27 problems

Let $f:S^2\to S^2$ be an area-preserving diffeomorphism that is Moser generic. Write $\operatorname{Per}(f)$ for the set of periodic points of $f$. Density conjecture. The set of p…

Height boundedness conjecture. The set of isolated periodic points of $f$ has bounded height.

Let $f$ be a $C^r$ diffeomorphism, with $r>1$, on a compact surface $M$. Write $h_{\rm top}(f)$ for its topological entropy and $\lambda^{+}(f)$ for the relevant positive Lyapunov-…

Let $\operatorname{Per}(\varphi,\mathbb{P}^1(K))$ denote the set of periodic points of a morphism $\varphi:\mathbb{P}^1\to\mathbb{P}^1$ defined over a number field $K$. Periodic-po…

Malkiewich–Ponto's conjecture. The fiberwise Fuller trace

Let $f$ be a $C^\infty$ mapping, and let $P_f(n)$ denote the number of isolated points of period $n$. In a typical sufficiently parameterized family $(f_a)$, interpret “almost all”…

Let $\varphi:\mathbb{P}^1(\mathbb{Q})\to\mathbb{P}^1(\mathbb{Q})$ be a morphism of degree $d$, and let $\operatorname{Per}(\varphi,\mathbb{P}^1(\mathbb{Q}))$ denote its set of peri…

Let $n\geq 1$ and $d\geq 2$ be integers. Let $K/\mathbb{Q}$ be a number field of degree at most $n$, and let $\varphi:\mathbb{P}^1(K)\to\mathbb{P}^1(K)$ be a morphism of degree $d$…

Uniform bounded-period conjecture. There exists an integer $N(d)\ge1$ such that every such map $f$ admits a rigid repelling periodic point of period at most $N(d)$.

Let $F$ be a field of characteristic zero and let $S_F$ be a normal affine surface over $F$. Suppose that $f,g\rbrace\in\operatorname{Aut}(S_F)$ are loxodromic automorphisms. Stron…

Let $(a,b)\in\Delta$, let $f$ be the corresponding heterochaos baker map on $[0,1]^3$, and let $\mu_\alpha$ and $\mu_\beta$ be the two ergodic measures of maximal entropy. Denote b…

Let $f_a\colon\mathbb R\to\mathbb R$ be the skew tent map and let the coupled tent map have coupling parameter $\omega$. Its topological attractor is the closed invariant set conta…

Generic period-two conjecture. For all but finitely many $\delta\in\mathbb{Q}$, the $\mathbb{Q}$-rational periodic points of every such map, with $c\in\mathbb{Q}$, have period divi…

Period classification conjecture. Over $\mathbb{Q}$, the map has no point of period $N$ except possibly when

Bogatyi–Shavgulidze conjecture. For every covering system $(I_1,\ldots,I_k;f)$, there exists $x_0\in I_1\cup\cdots\cup I_k$ and an integer $l\leq k$ such that

Let $f \colon {\mathbb{CP}}^n \to {\mathbb{CP}}^n$, with $n \ge 2$, be a post-critically finite endomorphism of degree $d \ge 2$, and let $\lambda$ be an eigenvalue of $f$ along a…

Let $B$ be a base space supporting a family of endomorphisms $f\colon E\to E$ over $B$. Write $R_B(f)$ and $R_B(f^2)$ for the corresponding fiberwise Reidemeister traces, and let…

Let $B$ be a finite-dimensional cell complex, and let $E\to B$ be a smooth closed manifold bundle with fiber $M$ of dimension at least $3+\dim B$. Let $f\colon E\to E$ be a family…

Let $\phi$ be a quadratic rational function defined over $\mathbb{Q}$, and let a critical $3$-cycle be the set of iterates of a periodic critical point of period $3$. Period bound…

Let $\phi$ be a quadratic rational function defined over $\mathbb{Q}$ having a $\mathbb{Q}$-rational periodic critical point of period $2$. Period bound conjecture for a rational c…

Let $M$ be the manifold under consideration, let $r\in\{1,\ldots,\infty,\omega\}$, and write $\operatorname{Diff}^r(M)$ for the space of $C^r$ diffeomorphisms of $M$. A family…

Let $F$ be a field of characteristic $p$, let $\varphi$ be an analytic map from the open unit polydisk in $n$ dimensions to itself, and let $x=(0,\ldots,0)$ be a fixed point whose…

Nonexistence conjecture. There are no rational values $c$ such that $\phi_c$ has a 5-periodic point in $\mathbb{Q}_{\mathrm{quad}}$.

Period-set characterization conjecture. There exist sets $E_c,E_d\subset\mathbb N$, each a finite union of tails of the orderings $\leso{2}$ and $\leso{3}$, such that

Let $M$ be a connected closed manifold, let $f\colon M\to M$ be a self-map, and let $n\geq 2$. For the homotopy $n$-periodic point space, let $\ell_n^k(f)$ denote the relevant comp…