Arnold Mathematical Journal最新文献

The billiard in an ellipse has a conserved quantity, the Joachimsthal integral. We show that the existence of such an integral characterizes conics. We extend this result to the spherical and hyperbolic geometries and to higher dimensions. We connect the existence of Joachimsthal integral with the Poritsky property, a property of billiard curves, called so after H. Poritsky whose important paper Poritsky (Ann Math 51:446–470, 1950) was one of the early studies of the billiard problem.

Given a system of analytic functions and an approximate zero, we introduce inflation to transform this system into one with a regular quadratic zero. This leads to a method for isolating a cluster of zeros of the given system.

In 1987, Yomdin proved a lemma on smooth parametrizations of semialgebraic sets as part of his solution of Shub’s entropy conjecture for (C^infty ) maps. The statement was further refined by Gromov, producing what is now known as the Yomdin–Gromov algebraic lemma. Several complete proofs based on Gromov’s sketch have appeared in the literature, but these have been considerably more complicated than Gromov’s original presentation due to some technical issues. In this note, we give a proof that closely follows Gromov’s original presentation. We prove a somewhat stronger statement, where the parametrizing maps are guaranteed to be cellular. It turns out that this additional restriction, along with some elementary lemmas on differentiable functions in o-minimal structures, allows the induction to be carried out without technical difficulties.

Dynamical billiards, or the behavior of a particle traveling in a planar region D undergoing elastic collisions with the boundary has been extensively studied and is used to model many physical phenomena such as a Boltzmann gas. Of particular interest are the dispersing billiards, where D consists of a union of finitely many open convex regions. These billiard flows are known to be ergodic and to possess the K-property. However, Turaev and Rom-Kedar proved that for dispersing systems permitting singular periodic orbits, there exists a family of smooth Hamiltonian flows with regions of stability near such orbits, converging to the billiard flow. They conjecture that systems possessing such singular periodic orbits are dense in the space of all dispersing billiard systems and remark that if this conjecture is true then every dispersing billiard system is arbitrarily close to a non-ergodic smooth Hamiltonian flow with regions of stability [6]. We present a partial solution to this conjecture by showing that any system with a near-singular periodic orbit satisfying certain conditions can be perturbed to a system that permits a singular periodic orbit. We comment on the assumptions of our theorem that must be removed to prove the conjecture of Turaev and Rom-Kedar.

An approach to interpolation of compact subsets of ({{mathbb {C}}}^n), including Brunn–Minkowski type inequalities for the capacities of the interpolating sets, was developed in [8] by means of plurisubharmonic geodesics between relative extremal functions of the given sets. Here we show that a much better control can be achieved by means of the geodesics between weighted relative extremal functions. In particular, we establish convexity properties of the capacities that are stronger than those given by the Brunn–Minkowski inequalities.

We introduce 50+ new invariants manifested by the dynamic geometry of N-periodics in the Elliptic Billiard, detected with an experimental/interactive toolbox. These involve sums, products and ratios of distances, areas, angles, etc. Though curious in their manifestation, said invariants do all depend upon the two fundamental conserved quantities in the Elliptic Billiard: perimeter and Joachimsthal’s constant. Several proofs have already been contributed (references are provided); these have mainly relied on algebraic geometry. We very much welcome new proofs and contributions.

Inou and Shishikura provided a class of maps that is invariant by near-parabolic renormalization, and that has proved extremely useful in the study of the dynamics of quadratic polynomials. We provide here another construction, using more general arguments. This will allow to extend the range of applications to unicritical polynomials of all degrees.

This article is devoted to the geometry of billiard trajectories in a regular polygon and geodesics on the surface of a regular polyhedron. Main results are formulated as conjectures based on ample computer experimentation.