29 problems

Let $f(z)=z^2-c$ with $c>2$, let $J(f)$ be its Julia set, and let $H_{\mu_{J(f)}}$ be the Jacobi matrix associated with the equilibrium measure $\mu_{J(f)}$. A Jacobi matrix is alm…

Let $T(z)$ be an expanding polynomial and let $\delta$ be a parameter for the renormalization equation. Factorize … by putting in the first factor all critical points related to…

Let $T$ be a sufficiently hyperbolic polynomial and consider the branches of solutions of its renormalization equation. Contraction conjecture. All branches of solutions of the ren…

The paper considers a Jacobi matrix generated by a balanced measure of a hyperbolic polynomial map. Bellissard's conjecture. This matrix should have an extremely strong periodicity…

Jacobi-matrix counterpart conjecture. Corresponding versions of Theorems 3 and 4 should hold for Jacobi matrices.

Let $T(z)$) be an expanding polynomial of degree $d$ with a real Julia set $E$, where $E\subset[-\xi,\xi]$ and $T^{-1}:[-\xi,\xi]\to[-\xi,\xi]$. For $x\in[-\xi,\xi]$, define the Ja…

Contraction conjecture. All branches of solutions of the renormalization equation are contractions.

Degree-eight distinct-diagonal test case. Every reducible connected degree-eight spectral curve is obtained from a constant branch or from reversal symmetry. In particular, there i…

Primitive codimension-growth conjecture. One has

No-connected-hypersurfaces conjecture. For every $n\ge4$, every codimension-one component of the reducible locus is one of the cutting hyperplanes $b_i=0$. Equivalently, the reduci…

Amended reducibility conjecture. Every reducible factorization of $\chi_n$ is expected to be obtained by iterating the following operations on connected blocks: cutting the chain a…

Let $J$ be a $7 \times 7$ Jacobi matrix with symmetric spectrum that realizes perfect state transfer (PST). Its early state exclusion (ESE) property means that no state is excluded…

Tangent-measure conjecture. For every … $\mu$ has a unique tangent measure at $$ $\xi$, and this tangent measure is Lebesgue measure.

Let $p\geq 3$, and let $\lambda_n$ and $\mu_n$ be birth and death rates that are polynomials of degree $p$ with … … Assume that $1<C-D<p-1$. Valent's conjecture. The order of the c…

Let $J$ be a reflectionless Jacobi matrix in the wider class for which the Borg–Hochstadt theorem is known, and suppose that its spectrum has finitely many gaps. Let Theorem … rema…

The paper considers natural boundaries of power series and their analogy with absolutely continuous spectra of Jacobi matrices. Spectral-theoretic stability conjecture. There are n…

Let $K(\gamma)$ be the Cantor-type set determined by $\gamma$, and let $H_{\mu_{K(\gamma)}}$ have recurrence-coefficient sequence $(a_n)_{n=1}^\infty$. An almost periodic sequence…

Let $\gamma=(\gamma_k)_{k=1}^\infty$ be the parameter sequence defining $K(\gamma)$, set $\varepsilon_k:=1-4\gamma_k$ for $k\in\mathbb{N}$, and let $(a_n)_{n=1}^\infty$ be the recu…

Let $\gamma=(\gamma_k)_{k=1}^\infty$ be the parameter sequence defining $K(\gamma)$, and let $\mu_{K(\gamma)}$ be its associated measure with recurrence coefficients…

A self-similar measure is a measure generated by an iterated self-similar construction, including the Cantor measure. A Jacobi matrix is asymptotically almost periodic when its rec…

ACET conjecture. For all Jacobi matrices $J$,

Let $J$ be a Jacobi matrix, let $T_E(n)$ be the transfer matrix from the first site to site $n$, and let $\Sigma_{\rm ac}(J)$ denote the essential support of the absolutely continu…

Let $J$ be a Jacobi matrix, let $u_E$ denote its generalized eigenfunction at energy $E$, and let $u{rm ac}$ be the absolutely continuous part of the associated spectral measure.…

The paper considers the Hermitian analytic vector bundle ${\mathfrak{K}}_z$ associated with an operator outside its spectrum and the model described in Section 2. Hermitian analyti…

Let a Jacobi matrix have coefficients $q_n$ and $\rho_n^2$ such that both $q_n$ and $\rho_n^2$ are polynomials. The order is the growth invariant associated with the corresponding…