26 problems

Abdenur–Diaz conjecture. A $C^1$-generic diffeomorphism with the shadowing property is structurally stable.

Let $M$ be a locally compact Riemannian manifold, let $X\in\mathfrak{X}(M)$ be a smooth vector field, and let $\phi_t$ be its flow. Write $\Omega(\phi_t)$ for the nonwandering set,…

Let a dynamical system be $C^r$ stable when its limit set is hyperbolic and its stable and unstable manifolds meet transversally at every point. The stability conjecture asserts th…

Let $X$ denote the relevant space of dynamical systems. The structural stability density conjecture asserts that structurally stable dynamical systems are dense among all dynamical…

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…

A polynomial $f$ is structurally stable if its dynamics are stable under small perturbations in the relevant parameter space. Density conjecture. The set of structurally stable pol…

Let a dynamical system be given, with structural stability, Axiom A, and the transversality condition understood in their standard dynamical-systems sense. Structural stability con…

Let $f$ be a transcendental entire map in the class $S_q$, meaning that the set of singularities of $f^{-1}$ contains at most $q$ points. Let $M_f$ be the set of all entire maps to…

A vector field is called structurally stable if its qualitative dynamics is unchanged under sufficiently small perturbations, and a Morse–Smale system is a vector field with finite…

Smale's conjecture. Typical dissipative dynamical systems should have dynamics reduced to a finite number of hyperbolic periodic orbits and, in particular, should be structurally s…

Instability conjecture. A rational map $R$ admitting a pseudoconformal measure with negative exponent must be unstable.

Let a weakly stable shock wave arise in isentropic elastodynamics. Weak-shock structural stability conjecture. Weakly stable shock waves in isentropic elastodynamics are structural…

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…

$C^{1}$-structural stability conjecture. Every $C^{1}$-structurally stable geodesic flow is an Anosov flow.

Generic structural stability conjecture. For a generic 5-tuple $(f,L,\psi,\phi,\bar x)\in\mathcal{X}$, the mean field game has finitely many solutions, all of which are structurall…

Consider the system $x^\Delta=V(t,x)$ on a time scale ${\mathbb T}$, satisfying the stated boundedness, derivative, uniform-continuity, and forward-uniqueness conditions. Let…

The paper considers non-degenerate singular orbits of real-analytic integrable systems, including the hyperbolic, focus-focus, and other singularities mentioned in the preceding di…

Axiom A characterization conjecture. Ω-stability is equivalent to Smale's Axiom A and the no-cycle condition, while structural stability is equivalent to Smale's Axiom A and the st…

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 $1\le r\le\infty$, and let a $C^r$-diffeomorphism be weakly stable when it remains in the weak-stability class under the relevant perturbations. A diffeomorphism satisfies axio…

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.