On polynomially integrable Birkhoff billiards on surfaces of constant curvature

We present a solution of the algebraic version of Birkhoff Conjecture on integrable billiards. Namely we show that every polynomially integrable real bounded convex planar billiard with smooth boundary is an ellipse. We extend this result to billiards with piecewise-smooth and not necessarily convex boundary on arbitrary two-dimensional surface of constant curvature: plane, sphere, Lobachevsky (hyperbolic) plane; each of them being modeled as a plane or a (pseudo-) sphere in $\mathbb R^3$ equipped with appropriate quadratic form. Namely, we show that a billiard is polynomially integrable, if and only if its boundary is a union of confocal conical arcs and appropriate geodesic segments. We also present a complexification of these results. These are joint results of Mikhail Bialy, Andrey Mironov and the author. The proof is split into two parts. The first part is given by Bialy and Mironov in their two joint papers. They considered the tautological projection of the boundary to $\mathbb{RP}^2$ and studied its orthogonal-polar dual curve, which is piecewise algebraic, by S.V.Bolotin's theorem. By their arguments and another Bolotin's theorem, it suffices to show that each non-linear complex irreducible component of the dual curve is a conic. They have proved that all its singularities and inflection points (if any) lie in the projectivized zero locus of the corresponding quadratic form on $\mathbb C^3$. The present paper provides the second part of the proof: we show that each above irreducible component is a conic and finish the solution of the Algebraic Birkhoff Conjecture in constant curvature.Comment: To appear in the Journal of the European Mathematical Society (JEMS), 69 pages, 2 figures. A shorter proof of Theorem 4.24. Minor precisions and misprint correction

On density of horospheres in dynamical laminations

In 1985 D.Sullivan had introduced a dictionary between two domains of complex dynamics: iterations of rational functions on the Riemann sphere and Kleinian groups. The latters are discrete subgroups of the group of conformal automorphisms of the Riemann sphere. This dictionary motivated many remarkable results in both domains, starting from the famous Sullivan's no wandering domain theorem in the theory of iterations of rational functions. One of the principal objects used in the study of Kleinian groups is the hyperbolic 3- manifold associated to a Kleinian group, which is the quotient of its lifted action to the hyperbolic 3- space. M.Lyubich and Y.Minsky have suggested to extend Sullivan's dictionary by providing an analogous construction for iterations of rational functions. Namely, they have constructed a lamination by three-dimensional manifolds equipped with a continuous family with hyperbolic metrics on them (may be with singularities). The action of the rational mapping on the sphere lifts naturally up to homeomorphic action on the hyperbolic lamination that is isometric along the leaves. The action on the hyperbolic lamination admits a well-defined quotient called {\it the quotient hyperbolic lamination}. We study the arrangement of the horospheres in the quotient hyperbolic lamination. The main result says that if a rational function does not belong to a small list of exceptions (powers, Chebyshev and Latt\`es), then there are many dense horospheres, i.e., the horospheric lamination is topologically-transitive. We show that for "many" rational functions (hyperbolic or critically-nonrecurrent nonparabolic) the quotient horospheric lamination is minimal: each horosphere is dense.Comment: The complete versio

On quadrilateral orbits in complex algebraic planar billiards

The famous conjecture of V.Ya.Ivrii (1978) says that {\it in every billiard with infinitely-smooth boundary in a Euclidean space the set of periodic orbits has measure zero}. In the present paper we study the complex algebraic version of Ivrii's conjecture for quadrilateral orbits in two dimensions, with reflections from complex algebraic curves. We present the complete classification of 4-reflective algebraic counterexamples: billiards formed by four complex algebraic curves in the projective plane that have open set of quadrilateral orbits.Comment: 64 pages. To appear in Moscow Math. Journal, No 2 (2014

On odd-periodic orbits in complex planar billiards

The famous conjecture of V.Ya.Ivrii (1978) says that {\it in every billiard with infinitely-smooth boundary in a Euclidean space the set of periodic orbits has measure zero}. In the present paper we study the complex version of Ivrii's conjecture for odd-periodic orbits in planar billiards, with reflections from complex analytic curves. We prove positive answer in the following cases: 1) triangular orbits; 2) odd-periodic orbits in the case, when the mirrors are algebraic curves avoiding two special points at infinity, the so-called isotropic points. We provide immediate applications to the real piecewise-algebraic Ivrii's conjecture and to its analogue in the invisibility theory

On germs of constriction curves in model of overdamped Josephson junction, dynamical isomonodromic foliation and Painlev\'e 3 equation

B.Josephson (Nobel Prize, 1973) predicted tunnelling effect for a system (called Josephson junction) of two superconductors separated by a narrow dielectric: existence of a supercurrent through it and equations governing it. The overdamped Josephson junction is modeled by a family of differential equations on 2-torus depending on 3 parameters: $B$, $A$, $\omega$. We study its rotation number $\rho(B,A;\omega)$ as a function of parameters. The three-dimensional phase-lock areas are the level sets $L_r:=\{\rho=r\}$ with non-empty interiors; they exist for $r\in\mathbb Z$ (Buchstaber, Karpov, Tertychnyi). For every fixed $\omega>0$ and $r\in\mathbb Z$ the planar slice $L_r\cap(\mathbb R^2_{B,A}\times\{\omega\})$ is a garland of domains going vertically to infinity and separated by points; those separating points for which $A\neq0$ are called constrictions. In a joint paper by Yu.Bibilo and the author, it was shown that 1) at each constriction the rescaled abscissa $\ell:=\frac B\omega$ is equal to $\rho$; 2) the family of constrictions with given $\ell\in\mathbb Z$ is an analytic submanifold $Constr_\ell$ in $(\mathbb R^2_+)_{a,s}$, $a=\omega^{-1}$, $s=\frac A\omega$. Here we show that the limit points of $Constr_\ell$ are $\beta_{\ell,k}=(0,s_{\ell,k})$, where $s_{\ell,k}>0$ are zeros of the Bessel function $J_\ell(s)$, and it lands at them regularly. Known numerical pictures show that high components of $Int(L_r)$ look similar. In his paper with Bibilo, the author introduced a candidate to the self-similarity map between neighbor components: the Poincar\'e map of the dynamical isomonodromic foliation governed by Painlev\'e 3 equation. Whenever well-defined, it preserves $\rho$. We show that the Poincar\'e map is well-defined on a neighborhood of the plane $\{ a=0\}\subset\mathbb R^2_{\ell,a}\times(\mathbb R_+)_s$, and it sends $\beta_{\ell,k}$ to $\beta_{\ell,k+1}$ for integer $\ell$.Comment: 41 page, 5 figure

