arXiv subjects

Marco Mazzucchelli

Publications and source records attributed to Marco Mazzucchelli.

At least 19 recordsLinked to original sources

arXiv2026

Klingenberg's theorem on Anosov geodesic flows

In 1970, Klingenberg proved a fundamental result on closed Riemannian manifolds with Anosov geodesic flows, asserting that such manifolds are without conjugate points. The proof, based on the Morse theory of the energy functional, had crucial gaps pointed out by Anosov, who saved the theorem by providing a completely independent proof. In this article, we fill the gaps in the original argument, and thus provide a full Morse theoretic proof of Klingenberg's theorem.

math.DS↗

arXiv2026

On flat translated chains of contactomorphisms of $\mathbb{R}^{2n+1}$ and $\mathbb{R}^{2n} \times S^1$

We introduce the notions of flat translated chains of contactomorphisms and periodic flat translated chains of finite sequences of contactomorphisms, and extend to these notions the theorem of Viterbo (1992) on the multiplicity of periodic points of compactly supported Hamiltonian diffeomorphisms of $\mathbb{R}^{2n}$. More precisely, we show that every non-trivial compactly supported contactomorphism of either $\mathbb{R}^{2n+1}$ or $\mathbb{R}^{2n} \times S^1$ that is contact isotopic to the identity has infinitely many geometrically distinct flat translated chains in the interior of the support with respect to the standard contact form, that the growth-rate of such translated chains is at least linear if the contactomorphism is non-negative, as well as similar statements for periodic flat translated chains of finite sequences of contactomorphisms.

math.SG↗

arXiv2026

Topics in Symplectic Dynamics: Barcode Entropy

Barcode entropy is an invariant of a Hamiltonian system -- a Hamiltonian diffeomorphism or a Reeb flow -- measuring its Morse or Floer theoretic complexity at a small scale. More specifically, it is the exponential growth rate of the number of not-too-short bars in the Floer or symplectic homology persistence module. Barcode entropy is closely related to topological entropy, even though they originate in different contexts, and in low dimensions they coincide. In these notes, we study barcode entropy and related invariants in various settings and explore their connections with pure dynamics features and, in particular, topological entropy. The methods build on techniques from symplectic topology and Floer theory, dynamical systems, and smooth integral geometry. We also touch upon some other applications of the machinery we develop. These notes are based on the mini-course given by the second author at the CIME summer school "Symplectic Dynamics and Topology" (Cetraro, Italy, June 16-20, 2025).

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arXiv2026

Equivariant critical point theory and bifurcation of $3d$ gravity-capillary Stokes waves

We establish novel existence results of $3d$ gravity-capillary periodic traveling waves. In particular we prove the bifurcation of multiple, geometrically distinct truly $3d$ Stokes waves having the same momentum of any non-resonant $2d$ Stokes wave. This unexpected clustering phenomenon of Stokes waves, observed in physical fluids, is a fundamental consequence of the Hamiltonian nature of the water waves equations, their symmetry groups, and novel topological arguments. We employ a variational Lyapunov-Schmidt reduction combined with equivariant Morse-Conley theory for a functional defined on a joined topological space invariant under a $2$-torus action. Although the reduction is a priori singular near the hyperplanes of $2d$-waves, we circumvent this difficulty by exhaustive use of the symmetry groups. This approach yields a complete bifurcation picture of $3d $ gravity-capillary Stokes waves.

math.AP↗

arXiv2026

Length spectrum rigidity and flexibility of spheres of revolution with one equator

We define a notion of marked length spectrum for $S^1$-symmetric Riemannian metrics on the two-sphere having only one equator. We prove that isospectral metrics in this class have conjugate geodesic flows. Under a further $\mathbb{Z}_2$-symmetry assumption, we show that the marked length spectrum determines the metric. Finally, we show that every isospectral class of metrics contains a unique $\mathbb{Z}_2$-symmetric metric and give an explicit description of this isospectral class as an infinite dimensional convex set, generalizing the known description of $S^1$-symmetric Zoll metrics. This paper contains also two appendices, in which we provide an elementary proof of the fact that a $C^2$ real valued function on an interval is determined by the set of tangent lines to its graph, and we classify a class of $S^1$-invariant contact forms on three-manifolds.

math.DG↗

arXiv2024

From curve shortening to flat link stability and Birkhoff sections of geodesic flows

We employ the curve shortening flow to establish three new results on the dynamics of geodesic flows of closed Riemannian surfaces. The first one is the stability, under $C^0$-small perturbations of the Riemannian metric, of certain flat links of closed geodesics. The second one is a forced existence theorem for closed connected orientable Riemannian surfaces: for surfaces of positive genus, the existence of a contractible simple closed geodesic $\gamma$ forces the existence of infinitely many closed geodesics intersecting $\gamma$ in every primitive free homotopy class of loops; for the 2-sphere, the existence of two disjoint simple closed geodesics forces the existence of a third one intersecting both. The final result asserts the existence of Birkhoff sections for the geodesic flow of any closed connected orientable Riemannian surface.

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arXiv2024

Closed geodesics and the first Betti number

We prove that, on any closed manifold of dimension at least two with non-trivial first Betti number, a $C^\infty$ generic Riemannian metric has infinitely many closed geodesics, and indeed closed geodesics of arbitrarily large length. We derive this existence result combining a theorem of Ma\~n\'e together with the following new theorem of independent interest: the existence of minimal closed geodesics, in the sense of Aubry-Mather theory, implies the existence of a transverse homoclinic, and thus of a horseshoe, for the geodesic flow of a suitable $C^\infty$-close Riemannian metric.

