10 problems

Let a stable $p$-cycle belong to an invariant closed curve, and consider the three eigendirections of the Jacobian matrix of the $p$th iterate at each point of the cycle. Assume th…

Let $P(x,y)$ and $Q(x,y)$ define a polynomial system … of degree $d$, and suppose the system has an invariant algebraic curve whose degree is sufficiently high, with the bound depe…

Boundary invariant-curve persistence conjecture. Invariant curves persist close to the boundary for arbitrarily large values of $T$.

A rotational invariant curve is a continuous curve $\gamma$ homotopic to the line $y=0$ and invariant under the map, so that $T(\gamma)=\gamma$. For an unperturbed family of interv…

Consider a perturbed family of symmetric $d$--interval exchange maps and call a continuous curve homotopic to the line $y=0$ that is invariant under the map a rotational invariant…

Let an invariant curve with frequency $\omega$ be denoted by $\mathcal C(\omega)$, and let $\mathcal P\left(\frac{p_j}{q_j}\right)$ be periodic orbits with frequencies…

Let $\epsilon\in (0,1/2)$ and $mu>1$, and let $F_{\mu,\epsilon}$ be the coupled logistic map. Let $\mu_1(\epsilon)$ be the parameter threshold and let $\Gamma$ be a curve as in The…

Let $X$ be a smooth germ of vector field at the origin of $\mathbb{R}^{n}$ with an algebraically isolated singularity, and suppose it has a formal invariant curve…

Let $(\alpha,\beta)$ be the pair consisting of the base rotation and the prescribed fibered rotation number for an analytic family of fibered holomorphic transformations. Let…

Let $\alpha$ be the irrational rotation on the base and let the fibered rotation number be prescribed. Let $\mathbb{B}_1$ denote the Brjuno-type arithmetic condition imposed on the…