In this paper, we provide a complete answer to the question posed in its title. For all closed surfaces for which this question has not yet been resolved, namely, closed nonorientable surfaces of odd genus greater than 5, a series of examples of pseudo-Anosov homeomorphisms are constructed. Each of these homeomorphisms is defined by means of the so-called code, which implies using the construction of a band surface. Combined with previously obtained results, this allows us to fill in the gap and to formulate a general result: pseudo-Anosov homeomorphisms exist on a closed nonorientable surface if and only if its genus is at least 4.

The triple linkage of Thurston and Weeks exhibits Anosov behavior for certain parameter values, which can be shown by examining the Gauss curvature of the configuration space equipped with the metric induced by kinetic energy. In this paper, we consider a spatial linkage that can be viewed as a conversion of the triple linkage. We show that the configuration space asymptotically becomes a Riemannian submanifold of the four-dimensional torus $\mathbb{T}^4$ taking the limit of the parameters. Through verified numerical computation, we demonstrate that the asymptotic configuration space has negative curvature, and hence that for parameters close to the limit the linkage is Anosov by structural stability of an Anosov flow.

We study central configurations when the set of positions is symmetric. We use a theorem (proved in [33]) that allows us to use the representation theory of finite groups to explore the symmetry properties of equations for central configurations. This approach simplifies equations for central configurations by considering arbitrary numbers of bodies, symmetry groups, and dimensions. We discuss how to use this theorem to obtain a more refined decomposition of the equations than that given before. The decomposition presented here uses the symmetry-adapted basis method.
As an application, we give a complete description of the existence and which masses are possible for central configurations of two nested regular tetrahedra, two nested regular octahedrons, and two nested regular cubes. To do this, we employ some methods of rational parameterizations and isolation of zeros of multivariate polynomials. The decomposition obtained allows symbolic calculations to be used to study the expressions. In this way, we summarized the same discussions of works done in [11, 25, 45] and extended them by completing the discussion on the cube case, in the inverse and direct problems.

For any given positive masses, we prove that the number of $\mathbf{S}$-balanced configurations of four bodies in the plane is finite up to similitudes, provided that the symmetric matrix $\mathbf{S}$ is sufficiently close to a numerical matrix. To establish this result, we utilize singular sequences to analyze the possible degenerate algebraic varieties defined by $\mathbf{S}$-balanced configurations. We derive all potential singular diagrams, encompassing both equal-order and non-equal-order cases. In the equal-order case, we obtain the necessary mass equations, while for $\mathbf{S}$ approaching the identity matrix, we demonstrate the absence of non-equal-order singular sequences, thereby rigorously rule out all non-generic scenarios. Furthermore, we extend this conclusion to the five-body scenario.

We consider a new type of billiard trajectories of point-particles moving freely in $d$-dimensional space until collision. At collisions of two or more particles we have scattering at a subspace with a co-dimension of at least $d$ preserving only the total momentum of the colliding particles, but the internal direction and kinetic energy can change arbitrary and even an exchange of mass is possible. Hence the future of the trajectory is nondeterministic.
Motivated by questions concerning non-collision singularities in the $n$-body problem, for which these systems might serve as approximations, we are mainly interested in the asymptotic growth rate for trajectories that have infinitely many collisions and are expanding. For this case we provide as our main results exponential lower bounds for the diameter and the kinetic energy of the system in the number of so-called chain-closing collisions.

We consider the Novikov problem, namely, the problem of describing the global geometry of level lines of quasiperiodic functions on a plane, for a special class of twodimensional potentials. Potentials of this class play an important role in the physics of twodimensional systems and are defined by superpositions of two periodic potentials with the same rotational symmetry. For different orientations of the periods of the original potentials, the resulting potential can have 4 quasiperiodes or be periodic. The main result of the paper is a proof that quasiperiodic potentials of this class can have open level lines at only one energy level. This property brings these potentials closer to random potentials on a plane, as well as to potentials with 3 quasiperiodes possessing “chaotic” level lines. The paper also presents an estimate for the energy interval containing open level lines of periodic potentials arising at “magic” angles of rotation of the original potentials relative to each other.