13 problems

Let $f$ and $g$ be two tent maps on $[0,1]$, with $f\neq g$, and write … for the inverse limit space with bonding map $f$. Ingram's conjecture. The inverse limit spaces … are not h…

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

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 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 $q_k$ be the limiting cluster probabilities defined by … Vanishing cluster-probability conjecture. We conjecture that $q_k=0$ for all $k$. The contrary could happen if the ball…

Let $\mathfrak{X}$ be the class of maps under consideration, and for each $f\in\mathfrak{X}$ let the acip denote its absolutely continuous invariant probability measure, with densi…

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

Let $h$ be the entropy function for the quadratic family, and let … For a function $f$ on an interval, being $\beta$-Hölder continuous means that there is a constant $C>0$ such tha…

Generic slow-recurrence conjecture. This conclusion should also hold when $t_0$ satisfies a more generic slow recurrence condition, such as polynomial recurrence, possibly after in…

Slow-recurrence conjecture. Corollary should hold for every Collet–Eckmann parameter $t_0$ with sufficiently slow recurrence.

Slow-recurrence conjecture. Theorem should hold for $\Delta$ equal to the set of all sufficiently slowly recurrent parameters, including parameters with polynomial recurrence of ex…

Feigenbaum's transversality conjecture. The manifolds $\Sigma_n$ intersect $W^u(\Phi,\mathcal{R})$ transversally.

Feigenbaum's conjectures. The operator $\mathcal{R}$ has a fixed point $\Phi$ and is differentiable in a neighborhood of $\Phi$. Furthermore, the spectrum of $D\mathcal{R}(\Phi)$ h…