18 problems

Let an AM-system be a pair of increasing affine interval homeomorphisms $\{f_-,f_+\}$ of $[0,1]$, chosen with probabilities $p_-$ and $p_+$, and let $\mu$ be its stationary measure…

Let $g:[0,1]\rightarrow [0,1]$ be a mixing map. For any open set $U\subseteq[0,1]$, suppose there exists an integer $n\in\mathbb N$ such that $g^n(U)=[0,1]$. Let $M_g$ denote the s…

A continuous map $f$ of a compact interval is $2$-fold if every point has at least two preimages under $f$. For a continuous map $f$ of a compact interval that is $2$-fold on the w…

Conjecture on DC2 pairs for interval nads. For a nads on an interval, the limit map must have a DC2 pair.

Let $f_0$ be a piecewise monotone interval map, and let $(g_n)$ be the associated sequence of constant-slope maps produced by the $Theta_{CS}$ tower algorithm. Write $s(g)$ for the…

Let $f:[0,1]\to[0,1]$ be continuously differentiable, and let $s\alpha(x)$ denote a special $\alpha$-limit set. Differentiable-map conjecture. Every special $\alpha$-limit set of…

Let $f:[0,1]\to[0,1]$ be continuous, let $x\in[0,1]$, and let $s\alpha(x)$ be the special $\alpha$-limit set of $x$. Period-three coexistence conjecture. If $s\alpha(x)$ contains a…

Let $f:[0,1]\to[0,1]$ be continuous. A special $\alpha$-limit set $s\alpha(x)$ is the union of the accumulation-point sets of all backward orbit branches of $x$. Kolyada–Misiurewic…

Let the boundary of the mappings with positive topological entropy be regarded as a parameter space of interval mappings, and let a cell mean a connected cell in this boundary. Bou…

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

Conjecture. If

Let $I=[0,1]$ and let $C(I,I)$ denote the space of continuous maps from $I$ to itself. Equip $C(I,I)\times C(I,I)$ with its usual product topology, and call a pair $(f,g)$ chaotic…

A coding space associated to a piecewise monotonic interval map is the symbolic space obtained by coding the map's orbits according to its monotonicity partition. A potential is hy…

Let $\{f_t\}_{t\in[0,1]}$ be a smooth family of interval maps $f_t\colon[0,1]\to[0,1]$. The family has robust chaos when there is no parameter $t\in[0,1]$ for which $f_t$ has a per…

Dissipativity conjecture. The product system $(I^2,f_2,\lambda_2)$ is dissipative.

Świątek's conjecture. The $RCE$ condition is topologically invariant for multimodal analytic maps with negative Schwarzian derivative.

Świątek's conjecture. The $RCE$ condition is topologically invariant for multimodal analytic maps with negative Schwarzian derivative: if two such maps are topologically conjugate,…

Derivative-growth insufficiency conjecture. It is not possible to obtain a general result guaranteeing the existence of an acip for interval maps with both critical points and sing…