31 problems

Analyticity conjecture. The map $t\mapsto k(t)$ is analytic everywhere except for at most $d-1$ points.

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 $\boldsymbol{r}_n$ be the return time to the $n$-cylinder of a point, and let $\boldsymbol{w}_n$ be the corresponding hitting time between two points. Let $\boldsymbol{ w}_n$ d…

Let $\boldsymbol{r}_n$ denote the return time to the $n$-cylinder of a point, and let $\boldsymbol{w}_n$ denote the corresponding hitting time between two points. For a Bowen–Gibbs…

Second Legendre equation. For every $\alpha$ in the segment

Existence conjecture. In general, for each $q>0$, there is an open set of potentials $A$ for which such a pair $(\varphi,c)$ exists.

Faure–Weich–Dolgopyat–Pollicott conjecture.

Analyticity conjecture. The map $t\mapsto k(t)$ is analytic everywhere except possibly at $t_0$.

Let $(\mu_{t\mathcal F})_{t>1}$ be the family of equilibrium states associated with the almost-additive potential $\mathcal F$. Zero-temperature limit conjecture. The weak limit ……

Let $(f_t)$ be a smooth family of degree-$d$ Blaschke products, and let $||\cdot||_{WP}$ denote the Weil–Petersson semi-norm on $\mathcal{B}_d$ defined by the second variation of t…

Let $(Ai){i e i}$ be a completely reducible family in $operatorname{GL}d(mathbb{R})^{mathcal{I}}$, with $kin{0,ldots,d-1}$ and $sin(k,k+1)$. Let $p$ be the number of functions in t…

Let $T$) be in a suitable class of correspondences on a compact metric space $(X,d)$, and let $\phi:\mathcal{O}_2(T)\to\mathbb{R}$ be a sufficiently regular function. A transition…

Let $M$ be a compact Riemannian manifold and let $f:M\rightarrow M$ be a $C^{2}$ local diffeomorphism. A transfer operator associated to a complex continuous potential…

Let $T$ be the piecewise expanding map, let $g\ge 0$ be its weight, let $\mathcal{H}^t_p$ be the associated fractional Sobolev space, let $\alpha$ be the regularity parameter, and…

Let $S$ be a compact surface, let $f\in \operatorname{Diff}^{1}(S)$ be a diffeomorphism, and let the potential be Holder continuous. A phase transition is a failure of analyticity…

Local fitting conjecture. Any reasonable analytic germ of finite level can be fitted locally by the pressure function of some locally constant potential, provided that the number o…

Let $f:M\rightarrow M$ be a $C^{2}$ local diffeomorphism on a Riemannian manifold $M$. For a potential $\phi$, define the transfer operator by … A suitable Banach space may consist…

Let $\theta_0\in\Theta$ be the true parameter, let $y\in\Omega$ be sampled according to $\mu_{\theta_0}$, and let $\pi_n(\cdot,y)$ denote the posterior probability measure on…

Let $(G,\mathcal W)$ be a loaded graph, where $G$ is a countable graph and $\mathcal W$ is a weight function. Suppose that $(G,\mathcal W)$ is an unstable positive loaded graph. An…

Let $(\Sigma,\sigma)$ be a finite entropy countable Markov shift, let $\rho$ be a totally bounded metric on its vertex set, and let $d$ be the corresponding metric on $\Sigma$. Wri…

Let $T:X\to X$ be the two-sided full shift, let $\phi:X\to{\mathbb R}$ be the continuous potential constructed in the preceding theorem, and let an equilibrium state for…

Let $T$ be the dynamical system and let $\phi$ be a potential with variations $\operatorname{var}_n(\phi)$ such that … Assume that $(x,Tx,\ldots)$ is distributed according to the e…

Let $\Upsilon$ be the space of polygonal paths equipped with the push-forward a priori probability measure $p$, let $f$ be the potential … where $J(n)\geq 0$ tends to zero sufficie…

Let $(M,g)$ be a pinched negatively curved manifold. Let $(\mu_n)_n$ be a sequence of invariant probability measures converging vaguely to $\mu$, and let $F\in C_b(T^1M)$ be a unif…

Let $(A_1,\ldots,A_N)\in GL_d(\mathbb{R})$ and let $s\in(0,d)\smallsetminus\mathbb{Z}$. For a tuple of matrices, let $\varphi^s$ denote Falconer's singular value function, and call…