The return-time dimension conjecture for a large class of dynamical systems

Let $(T,X, u)$ be a dynamical system, and let $D^7_mu(q)$ and $D^7_tau(q)$ denote its generalized measure and return-time dimensions. Let $q_c$ be the lowest value of $q$ for which the partition functions of return times are finite.

Return-time dimension conjecture. For a large class of dynamical systems,

$$D^\pm_\tau(q)=D^\pm_\mu(q)\qquad\text{for }q_c<q\leq 2,$$

and

$$D^\pm_\tau(q)=\frac{D^\pm_\mu(2)}{q-1}\qquad\text{for }q\geq 2,$$

exactly, or at least asymptotically for large $q$.

The conjecture proposes that the behavior observed for the von Neumann--Kakutani map is typical: return-time and measure dimensions agree below the transition value $q=2$, while short returns produce a universal reciprocal dependence above it. The statement remains open in the supplied source.