10 problems

Let $n\in\mathbb{Z}_{>0}$ be odd. Let $\mathbb{T}=\mathbb{R}/\mathbb{Z}$ be the circle, and let $E_2:\mathbb{T}\to\mathbb{T}$ be the doubling map. For observables … consider measur…

Let $(X,T)$ be an open expanding multi-valued dynamical system, and let $\operatorname{Lip}(X)$ denote the Banach space of Lipschitz real-valued functions on $X$. Let…

For integers $1\le p<q$, let $F_{0,p,q}$ be the multi-valued dynamical system on $S^1$ appearing in the paper, and let a Sturmian measure mean a probability measure whose support i…

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 $A$ and $T(x)=2x\pmod 1$ be as in the preceding construction, with inverse branches $\tau_1(x)=x/2$ and $\tau_2(x)=(x+1)/2$. Define $\eta=\tau_2\circ\tau_1$ and … A subaction i…

Let $K$ be the set used in the source, let $d(x,K)$ denote the distance from $x$ to $K$, and set $A(x)=-d(x,K)$. Let $T(x)=2x\pmod 1$ on $[0,1]$, with inverse branch…

Typically periodic optimization (TPO) conjecture. The set $V_{Per}$ contains an open dense subset of $V$.

Mañé's strong conjecture. The set $\mathcal{O}_1(L)$ is residual in $C^{\infty}(M)$. Equivalently, for a generic potential perturbation, there is a unique minimizing measure and it…