42 problems

Let $\Gamma_\lambda$ be the Hecke triangle group, let $\chi$ be a unitary finite-dimensional representation, and let $\mathcal L_{s,\chi}$ be the associated transfer operator. At a…

Let $\lambda_n$ be the eigenvalues of the transfer operator for the Gauss continued fraction map, arranged so that $|\lambda_1|\geq |\lambda_2|\geq\cdots$, and let…

Let ${\cal H}_0$ be the space defined in the paper, and let ${\cal P}:{\cal H}_0\to{\cal H}_0$ be the associated operator. Unique nonzero eigenvalue conjecture. The only nonzero ei…

Let $\Gamma_0(N)$ be a congruence subgroup of Hecke type of level $N\ge 2$. For even $n\ge 2$, let $\wp_n(\Gamma_0(N))$ be the space of polynomial eigenfunctions of degree $n$ of t…

Let $\Gamma(N)$ be the principal congruence subgroup of $PSL(2,\mathbb Z)$ of level $N\ge 2$. For even $n\ge 2$, let $\wp_n(\Gamma(N))$ be the space of polynomial eigenfunctions of…

Let $L_s$ be the transfer operator for the geodesic flow associated with a modular group, and let the Lewis equation be the functional equation characterizing its polynomial eigenf…

Let $\mu$ be an initial distribution in the equivalence class indexed by $\sigma$, and let $M^*$ be the normalised transfer operator. A convergence conjecture. If there exists a qu…

Let $d\geq 2$, and let ${\mathscr L}_N$ be the randomly twisted transfer operator defined using independently uniform permutation matrices. Write $\mathrm{Sp}({\mathscr L}_N)$ for…

Finite-attractor conjecture. For Lebesgue-almost every $r$, the map $\widehat{\mathfrak{P}}_r$ has finitely many attractors.

Let $\Phi$ be the Dyson interaction with standard couplings of the form $J(k)=O(k^{-\alpha})$, and let $\mathcal{L}_{\beta\phi}$ be its transfer operator. In the regime…

Johansson–Öberg–Pollicott's conjecture. The critical inverse temperatures are equal:

Let $d53f$ be the transfer operator associated with the greedy $d6fd$-expansion, and let $N$ be the differentiability order in the estimate … When $a_1=1$, the regularity-threshold…

Let $\beta>1$, $0<a1 eq 1$, and let $d53f$ be the transfer operator in the asymptotic estimate … For $F\in C^\infty([0,1])$, the optimal-exponent conjecture. The estimate above onl…

Let $L$ be the transfer operator for the Dyson potential, let $alpha>1$ be the interaction exponent, and let $\beta_c(alpha)$ be the critical inverse temperature. Dyson eigenfuncti…

Let $\beta_c(alpha)$ be the critical inverse temperature of the Dyson model, and let $L$ be its transfer operator. Subcritical eigenfunction conjecture. In the regime corresponding…

Let $L$ be the transfer operator associated with the Dyson model, with interaction exponent $alpha$. Sharpness conjecture. For $alpha greater than or equal to 0$?

Analytic-extension conjecture. In all cases, the function

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 $J_V(ij)=\mathrm{J}(|i-j|)$ be the interactions of the long-range Ising models on $V=\mathbb{N}$ and $V=\mathbb{Z}$, and let $\beta_c(J_\mathbb{N})$ and $\beta_c(J_\mathbb{Z})$…

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…

The point cloud is sampled from the alanine-dipeptide model, and let $\Gamma^{N,\varepsilon}$ denote the entropic transfer operator with sample size $N$ and parameter $\varepsilon$…

Spectral conjecture. In fact, for

Eigenfunction-form conjecture. The eigenfunctions of the Perron–Frobenius operator for $T_{N,a}$ are of the form

Mean-map injectivity conjecture. For the stochastic dynamical system considered here, no two transition densities $p^\tau(x_1,\cdot)$ and $p^\tau(x_2,\cdot)$ possess the same mean;…