8 problems

Let $T\colon[0,1)\to[0,1)$ be the Bolyai–Rényi transformation, and let $\mu$ be its unique $T$-invariant probability measure. The Bosma–Dajani–Kraaikamp conjecture. The metric entr…

Let $(M,g)$ be a closed Riemannian manifold and let $\omega$ be its volume form. For an integer $r$, let $\Lambda^{r}_{\omega}(M)$ denote the space of $C^1$ expanding $r$-folds of…

Coalescing-orbit model conjecture. This model should capture sufficiently well the relevant features of discretizations in the middle-term range to approximate well the evolution o…

Minimizing upper-Lyapunov measure conjecture. For a generic expanding self-map $T$ on a manifold $M$, the minimizing measure of the upper Lyapunov exponent $\lambda_1$ exists, is u…

Upper Lyapunov exponent maximizing-measure conjecture. If $k\geq 2$, then for a $C^k$-generic expanding self-map $T$, the upper Lyapunov exponent has a unique maximizing measure, s…

Jenkinson–Morris conjecture. For a generic $T\in\mathcal{E}^k$, the Lyapunov minimizing measure is unique and supported on a periodic orbit of $T$.

Let $T$ be a uniformly expanding map and let the observables be real-valued functions in a specified smooth function space, such as the space of H"{o}lder continuous, Lipschitz con…

Let $f$ be a $C^1$ expanding map on a compact manifold of arbitrary dimension. The results proved in this paper for $C^1$ expanding maps of the circle should also hold for $f$ in a…