33 problems

Let $T$ be a suitably chaotic map, let $\phi$ belong to a space of suitably regular functions, and let $U$ be a subset of that function space. Typical periodic optimization conject…

Hunt–Ott and Yuan–Hunt conjectures. For suitable chaotic maps $T$, the periodic optimization property is typical in a probabilistic sense; topologically, it should be typical when…

Let $T$ be an expanding map or an Axiom A diffeomorphism, and let $\phi$ be a Lipschitz function. Yuan–Hunt conjecture. There is an open and dense subset of Lipschitz functions…

Yuan–Hunt conjecture. For a topologically generic Lipschitz continuous or a topologically generic $C^{1}$-smooth potential $\psi$, there is a $\psi$-maximizing measure and it is su…

Let a Lagrangian be chosen from the generic class considered in the ergodic optimization setting, and let minimizing measures be invariant probability measures minimizing the space…

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…

Typically periodic optimization conjecture. The periodic optimization property may be typical for a considerably wider class of chaotic dynamical systems than those covered by the…

Let $I=[0,1]$, let $G:I\to I$ be Gauss's continued fraction map, and let $\phi$ be a Lipschitz function on $I$. A measure in $\mathcal{M}_{\max}^{*}(G,\phi)$ is a $(G,\phi)$-limit-…

Let $(X,f)$ be a dynamical system, let $\varphi\in C(X)$ be an observable, and define its maximum ergodic average by … A measure $\mu\in\mathcal{M}_f(X)$ is maximizing for…

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…

Let $(X,T)$ and $(Y,S)$ be transitive shifts of finite type, and let $f\in C(X\times Y)$ be locally constant on $X\times Y$. Let $\mathcal{M}_S^{co}(Y)$ denote the specified class…

Hunt–Ott–Yuan conjecture. For a typical chaotic system $T$, a typical smooth function $f$ admits a maximizing measure supported on a periodic orbit, with typicality of $f$ understo…

Maesumi's conjecture. Lebesgue-almost every finite family of matrices has an SMP. The conjecture was proposed after the finiteness conjecture was refuted by Bousch and Mairesse; th…

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 $M$ be a manifold, let $X$ be a $C^1$-generic vector field on $M$, and let $bb$ be a homoclinic class that is away from homoclinic tangencies and contains singularities. A subs…

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 $\mathcal{A}$ be a cocycle, let $\beta(\mathcal{A})$ denote a boundary value of its Lyapunov spectrum, and let $E(\beta(\mathcal{A}))$ be the corresponding Lyapunov level set w…

Let $(X,T)$ be a chaotic dynamical system and let $u$ be a real-valued observable. A measure in $\mathcal{M}(X,T)$ is $u$-minimizing if it minimizes the ergodic average of $u$ amon…

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…

Let $T(x)=2x\pmod 1$ act on $[0,1]$, let $A(x)=\sin(2\pi x)$, and let $m(A)$ denote the maximal ergodic average of $A$. Conze's conjecture. … The equality identifies the maximal av…

Let $T(x)=2x\pmod 1$ act on $[0,1]$, with inverse branches $\tau_1(x)=x/2$ and $\tau_2(x)=(x+1)/2$, and let $A(x)=\sin(2\pi x)$. A maximizing probability is a $T$-invariant probabi…