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Maxim Arnold

Publications and source records attributed to Maxim Arnold.

16 recordsLinked to original sources
arXiv2026

Integrability of Cauchy problems for discrete conformal maps and circle patterns

A map from a square lattice to the Riemann sphere is called discrete conformal if the image of every elementary square is a harmonic quadrilateral. We prove that the initial value problem for discrete conformal maps with quasi-periodic boundary conditions is Liouville integrable. We also show that the image of the embedding of Schramm's orthogonal square grid circle patterns into the space of discrete conformal maps is the real part of a symplectic leaf. As a consequence, we obtain the integrability of the corresponding Cauchy problem for circle patterns.

math.DS↗
arXiv2025

Snell meets Fagnano. Path optimization through an imperfect mirror

The renowned Fagnano problem asks for the inscribed triangle of minimal perimeter within a given reference triangle. Equivalently, it seeks a billiard trajectory inside the triangle that closes after three reflections. In this note, we consider a modification of this classical problem: finding the inscribed triangle of minimal weighted perimeter -- or, equivalently, the periodic trajectory of a Snell billiard.

math.DS↗
arXiv2025

On Euclidean Algorithms for oriented linear Grassmanians

In this paper we study Euclidean algorithms and the corresponding continued fractions for oriented linear Grassmanians $G(k,n)$. We propose two algorithms: Maximal Element Elimination algorithm and Minimal Element Elimination algorithm. The first algorithm reduces the absolute maximal value of the Pl\"ucker coordinates; the algorithm works only in $G(2,n)$. The second algorithm eliminates the Pl\"ucker coordinate with the smallest absolute values, while all other coordinates may increase; the algorithm works for arbitrary $G(2,n)$. We discuss basic features of these algorithms and formulate several natural open questions for further studies.

math.NT↗
arXiv2024

A family of maps and a vector field on plane polygons

We study, theoretically and experimentally, a 1-parameter family of transformations and their limiting vector field on the space of plane polygons. These transformations are discrete analogs of completely integrable transformation on closed plane curves, known as the bicycle correspondence, that is a geometric realization of the B\"acklund transformation of the planar filament equation. For odd-gons, we construct a symplectic form on the quotient space by parallel translations and show that the transformations are symplectic, and the vector field is Hamiltonian. In the case of triangles, we prove complete integrability of the respective vector field and provide evidence for the conjecture that the transformations are integrable as well.

math.DS↗
arXiv2023

Symmedians as Hyperbolic Barycenters

The symmedian point of a triangle enjoys several geometric and optimality properties, which also serve to define it. We develop a new dynamical coordinatization of the symmedian, which naturally generalizes to other ideal hyperbolic polygons beyond triangles. We prove that in general this point still satisfies analogous geometric and optimality properties to those of the symmedian, making it into a hyperbolic barycenter. We initiate a study of moduli spaces of ideal polygons with fixed hyperbolic barycenter, and of some additional optimality properties of this point for harmonic (and sufficiently regular) ideal polygons.

math.DG↗
arXiv2022

On projective evolutes of polygons

The evolute of a curve is the envelope of its normals. In this note we consider a projectively natural discrete analog of this construction: we define projective perpendicular bisectors of the sides of a polygon in the projective plane, and study the map that sends a polygon to the new polygon formed by the projective perpendicular bisectors of its sides. We consider this map acting on the moduli space of projective polygons. We analyze the case of pentagons; the moduli space is 2-dimensional in this case. The second iteration of the map has one integral whose level curves are cubic curves, and the transformation on these level curves is conjugated to the map $x\mapsto -4x$ mod 1. We also present the results of an experimental study in the case of hexagons.

math.DS↗
arXiv2020

Remarks on Joachimsthal integral and Poritsky property

The billiard in an ellipse has an integral linear in momentum, 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 1950 paper was one of the early studies of the billiard problem.

math.DG↗
arXiv2018

Cross-ratio dynamics on ideal polygons

Two ideal polygons, $(p_1,\ldots,p_n)$ and $(q_1,\ldots,q_n)$, in the hyperbolic plane or in hyperbolic space are said to be $α$-related if the cross-ratio $[p_i,p_{i+1},q_i,q_{i+1}] = α$ for all $i$ (the vertices lie on the projective line, real or complex, respectively). For example, if $α= -1$, the respective sides of the two polygons are orthogonal. This relation extends to twisted ideal polygons, that is, polygons with monodromy, and it descends to the moduli space of Möbius-equivalent polygons. We prove that this relation, which is, generically, a 2-2 map, is completely integrable in the sense of Liouville. We describe integrals and invariant Poisson structures, and show that these relations, with different values of the constants $α$, commute, in an appropriate sense. We investigate the case of small-gons, describe the exceptional ideal polygons, that possess infinitely many $α$-related polygons, and study the ideal polygons that are $α$-related to themselves (with a cyclic shift of the indices).

math.DS↗
arXiv2017

On Finite Gauss Transform

We present an invariant density for the finite Gauss transformation of the unit interval and discuss some properties of this transformation.

math.DS↗
arXiv2017

Non-smooth convex caustics for Birkhoff billiard

This paper is devoted to the examination of the properties of the string construction for the Birkhoff billiard. Based on purely geometric considerations, string construction is suited to provide a table for the Birkhoff billiard, having the prescribed caustic. Exploiting this framework together with the properties of convex caustics, we give a geometric proof of a result by Innami first proved in 2002 by means of Aubry-Mather theory. In the second part of the paper we show that applying the string construction one can find a new collection of examples of $C^2$-smooth convex billiard tables with a non-smooth convex caustic.

math.DS↗
arXiv2016

On a Very Steep Version of the Standard Map

We consider the long time behavior of the trajectories of the discontinuous analog of the standard Chirikov map. We prove that for some values of parameters all the trajectories remains bounded for all time. For other set of parameters we provide an estimate for the escape rate for the trajectories and present a numerically supported conjecture for the actual escape rate.

math.DS↗
arXiv2016

Dynamics of Discrete Time Systems with a Hysteresis Stop Operator

We consider a piecewise linear two-dimensional dynamical system that couples a linear equation with the so-called stop operator. Global dynamics and bifurcations of this system are studied depending on two parameters. The system is motivated by modifications to general-equilibrium macroeconomic models that attempt to capture the frictions and memory-dependence of realistic economic agents.

math.DS↗
arXiv2016

Typical representatives of free homotopy classes in a multi-punctured plane

We show that a uniform probability measure supported on a specific set of piecewise linear loops in a non-trivial free homotopy class in a multi-punctured plane is overwhelmingly concentrated around loops of minimal lengths. Our approach is based on extending Mogulskii's theorem to closed paths, which is a useful result of independent interest. In addition, we show that the above measure can be sampled using standard Markov Chain Monte Carlo techniques, thus providing a simple methods for approximating shortest loops.

math.PR↗
arXiv2013

Pinball dynamics: unlimited energy growth in switching Hamiltonian systems

A family of discontinuous symplectic maps on the cylinder is considered. This family arises naturally in the study of nonsmooth Hamiltonian dynamics and in switched Hamiltonian systems. The transformation depends on two parameters and is a canonical model for the study of bounded and unbounded behavior in discontinuous area-preserving mappings due to nonlinear resonances. This paper provides a general description of the map and points out its connection with another map considered earlier by Kesten. In one special case, an unbounded orbit is explicitly constructed.

math-ph↗