36 problems

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…

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 $D\subset\mathbb R^d$ 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 the sped-up Molecule map be the map associated with the Molecule Renormalization, as in the source's Figure, and let a renormalization be parametrized by this map. Molecule ren…

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…

For $c\in\partial\mathcal{M}$ and $\eta>0$, let $p_c(z)=z^2+c$, and let $\mathcal{R}_\eta p_c$ be the rescaled first-return renormalization defined by … where…

Uniform critical-orbit comparison conjecture. There exists a uniform constant $C_{}>0$ such that, for all $k\geq 0$,

Universal bifurcation diagram conjecture. There exists a homotopy

Let $f_{a_*} \in \mathcal Q$ be an infinitely renormalizable quadratic polynomial with bounded type combinatorics. For $a,b \in \mathbb R$, let $F_{a,b}(x,y)=(x^2+a-by,x)$ be a Hén…

Let $\mathcal{R}$ be a compact analytic renormalization operator with a hyperbolic fixed point, and let $\mathcal{W}^u_{\mathrm{loc}}$ be its local unstable manifold. Assume that e…

Let $d_0$ and $d_\infty$ be the local degrees of the critical fixed points at $0$ and $\infty$, respectively, and let $\{F_c\}_{c\in\mathbb{C}^*}$ be the one-parameter family of ra…

Let $f:\mathcal{H}_q\to\mathcal{H}_q$ be a critical quasicircle map with irrational rotation number, and let $\mathcal{N}$ be the Banach neighborhood of $f$ consisting of nearby un…

Self-similarity conjecture. For every stationary type irrational number $\theta=[0;N,N,N,\ldots]$, the non-escaping locus $\mathcal{M}_{d_0,d_\infty}$ is asymptotically self-simila…

Predictivity conjecture. (Effective) Quantum General Relativity, possibly coupled to matter from the Standard Model, is predictive.

Renormalisation hyperbolicity conjecture. The renormalisation operator $\mathcal{R}$, restricted to $\mathcal{E}_{\mathbf{c}}$, is hyperbolic when restricted to its attractor.

Rigidity conjecture. There is a full-measure set of decorated rotation numbers $\gamma$ such that, whenever $T_1$ and $T_2$ are circle diffeomorphisms with breaks having breaks of…

Let $f$ and $\tilde f$ be maps in the setting of the main theorems, and let $c$ be a critical point. Write $\omega(c)$ for the omega-limit set of $c$, and call it of bounded geomet…

Let $\mathcal{T}_w$ be the set of infinitely $w$-renormalizable Lorenz maps. A rigidity class of $f\in\mathcal{T}_w$ is the set of maps in $\mathcal{T}_w$ smoothly conjugate to $f$…

Let $\mathcal{T}_w$ be the set of infinitely $w$-renormalizable Lorenz maps, and let $\mathcal{L}$ denote the space of Lorenz maps. The Lorenz rigidity conjecture. The set…

Let $f$ be a Lorenz map, and let $w=(w_0,w_1)$ be a renormalization type. A map is infinitely $w$-renormalizable when every successive renormalization is $w$-renormalizable. Denote…

Let $R$ be the renormalisation transformation on point fields of concentration points and shocks, and suppose the density decays as $n^{-\alpha}$ for a critical exponent…

Strange-attractor abundance conjecture. For each natural number $n$ there exists an interval $I_n\subset (1/\sqrt{2},t_n)$ such that $\Lambda_t$ displays, at least, $2^{n-1}$ diffe…

Fix a topological class $\mathcal{T}$ of infinitely $(a,b)$-renormalizable Lorenz maps, with $a$ and $b$ sufficiently large. Let $\mathcal{U}\subset\mathcal{T}$ be the nonempty ope…