arXiv subjects

Luca Baracco

Publications and source records attributed to Luca Baracco.

At least 19 recordsLinked to original sources

arXiv2026

Generic properties of planar symplectic billiards

We study generic properties of planar symplectic billiards, a symplectic analogue of classical Birkhoff billiards introduced by P. Albers and S. Tabachnikov. We prove that, for a residual set of $C^\infty$ strongly convex domains, periodic symplectic billiard trajectories are in general position and non-degenerate. We also prove a Franks' lemma for symplectic billiards and deduce that stably hyperbolic periodic points form a hyperbolic set given by the closure of the set of hyperbolic periodic points. We further show that, for a residual set of $C^\infty$ strongly convex domains, elliptic periodic points are stable and hyperbolic periodic points have transverse homoclinic points. Based on these results, we conclude that --generically-- $C^\infty$ strongly convex symplectic billiard has positive topological entropy.

math.DS↗

arXiv2026

Area spectral rigidity for axially symmetric symplectic billiard tables

We prove that any finitely smooth axially symmetric strictly convex domain, with everywhere positive curvature and sufficiently close to an ellipse is area spectrally rigid. This means that any area-isospectral family of domains in this class is necessarily equi-affine. We use techniques, adapted to symplectic billiards, inspired to the paper by J. De Simoi, V. Kaloshin and Q. Wei (2017).

math.DS↗

arXiv2025

Birkhoff attractors for dissipative symplectic billiards

The aim of the present paper is to propose and study a dissipative variant of symplectic billiards within planar strictly convex domains. The associated billiard map is dissipative, thus it admits a compact invariant set, the so-called Birkhoff attractor. Its complexity depends on the rate of the dissipation as well as on the geometry of the billiard table. We prove that (a) for strong dissipation, the Birkhoff attractor is a normally contracted graph over the zero section; (b) for mild dissipation, the Birkhoff attractor within a centrally symmetric domain is an indecomposable continuum whose restricted dynamics has positive topological entropy. We compare these results with the case of dissipative Birkhoff billiards, studied in a paper by Bernardi-Florio-Leguil

math.DS↗

arXiv2025

Starting the study of outer length billiards

We focus on the outer length billiard dynamics, acting on the exterior of a strictly-convex planar domain. We first show that ellipses are totally integrable. We then provide an explicit representation of first order terms for the formal Taylor expansion of the corresponding Mather's $β$-function. Finally, we provide explicit Lazutkin coordinates up to order 4.

math.DS↗

arXiv2024

Bialy-Mironov type rigidity for centrally symmetric symplectic billiards

The aim of the present paper is to establish a Bialy-Mironov type rigidity for centrally symmetric symplectic billiards. For a centrally symmetric $C^2$ strongly-convex domain $D$ with boundary $\partial D$, assume that the symplectic billiard map has a (simple) continuous invariant curve $δ\subset \mathcal{P}$ of rotation number $1/4$ (winding once around $\partial D$) and consisting only of $4$-periodic orbits. If one of the parts between $δ$ and each boundary of the phase-space is entirely foliated by continuous invariant closed (not null-homotopic) curves, then $\partial D$ is an ellipse. The differences with Birkhoff billiards are essentially two: it is possible to assume the existence of the foliation in one of the parts of the phase-space detected by the curve $δ$, and the result is obtained by tracing back the problem directly to the totally integrable case.

math.DS↗

arXiv2023

Higher order terms of Mather's $β$-function for symplectic and outer billiards

We compute explicitly the higher order terms of the formal Taylor expansion of Mather's $β$-function for symplectic and outer billiards in a strictly-convex planar domain $C$. In particular, we specify the third terms of the asymptotic expansions of the distance (in the sense of the symmetric difference metric) between $C$ and its best approximating inscribed or circumscribed polygons with at most $n$ vertices. We use tools from affine differential geometry.

math.DS↗

arXiv2023

Totally integrable symplectic billiards are ellipses

In this paper we prove that a totally integrable strictly-convex symplectic billiard table, whose boundary has everywhere strictly positive curvature, must be an ellipse. The proof, inspired by the analogous result of Bialy for Birkhoff billiards, uses the affine equivariance of the symplectic billiard map.

math.DS↗

arXiv2023

On the local maximizers of higher capacity ratios

We prove an analogue of the 4-dimensional local Viterbo conjecture for the higher Ekeland-Hofer capacities: on the space of 4-dimensional smooth star-shaped domains of unitary volume, endowed with the $C^3$ topology, the local maximizers of the $k$-th Ekeland-Hofer capacities are those domains symplectomorphic to suitable rational ellipsoids.

math.SG↗

arXiv2021

Fractional powers of higher order vector operators on bounded and unbounded domains

