134 problems

In perturbative $phi^4$ theory, consider the beta function renormalized in the minimal-subtraction (MS) scheme, and call a Feynman graph primitive if it has no subdivergences. Prim…

Kontsevich's conjecture. Every graph hypersurface $\hat X_\Gamma$ should be a mixed Tate motive.

Let $D\subset\mathbb R^d$ be the domain of a renormalization class $\mathcal W$, let $(c_k)_{k\geq0}$ be positive migration constants, and define $s_n=\sum_{k=0}^{n-1}c_k^{-1}$ and…

Let $\overline{\Theta}$ be the parameter space of the cylinder renormalization considered in the source, and let $\mathcal{R}_{\operatorname{cyl}}$ denote a cylinder renormalizatio…

Renormalization is a microscope describing the dynamics of a system on a smaller scale, and classical systems such as unimodal maps and critical circle maps exhibit rigidity and pa…

Let the period-doubling renormalization operator act on a space of quadratic unimodal maps. A Feigenbaum–Coullet–Tresser conjecture. It has a unique fixed point, this fixed point i…

Let $b$ be a bounded, nearly incompressible vector field in $\operatorname{BV}_{\rm loc}(\mathbb{R} \times \mathbb{R}^{d})$. A bounded, nearly incompressible vector field…

Let $\mathcal{B}_1'$ be the space in which the linearized renormalization dynamics is considered, let $L_1$ be the map given by the referenced equation, and let…

Let $\Omega=\Omega_{\gamma,\tau}$ be the set of Diophantine rotation numbers, let $\mathcal{B}$ be the Banach space of maps, and let $\mathcal{D}(\mathcal{T})$ be the domain of the…

Let $R$ denote a critical invariant circle, and let $\kappa(R)$ denote its regularity. A numerical characteristic is universal when it takes the same value on an open set of functi…

Consider the renormalization operator acting on the class of twist maps under discussion, and call a property universal when it holds on an open set of functions. A fixed point is…

Let $\rho$in$(0,1)$ be a Diophantine number, and let $F$ be a twist map of the form given in the source. Denote by $\Lambda(\rho)$ the parameter threshold at which the invariant ci…

A critical circle mapping is represented by a commuting pair, and renormalization acts on the infinite-dimensional space of such commuting pairs while inducing the Gauss map on rot…

Secondary limb conjecture. The secondary limb condition implies a priori bounds.

Folklore conjecture. The map $f$ can be approximated by polynomial maps with positive entropy and by polynomial maps with finitely many periodic orbits.

Universality conjecture. When $\nu>2$, the parameter scalings $\delta_n$ tend to a universal limit depending only on $\nu$.

Let $R$ be the renormalization operator on once-renormalizable symmetric analytic unimodal mappings, and let $\rho=(w_{1}*w_{2}*\cdots*w_{k})^{*\infty}$ be a periodic kneading sequ…

Let $f:[-1,1]\rightarrow[-1,1]$ be a symmetric analytic folding mapping with a unique non-degenerate critical point, and let $R$ be the period-doubling operator defined by restrict…

Let the period-doubling operator be the nonlinear operator acting on the space of unimodal maps that describes period-doubling renormalization. Feigenbaum–Coullet–Tresser conjectur…

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…

Let $\theta^*=(\sqrt{5}-1)/2$ be the inverse golden mean, and consider holomorphic maps with a fixed point at the origin having multiplier $e^{2\pi i\theta^*}$. For such a map, let…

Let $D\subset\mathbb R^d$ be open, bounded, and convex, let ${\cal W}$ be a renormalization class on $\overline D$, and suppose that the asymptotic fixed-point equation in the zero…

Let ${\cal W}^{0,1}_{\rm cat}$ be the catalytic renormalization class, and let $p^\ast_{0,1,\gamma^\ast}$ denote the corresponding solution parameter for asymptotic ratio…

Fibonacci renormalization conjecture. For every $l>2$ there exists a unique fixed point of the generalized renormalization with the Fibonacci combinatorics.