On polynomially integrable planar outer billiards and curves with symmetry property

We show that every polynomially integrable planar outer convex billiard is elliptic.Comment: To appear in Mathematische Annalen. 26 pages. Minor improvement of presentatio

On 4-reflective complex analytic planar billiards

The famous conjecture of V.Ya.Ivrii says that {\it in every billiard with infinitely-smooth boundary in a Euclidean space the set of periodic orbits has measure zero}. In the present paper we study its complex analytic version for quadrilateral orbits in two dimensions, with reflections from holomorphic curves. We present the complete classification of 4-reflective complex analytic counterexamples: billiards formed by four holomorphic curves in the projective plane that have open set of quadrilateral orbits. This extends the previous author's result classifying 4-reflective complex planar algebraic counterexamples. We provide applications to real planar billiards: classification of 4-reflective germs of real planar $C^4$-smooth pseudo-billiards; solutions of Tabachnikov's Commuting Billiard Conjecture and the 4-reflective case of Plakhov's Invisibility Conjecture (both in two dimensions; the boundary is required to be piecewise $C^4$-smooth). We provide a survey and a small technical result concerning higher number of complex reflections.Comment: To appear in Journal of Geometric Analysis. 69 pages, 14 figures. New changes: simplifying and strengthening the last proposition in Subsection 4.2; generalizing and simplifying definitions of billiard combinations in Section 6; updating list of known k-reflective billiards in Section 6; new bibliographic references; minor polishing of some part

On curves with Poritsky property

For a given closed convex planar curve γ with smooth boundary and a given p > 0, the string construction is obtained by putting a string surrounding γ of length p + |γ| to the plane. Then we pull some point of the string "outwards from γ" until its final position A, when the string becomes stretched completely. The set of all the points A thus obtained is a planar convex curve Γ p. The billiard reflection T p from the curve Γ p acts on oriented lines, and γ is a caustic for Γ p : that is, the family of lines tangent to γ is T p-invariant. The action of the reflection T p on the tangent lines to γ ≃ S 1 induces its action on the tangency points: a circle diffeomorphism T p : γ → γ. We say that γ has string Poritsky property, if it admits a parameter t (called Poritsky-Lazutkin string length) in which all the transformations T p are translations t → t + c p. These definitions also make sense for germs of curves γ. Poritsky property is closely related to the famous Birkhoff Conjecture. It is classically known that each conic has string Poritsky property. In 1950 H.Poritsky proved the converse: each germ of planar curve with Poritsky property is a conic. In the present paper we extend this Poritsky's result to germs of curves to all the simply connected complete Riemannian surfaces of constant curvature and to outer billiards on all these surfaces. We also consider the general case of curves with Poritsky property on any two-dimensional surface with Riemannian metric and prove a formula for the derivative of the Poritsky-Lazutkin length as a function of the natural length parameter. In this general setting we also prove the following uniqueness result: a germ of curve with Poritsky property is uniquely determined by its 4-th jet. In the Euclidean case this statement follows from the above-mentioned Poritsky’s result

On determinants of modified Bessel functions and entire solutions of double confluent Heun equations

We investigate the question on existence of entire solutions of well-known linear differential equations that are linearizations of nonlinear equations modeling the Josephson effect in superconductivity. We consider the modified Bessel functions $I_j(x)$ of the first kind, which are Laurent series coefficients of the analytic function family $e^{\frac x2(z+\frac 1z)}$. For every $l\geq1$ we study the family parametrized by $k, n\in\mathbb Z^l$, $k_1>\dots>k_l$, $n_1>\dots>n_l$ of $(l\times l)$-matrix functions formed by the modified Bessel functions of the first kind $a_{ij}(x)=I_{k_j-n_i}(x)$, $i,j=1,\dots,l$. We show that their determinants $f_{k,n}(x)$ are positive for every $l\geq1$, $k,n\in\mathbb Z^l$ as above and $x>0$. The above determinants are closely related to a sequence (indexed by $l$) of families of double confluent Heun equations, which are linear second order differential equations with two irregular singularities, at zero and at infinity. V.M.Buchstaber and S.I.Tertychnyi have constructed their holomorphic solutions on $\mathbb C$ for an explicit class of parameter values and conjectured that they do not exist for other parameter values. They have reduced their conjecture to the second conjecture saying that if an appropriate second similar equation has a polynomial solution, then the first one has no entire solution. They have proved the latter statement under the additional assumption (third conjecture) that $f_{k,n}(x)\neq0$ for $k=(l,\dots,1)$, $n=(l-1,\dots,0)$ and every $x>0$. Our more general result implies all the above conjectures, together with their corollary for the overdamped model of the Josephson junction in superconductivity: the description of adjacency points of phase-lock areas as solutions of explicit analytic equations.Comment: 19 pages, 1 figure. To appear in Nonlinearity. Minor changes. The present version includes additional historical remarks and bibliograph