Topological invariance of the sign of upper pointwise Lyapunov exponents

Let $f$ and $g$ be topologically conjugate one-dimensional maps, and let $x$ be a point of $f$ that is not attracted to a periodic orbit. The upper pointwise Lyapunov exponent of $x$ is the limit superior of its finite-time Lyapunov exponents. Topological invariance conjecture. Topological conjugacy preserves the sign of the upper pointwise Lyapunov exponents of all points that are not attracted to a periodic orbit. The paper contrasts this conjecture with the failure of analogous invariance for lower pointwise Lyapunov exponents, while noting that the upper exponent has positive sign for every point of a unimodal Collet–Eckmann map; the conjecture is otherwise left unresolved here.