having a generating partition, and focus on the unique fixed point p (taken as 0) and its other preimage q.

Two main theorems are proved. Theorem 1.2 exhibits a map f∈D that is C^∞ on the whole circle except at the point q, where it is merely continuous, and whose unique fixed point p satisfies |f′(p)|>1. For this map the basin of the Dirac measure δ_p has full Lebesgue measure, i.e. |B_{δ_p}|=1. Theorem 1.3 provides a C¹‑smooth example (still C^∞ away from q, with f′(q)=0) for which the basin of δ_p has positive Lebesgue measure, |B_{δ_p}|>0. Both results answer affirmatively the question whether a hyperbolic repelling fixed point can be a statistical attractor in the setting of circle coverings.

The construction hinges on the first‑return (induced) map F on the right branch I₂=