Bowen–Walters inequality without uniform C₀ continuity

Determine whether the Bowen–Walters inequality limsup_{T\to\infty}T^{-1}\log\nu(T)\leq(\phi) remains valid for a geometrically separating flow that is dynamically isolated at infinity when the uniformly C₀ hypothesis is removed.

Background

The paper proves a noncompact version of the Bowen–Walters inequality for flows that are geometrically separating, uniformly C₀, and dynamically isolated at infinity. Here, $\nu(T)$ counts distinct periodic orbits with period at most $T$, while $(\phi)$ denotes the supremum of the upper-capacity entropies over compact subsets of the phase space.

Uniform C₀ continuity provides uniform control of the spatial displacement caused by small time errors in the piecewise-linear reparametrizations used to compare periodic orbits. The authors explain that without this condition, points near infinity may undergo a fixed spatial displacement in arbitrarily short times, causing the periodic-orbit separation argument to fail. They explicitly leave unresolved whether the inequality itself nevertheless remains true without uniform C₀ continuity; the issue had also been considered in the cited work \cite{ym1}.