13 problems

Let $q$ be a parameter and let $\mathbf{E}_{q}(y)$ denote the function introduced through the Farey transfer operators. For the previously considered cases $q=1$ and $q=\frac12$, o…

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…

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 $f:M\rightarrow M$ 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 $d4d8_p$ be the operator on functions defined by … For every positive integer $p$, let $d4d8_p$ denote the $n$-th eigenvalue of the generalized Gauss-Kuzmin-Wirsing operator an…

Monotonicity conjecture. The function $F(s)$ is strictly increasing on $(1,\infty)$. In particular, $s=2$ is the unique zero of $F(s)$.

Sign-pattern conjecture. Lemma (b) is true for every $h_n$, $n\in\mathbb{N}$; that is, for $q\in Q_k$ and $0<t<1/2$, one has $(-1)^{k-1}h_n'(t)>0$.

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 $(V,\chi)$ be any unitary finite-dimensional representation of $\Gamma_\lambda$, and let $s$ satisfy $\operatorname{Re}s\in(0,1)$. Let $(\Gamma_\lambda,\chi)$-automorphic cusp…

Weak variant of Mayer's conjecture. All eigenvalues of $\mathcal{L}$ with sufficiently small modulus should be real.

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 $I$ be the underlying expanding dynamical system, let $\tau$ be a cocycle, and let $\boldsymbol{G}=?$. In the one-dimensional torus case $\boldsymbol{G}=?$, for $q e 0$ define…

Comparison conjecture. The space of $1$-eigenfunctions of the meromorphic extension of $\mathcal L_{H,s}^{+}$ is linearly isomorphic to…