Using the $H^\infty$-functional calculus for quaternionic operators, we show how to generate the fractional powers of some densely defined differential quaternionic operators of order $m\geq 1$, acting on the right linear quaternionic Hilbert space $L^2(Ω,\mathbb C\otimes\mathbb H)$. The operators that we consider are of the type $$ T=i^{m-1}\left(a_1(x) e_1\partial_{x_1}^{m}+a_2(x) e_2\partial_{x_2}^{m}+a_3(x) e_3\partial_{x_3}^{m}\right), \ \ \ x=(x_1,\, x_2,\, x_3)\in \overlineΩ, $$ where $\overlineΩ$ is the closure of either a bounded domain $Ω$ with $C^1$ boundary, or an unbounded domain $Ω$ in $\mathbb R^3$ with a sufficiently regular boundary which satisfy the so called property $(R)$, $\{e_1,\, e_2,\, e_3\}$ is an orthonormal basis for the imaginary units of $\mathbb H$, $a_1,\,a_2,\, a_3: \overlineΩ \subset\mathbb{R}^3\to \mathbb{R}$ are the coefficients of $T$. In particular it will be given sufficient conditions on the coefficients of $T$ in order to generate the fractional powers of $T$, denoted by $P_α(T)$ for $α\in(0,1)$, when the components of $T$, i.e. the operators $T_l:=a_l\partial_{x_l}^m$, do not commute among themselves.

math.SP↗

arXiv2021

Testing families of analytic discs in the unit ball

Let $a,b,c\in \mathbb{C}^2$ be three non collinear points such that their mutual joining complex lines do not intersect the unit ball $\mathbb{B}^2$ and such that the line through $a$ and $b$ is tangent to $\mathbb{B}^2$. Then the set of lines concurrent to $a,b$ and $c$ is a testing family for continuous functions on $\mathbb{S}^3$. This improves a result by the authors and solves a case left open in the literature as described by Globevnik.

math.CV↗

arXiv2021

Higher order gradients of monogenic functions

Given a monogenic function on the quaternionic algebra $\mathbb{H}$, the Clifford algebra $\mathbb{R}_n$ or the octonionic algebra $\mathbb{O}$ we prove that $|\nabla^m f|^α$ is subharmonic for some $α>0$ where $\nabla^m f$ is the $m$-th order gradient of $f$. We find also the optimal value of $α$. This is generalization of a result of Calderon and Zygmund.

math.CV↗

arXiv2020

A note on the closed range of $\bar\partial_b$ on q-convex manifolds

We prove that the tangential Cauchy-Riemann operator has closed range on Levi-pseudoconvex CR manifolds that are embedded in a q-convex complex manifold $X$. Our result generalizes the known case when $X$ is a Stein manifold.

math.CV↗

arXiv2019

An extension theorem for regular functions of two quaternionic variables

For functions of two quaternionic variables that are regular in the sense of Fueter, we establish a result similar in spirit to the Hanges and Trèves theorem. Namely, we show that a ball contained in the boundary of a domain is a propagator of regular extendability across the boundary.

math.CV↗

arXiv2019

On the Ekeland-Hofer symplectic capacities of the real bidisc

In $\mathbb{C}^2$ with the standard symplectic structure we consider the bidisc $D^2\times D^2$ constructed as the product of two open real discs of radius $1$. We compute explicit values for the first, second and third Ekeland-Hofer symplectic capacity of $D^2\times D^2$. We discuss some applications to questions of symplectic rigidity.

math.CV↗

arXiv2019

Orthogonal testing families and holomorphic extension from the sphere to the ball

Let $\mathbb{B}^2$ denote the open unit ball in $\mathbb{C}^2$, and let $p\in \mathbb{C}^2\setminus\overline{\mathbb{B}^2}$. We prove that if $f$ is an analytic function on the sphere $\partial\mathbb{B}^2$ that extends holomorphically in each variable separately and along each complex line through $p$, then $f$ is the trace of a holomorphic function in the ball.

math.CV↗

arXiv2017

Hölder regularity of the solution to the complex Monge-Ampère equation with $L^p$ density

On a smooth domain $Ω\subset\subset\mathbb C^n$, we consider the Dirichlet problem for the complex Monge-Ampère equation $((dd^cu)^n=fdV,\,u|_{bΩ}\equivϕ)$. We state the Hölder regularity of the solution $u$ when the boundary value $ϕ$ is Hölder continuous and the density $f$ is only $L^p$, $p>1$. Note that in former literature (Guedj-Kolodziej-Zeriahi) the weakness of the assumption $f\in L^p$ was balanced by taking $ϕ\in C^{1,1}$ (in addition to assuming $Ω$ strongly pseudoconvex).

math.CV↗

arXiv2017

The complex Monge-Ampère equation on weakly pseudoconvex domains

We show here a "weak" Hölder-regularity up to the boundary of the solution to the Dirichlet problem for the complex Monge-Ampère equation with data in the $L^p$ space and the boundary of the domain satisfying an $f$-property. The $f$-property is a potential-theoretical condition which holds for all pseudoconvex domains of finite type and many examples of infinite type.

math.CV↗

arXiv2015

Extension of L^2, di-bar-closed, forms

We prove extension of a di-bar-closed, smooth, form from the intersection of a pseudoconvex domain with a complex hyperplane to the whole domain. The extension form is di-bar-closed, has harmonic coefficients and its L^2-norm is estimated by the L^2-norm of the trace. For holomorphic functions this is proved by Ohsawa-Takegoshi [12]. For forms of higher degree, this is stated by Manivel [9]. It seems, however, that the proof contains a gap because of the use of a a singular weight and the failure of regularity for the solution of the related di-bar-equation. There is a rich literature on the subject (cf. among otheres [7], [14]) but it does not seem to contain complete answer to the question. Also, the problem of extending cohomology classes of di-bar of higher degree in a compact Kahler space is addressed in [8] and [3]. Apart from the formal analogy, this has little in common with our problem in which these classes are 0. We also believe, comparing to the preceding literature, that our approach is original and, somewhat, simpler.

math.CV↗