15 problems

Closed-orbit density conjecture. The density of closed orbit measures holds for any transitive piecewise monotonic map with positive topological entropy.

For a polynomial interval map with real critical points, let an isentropic level set be a set of maps having the same topological entropy. Milnor's monotonicity of entropy conjectu…

Let $f:I\to I$ be a continuous map of a compact interval, and let $x\in I$. The Kolyada–Misiurewicz–Snoha conjecture. The special $\alpha$-limit set $s\alpha_f(x)$ is closed.

Let $f$ be either a mixing, piecewise expanding, piecewise smooth unimodal interval map such that the critical point is not periodic, or a mixing smooth Collet-Eckmann unimodal int…

Let $C^r([0,1])$ denote the space of $C^r$ interval maps. For a generic $k$-parameter family $(f_t)_{t\in[-1,1]^k} \subset C^r([0,1])$, where $t\in[-1,1]^k$ is equipped with Lebesg…

Let $f:I\to I$ be a postcritically finite interval map of pseudo-Anosov type, and let $\operatorname{PC}(f)$ denote its set of postcritical points. Fix all but one postcritical orb…

Fix $m\geq 2$, and let $f_1,f_2\in\operatorname{PA}(m)$ be $m$-modal standard PCP zig-zags of pseudo-Anosov type. Let $\lambda_i$ be the growth rate of $f_i$, and let…

Milnor's conjecture. The isentropes are connected within families of polynomial interval maps whose critical points are all real.

Let $f_r$ be the $C^r$ interval transformation considered in the paper, with its associated countable-state Markov extension. Denote the topological entropy by $h_{\mathrm{top}}$ a…

Let $\Phi$ assign to each continuous topologically transitive piecewise monotone interval map its constant slope model, and fix a modality. Alsedà–Misiurewicz continuity conjecture…

Let $(X,f)$ be a dynamical system. It is minimally mean attractable when, for every $x\in X$, there is a minimal subsystem $(K_x,f)$ such that, for every $\varepsilon>0$, some…

Let $T_p:[0,1]\to[0,1]$ be the interval map with an indifferent fixed point at $x^*=0$ described above, where $p\in(0,1)$. For shrinking neighbourhoods $[0,\epsilon]$ of $x^*$, let…

Let $F_+$ be an essentially contracting random interval system over a Bernoulli base, and let $u$ be the measure on the interval such that the unique nontrivial invariant measure…

Let $(X,T)$ be a topological dynamical system and let $\mathcal{U}$ be an open cover of $X$. Define … A topological dynamical system $(X,T)$ is null when its maximal pattern entrop…

Let ${\mathcal F}$ be the set of quadratic maps with some growth along the critical orbit, and let the acip associated to each map in ${\mathcal F}$ be given by its absolutely cont…