12 problems

Let $f$ be a mapping (neural network) with sufficiently high dimension $d$, let $U$ be a bifurcation chain set as in the hyperbolicity violation conjecture, and let $V$ be the chai…

Let $f$ be a rational map. It is structurally stable if, within a holomorphic family containing it, all sufficiently nearby maps are topologically conjugate to it; it is hyperbolic…

Let $F$ be a piecewise Möbius transformation with component transformations $f_k$, and let structural stability, hyperbolicity, and $b1$-expansion have their meanings defined in th…

Let $F$ be a piecewise Möbius transformation (PMT), and let structural stability, hyperbolicity, and $b1$-expansion have their meanings defined in the paper. Structural-stability c…

Let $f$ be an endomorphism, with inverse limit space $\overleftarrow M_f$ and natural extension $\overleftarrow f$. It is $C^1$-inverse limit stable if every $C^1$ perturbation…

Let $f$ be a covering map. It satisfies axiom A and the strong transversality condition when its non-wandering set is locally maximal, its non-wandering set is the union of a hyper…

Let $1\le r\le\infty$ or let $\mathcal C=\mathcal H$, and let $\mathcal C$ be the corresponding category of maps. A $\mathcal C$-diffeomorphism is structurally stable if every…

Consider complex rational maps of the sphere, in particular quadratic maps. Such a map satisfies axiom A when its non-wandering set is a finite union of basic pieces, and its criti…

Let $1\le r\le\infty$ or let $\mathcal C=\mathcal H$, and let $\mathcal C$ be the corresponding category of maps. A $\mathcal C$-diffeomorphism is structurally stable if every…

Good-family conjecture for two parameters. Generic two-parameter families of vector fields in the two-sphere are good.

Good-family conjecture. Generic one-parameter families of vector fields in the two-sphere are good.

A star system is a dynamical system for which all sufficiently small perturbations have only hyperbolic closed orbits; its non-wandering set is denoted by $\Omega$. Star-system hyp…