14 problems

Let $a,b,c$ be a positive Pythagorean triple, so that … In the one-dimensional rotor-router model, start with $A=[-a,0]$, use the case $r=s=1$, and measure time in iterations. Prop…

Propp's conjecture. The quantity $\mu(t)$ is bounded independently of $t$. The paper proves this conjecture and further confines the limit points of the sequence $\mu$ to an interv…

Let $f$ and $g$ be topologically conjugate one-dimensional maps, and let $x$ be a point of $f$ that is not attracted to a periodic orbit. The upper pointwise Lyapunov exponent of…

Consider a generic one-parameter family of maps. Density conjecture. Maps with exponential decay of correlations are Lebesgue density points of other maps with uniform exponential…

Let $\Lambda$ be a compact forward-invariant set for a smooth map $f$. An invariant ergodic probability measure is hyperbolic if all its Lyapunov exponents are non-zero, and…

A multimodal Collet–Eckmann map is a multimodal interval map satisfying the Collet–Eckmann condition, and strong stochastic stability means stability of its statistical behaviour u…

Galatolo et al.'s conjecture. The bound $|\Gamma^{-}|\geq 0.0455$ is very close to the actual measure $|\Gamma^{-}|$. The conjecture concerns how much of the regular-parameter set…

Let $f$ be the one-dimensional map under consideration, and let a partition of its domain be chosen either uniformly or with smaller elements in regions where the derivative $f'$ i…

Finite-or-dense orbit conjecture. For every $x_0\in U$, the orbit $\mathcal O(f,x_0)$ is finite or dense in $U$.

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…

Consider analytic or smooth mappings of the interval whose periodic orbits are all hyperbolic. Sullivan's manifold conjecture. Their conjugacy classes are analytic or smooth connec…

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…

Periodic-attractor bound. There exists a neighbourhood ${\cal F} \subset C^3({\cal N})$ of $f$ such that for any $g\in {\cal F}$ the number of periodic attractors of $g$ is bounded…

Suppose a topological class of one-dimensional maps consists of maps with one critical value, and suppose that, for each critical exponent in a sequence tending to $\infty$, limits…