49 problems

A Hamiltonian diffeomorphism is a time-one map of a Hamiltonian flow on a symplectic manifold. For the standard symplectic $2$-torus, a periodic point is a point fixed by some posi…

A symplectic manifold is a manifold $(M,\omega)$ equipped with a symplectic form, and a Hamiltonian diffeomorphism is the time-one map of a Hamiltonian isotopy. A Hamiltonian diffe…

Bogatyi–Shavgulidze's conjecture. Every covering system has a periodic point of period at most $k$ in $I_1\cup\cdots\cup I_k$. This conjecture is the analogue, for covering systems…

Let ${\cal I}_n$ be the set of $n$-periodic points, let ${\cal I}'_n$ be the subset selected in the preceding construction, let $\Xi_m$ be the set of points considered when cycles…

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…

An invariant variety of periodic points is a variety determined uniquely by the invariants of a map, whose points are periodic with a fixed period. Invariant-variety conjecture. If…

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-…

Consider the discrete map $f\colon\mathbb{R}^4\to\mathbb{R}^4$ given by … where $\alpha<0$, $\tilde\alpha>0$, $\beta>0$, and $\gamma\ne0$. Periodic-point non-density conjecture. Th…

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 $n$ be a positive integer. Let $g_c:\mathbb T^n\to\mathbb T^n$ be a continuous map such that the characteristic polynomial of the induced map $(g_c)_{*1}$ is $t^n-c$, with…

Let $f$ be a holomorphic endomorphism of $cP^k$ of algebraic degree $d\geq 2$, and let $\mu$ be its equilibrium measure. For $n\geq 1$, let $P_n$ be the set of periodic points of p…

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

Uniform boundedness conjecture. As $f$ varies over generalized Hénon maps of degree $d$ over $K$, the quantity

For each $n\in\mathbb{N}$, let $g(n)$ denote the least universal upper bound such that every covering system with $n$ intervals has a periodic point of period at most $g(n)$. Bogat…

Let $X$ be a smooth complex projective variety of dimension $d$, and let $f:X\dashrightarrow X$ be a dominant rational map. Assume that $f$ is cohomologically hyperbolic, and defin…