In this paper we extend the novel approach to discrete Painlevé equations initiated in our previous work [2]. A classification scheme for discrete Painlevé equations proposed by Sakai interprets them as birational isomorphisms between generalized Halphen surfaces (surfaces obtained from ℙ^1×ℙ^1 by blowing up at eight points). Sakai's classification is thus based on the classification of generalized Halphen surfaces. In our scheme, the family of generalized Halphen surfaces is replaced by a pencil of quadrics in ℙ^3. A discrete Painlevé equation is viewed as an autonomous transformation of ℙ^3 that preserves the pencil and maps each quadric of the pencil to a different one. Thus, our scheme is based on the classification of pencils of quadrics in ℙ^3. Compared to our previous work, here we consider a technically more demanding case where the characteristic polynomial Δ(λ) of the pencil of quadrics is not a complete square. As a consequence, traversing the pencil via a 3D Painlevé map corresponds to a translation on the universal cover of the Riemann surface of √(Δ(λ)), rather than to a Möbius transformation of the pencil parameter λ as in [2].
We prove that the classification of real-analytic vector fields on the two-torus up to orbital topological equivalence does not admit a complete numerical invariant that is a Borel function. Moreover, smooth vector fields that are difficult to classify appear in generic smooth 7-parameter families. In dimension 2, this improves the recent result of Gorodetski and Foreman (arXiv:2206.09322) for non-classifiability of smooth diffeomorphisms up to continuous conjugacy.
A Morse-Bott volume form on a manifold is a top-degree form which vanishes along a non-degenerate critical submanifold. We prove that two such forms are diffeomorphic (by a diffeomorphism fixed on the submanifold) provided that their relative cohomology classes with respect to the submanifold coincide. For a zero submanifold of codimension at least 2, this means that two Morse-Bott volume forms with the same zero set are diffeomorphic if and only if they have equal total volumes. We show how "Moser's trick" for establishing equivalence of non-degenerate volume forms can be adapted to this setting.
We show that the Yajima-Oikawa (YO) equations, a model of short wave-long wave interaction, arise from a simple geometric flow on curves in the 3-dimensional sphere $S^3$ that are transverse to the standard contact structure. For the family of periodic plane wave solutions of the YO equations studied by Wright, we construct the associated transverse curves, derive their closure condition, and exhibit several examples with non-trivial topology.
In projective differential geometry, pre-geodesics of an affine connection are curves that are geodesics after a reparametrization (the analogous concept in Kähler geometry is known as J-planar curves). Similarly, dual-geodesics on a Riemannian manifold are curves along which the 1-forms associated to the velocity are preserved after a reparametrization. Superintegrable systems are Hamiltonian systems with a large number of independent constants of the motion. They are said to be second order if the constants of the motion can be chosen to be quadratic polynomials in the momenta. Famous examples include the Kepler-Coulomb and the harmonic oscillator systems, which are foundational models in the Sciences. We show that certain torsion-free affine connections which are naturally associated to certain second order superintegrable systems share the same dual-geodesics.
The deep diagonal map T_k acts on planar polygons by connecting the k-th diagonals and intersecting them successively. The map T_2 is the pentagram map, and T_k is a generalization. We study the action of T_k on two subsets of the so-called twisted polygons, which we term type-α and type-β k-spirals. For k ≥ 2, T_k preserves both types of k-spirals. In particular, we show that for k = 2 and k = 3, both types of k-spirals have precompact forward and backward T_k-orbits modulo projective transformations. We derive a rational formula for T_3, which generalizes the y-variables transformation formula of the corresponding quiver mutation by M. Glick and P. Pylyavskyy. We also present four algebraic invariants of T_3. These special orbits in the moduli space are partitioned into cells of a 3 × 3 tic-tac-toe grid. This establishes the action of T_k on k-spirals as a geometric generalization of T_2 on convex polygons.
A linear differential operator T=Q(z)d/dz+P(z) with polynomial coefficients defines a continuous family of Hutchinson operators when acting on the space of positive powers of linear forms. In this context, T has a unique minimal Hutchinson-invariant set M_CH^T in the complex plane. Using a geometric interpretation of its boundary in terms of envelopes of certain families of rays, we subdivide this boundary into local and global arcs (the former being portions of integral curves of the rational vector field Q(z)/P(z)∂_z), and singular points of different types which we classify below. The latter decomposition of the boundary of M_CH^T is largely determined by its intersection with the plane algebraic curve formed by the inflection points of trajectories of the field Q(z)/P(z)∂_z. We provide an upper bound for the number of local arcs in terms of degrees of P and Q. As an application of our classification, we obtain a number of global geometric properties of minimal Hutchinson-invariant sets.
We present conjectures on the scattering terms of cluster scattering diagrams of rank 2, supported by significant computational evidence.
We show that if $n$ functionally independent commutative quadratic in momenta integrals for the geodesic flow of a Riemannian or pseudo-Riemannian metric on an $n$-dimensional manifold are simultaneously diagonalisable at the tangent space to every point, then they come from the St\"ackel construction, so the metric admits orthogonal separation of variables.
This is the first in a series of papers where scissor congruence and K-theoretical invariants are related to cobordism groups of foams in various dimensions. A model example is provided where the cobordism group of weighted one-foams is identified, via the Sah-Arnoux-Fathi invariant, with the first homology of the group of interval exchange automorphisms and with the Zakharevich first K-group of the corresponding assembler. Several variations on this cobordism group are computed as well.
We study the (k+1,k) diagonal map for k=2,3,4,.... We call this map Δ_k. The map Δ_1 is the pentagram map and Δ_k is a generalization. Δ_k does not preserve convexity, but we prove that Δ_k preserves a subset B_k of certain star-shaped polygons which we call k-birds. The action of Δ_k on B_k seems similar to the action of Δ_1 on the space of convex polygons. We show that some classic geometric results about Δ_1 generalize to this setting.
We describe a model $\mathcal{M}_3^{comb}$ for the boundary of the connectedness locus $\mathcal{M}^{sy}_3$ of the parameter space of cubic symmetric polynomials $p_c(z)=z^3-3c^2z$. We show that there exists a monotone continuous function $\pi:\partial \mathcal{M}_c^{sy}\to \mathcal{M}_3^{comb}$ which is a homeomorphism if $\mathcal{M}^{sy}_3$ is locally connected.
F. Schweiger introduced the fibred system in , to unify and generalize many known continued fraction algorithms. An advantage of a fibred system is that it often provides a systematic construction of absolutely continuous invariant density. In this paper, we define and study the self-duality of fibred systems, a strong symmetry of a given system. We show that explicit algebraic self-duality holds in many systems and presents a curious system with "partial" self-duality.
We present a short, hopefully pedagogical construction of the field and ring of Witt vectors. It uses a natural binary operation on polynomials of one variable, which we call convolution.
Integer geometry on a plane deals with objects whose vertices are points in ℤ^2. The congruence relation is provided by all affine transformations preserving the lattice ℤ^2. In this paper we study circumscribed circles in integer geometry. We introduce the notions of integer and rational circumscribed circles of integer sets. We determine the conditions for a finite integer set to admit an integer circumscribed circle and describe the spectra of radii for integer and rational circumscribed circles.
We introduce a new class of billiard-like system, “bouncing outer billiards" which are 3-dimensional cousins of outer billiards of Neumann and Moser. We prove that bouncing outer billiard on a smooth convex body has at least four 1-parameter families of fixed points. We also fully describe dynamics of bouncing outer billiard on a line segment. Finally we carry out numerical experiments suggesting very complicated (non-ergodic) behavior for several shapes including the square and an ellipse.
