19 problems

Feigenbaum's universality conjecture. The period-doubling route to chaos in unimodal maps is characterized by universal constants $elta$ and $lpha$.

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 $(g_k^*,f_k^*)$ be the C-type renormalisation two-cycle, and let $R^2$ denote the second iterate of the renormalisation operator. Consider the relevant parts of the spectrum of…

Consider the renormalisation-group dynamics for non-invertible maps with a critical point of continuously varying degree $d$. A C-type two-cycle is a period-two orbit of the renorm…

Let $R$ be the Cvitanović–Feigenbaum operator and let $g^*$ be a nontrivial stationary point satisfying $R(g^*)=g^*$. Restrict attention to the dynamically relevant eigenvalues of…

Let $q_{2,\mu}(x)=1-\mu x^2$ be the quadratic family, and let $\mu_n$ denote the parameters at which successive period-doubling bifurcations occur. Write $\mu_\infty$ for their lim…

Consider the symbol sequences of periodic orbits of the map in the paper, with a period-doubled orbit obtained from a parent periodic orbit. When the sequences are ordered appropri…

Period-doubling analog of the FLB slope theorem. The relation above should hold for period-doubling bifurcations of rotational pendula.

Let $\rho$ be a positive integer, let $\bar{\lambda}$ be the average parameter value, and let $p_i(\bar{\lambda})$ for $1\leq i\leq 2^\rho$ denote the stable period-$2^\rho$ points…

Coullet–Tresser–Feigenbaum conjecture. Unimodal maps in the boundary of the zero entropy class should be limits of a period-doubling cascade with a universal metric property, under…

Let $J\in\mathbf{F}_4\backslash\{0,1\}$, and let $\mathbf{u}$ be the $(1,J)$-period-doubling sequence. Period-doubling Stieltjes transcendence conjecture. The Stieltjes continued f…

Non-universality of parameter bifurcations. The parameter scaling should satisfy

Boundary of Chaos Conjecture I. Every such $f$ with

In a renormalization-group description, distinguish relevant eigenvalues, singular-irrelevant eigenvalues associated with genuinely new universal corrections, and gauge-irrelevant…

Consider gauge eigenvalues arising from coordinate changes in a renormalization-group description, and universal eigenvalues associated with the renormalization group. Gauge-eigenv…

Let a critical point serve as a fixed point of a renormalization-group transformation, and distinguish gauge-irrelevant corrections arising from redundant coordinate changes from s…

Let $T$ be the period-doubling operator … and let $g$ be a solution of the universality equation $g=T(g)$. Write $dT(g)$ for the Fréchet derivative of $T$ at $g$, and let $\delta$…

Let $C$ be a cascade in an unbounded component, and suppose that outside a compact subset all orbits of $C$ have the same period; call this common period the stem period. Let the p…

Let $f:\mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n$ be a continuous family and let $U \subset \mathbb{R} \times \mathbb{R}^n$ be such that the periodic orbits in $U$ are contai…