7 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…

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…

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 $T(x)=2x\pmod 1$ 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…

Periodic-orbit conjecture. The maximizing measure of $f$ should be supported on a single periodic orbit.

Let $\Sigma=\{1,2,3,\ldots,d\}^{\mathbb{N}}$ be the Bernoulli space with its shift, and let $A:\Sigma\to\mathbb{R}$ be a Hölder potential. A probability measure maximizing…