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arXiv2024

On potentials whose level sets are orbits

A level orbit of a mechanical Hamiltonian system is a solution of Newton equation that is contained in a level set of the potential energy. In 2003, Mark Levi asked for a characterization of the smooth potential energy functions on the plane with the property that any point on the plane lies on a level orbit; we call such functions Levi potentials. The basic examples are the radial monotone increasing smooth functions. In this paper we show that any Levi potential that is analytic or has totally path-disconnected critical set must be radial. Nevertheless, we show that every compact convex subset of the plane is the critical set of a Levi potential. A crucial observation for these theorems is that, outside the critical set, the family of level sets of a Levi potential forms a solution of the inverse curvature flow.

math.DG↗

arXiv2024

On the Barcode Entropy of Reeb Flows

In this paper we continue investigating connections between Floer theory and dynamics of Hamiltonian systems, focusing on the barcode entropy of Reeb flows. Barcode entropy is the exponential growth rate of the number of not-too-short bars in the Floer or symplectic homology persistence module. The key novel result is that the barcode entropy is bounded from below by the topological entropy of any hyperbolic invariant set. This, combined with the fact that the topological entropy bounds the barcode entropy from above, established by Fender, Lee and Sohn, implies that in dimension three the two types of entropy agree. The main new ingredient of the proof is a variant of the Crossing Energy Theorem for Reeb flows.

math.SG↗

arXiv2023

Invariant Sets and Hyperbolic Closed Reeb Orbits

We investigate the effect of a hyperbolic (or, more generally, isolated as an invariant set) closed Reeb orbit on the dynamics of a Reeb flow on the $(2n-1)$-dimensional standard contact sphere, extending two results previously known for Hamiltonian diffeomorphisms to the Reeb setting. In particular, we show that under very mild dynamical convexity type assumptions, the presence of one hyperbolic closed orbit implies the existence of infinitely many simple closed Reeb orbits. The second main result of the paper is a higher-dimensional Reeb analogue of the Le Calvez-Yoccoz theorem, asserting that no closed orbit of a non-degenerate dynamically convex Reeb pseudo-rotation is locally maximal, i.e., isolated as an invariant set. The key new ingredient of the proofs is a Reeb variant of the crossing energy theorem.

math.SG↗

arXiv2023

A general support theorem for analytic double fibration transforms

We develop a systematic approach for resolving the analytic wave front set for a class of integral geometry transforms appearing in various tomography problems. Combined with microlocal analytic continuation, this leads to uniqueness and support theorems for analytic integral transforms which are in the microlocal double fibration framework introduced by Guillemin. For the case of ray transforms, we show that the double fibration setup has a concrete interpretation in terms of curve families obtained by projecting integral curves of a fixed vector field on some fiber bundle down to the base. This setup includes geodesic X-ray type transforms, null bicharacteristic ray transforms and transforms related to real principal type systems. We also study transforms integrating over submanifolds of any codimension, and give geometric characterizations for the Bolker condition required for recovering singularities. Our approach is based on a general result related to recovering the analytic wave front set of a function from its transform given by a suitable analytic elliptic Fourier integral operator. This approach extends and unifies a number of previous works. We use wave packet transforms to extrapolate the geometric features of wave front set propagation for such operators when their canonical relation satisfies the Bolker condition.

math.AP↗

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↗

arXiv2022

Barcode entropy of geodesic flows

We introduce and study the barcode entropy for geodesic flows of closed Riemannian manifolds, which measures the exponential growth rate of the number of not-too-short bars in the Morse-theoretic barcode of the energy functional. We prove that the barcode entropy bounds from below the topological entropy of the geodesic flow and, conversely, bounds from above the topological entropy of any hyperbolic compact invariant set. As a consequence, for Riemannian metrics on surfaces, the barcode entropy is equal to the topological entropy. A key to the proofs and of independent interest is a crossing energy theorem for gradient flow lines of the energy functional.

math.SG↗

arXiv2022

Surfaces of section for geodesic flows of closed surfaces

We prove several results concerning the existence of surfaces of section for the geodesic flows of closed orientable Riemannian surfaces. The surfaces of section $\Sigma$ that we construct are either Birkhoff sections, meaning that they intersect every sufficiently long orbit segment of the geodesic flow, or at least they have some hyperbolic components in $\partial\Sigma$ as limit sets of the orbits of the geodesic flow that do not return to $\Sigma$. In order to prove these theorems, we provide a study of configurations of simple closed geodesics of closed orientable Riemannian surfaces, which may have independent interest. Our arguments are based on the curve shortening flow.

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arXiv2021

Existence of Birkhoff sections for Kupka-Smale Reeb flows of closed contact 3-manifolds

A Reeb vector field satisfies the Kupka-Smale condition when all its closed orbits are non-degenerate, and the stable and unstable manifolds of its hyperbolic closed orbits intersect transversely. We show that, on a closed 3-manifold, any Reeb vector field satisfying the Kupka-Smale condition admits a Birkhoff section. In particular, this implies that the Reeb vector field of a $C^\infty$-generic contact form on a closed 3-manifold admits a Birkhoff section, and that the geodesic vector field of a $C^\infty$-generic Riemannian metric on a closed surface admits a Birkhoff section.

math.DG↗

arXiv2021

Proof of the $C^2$-stability conjecture for geodesic flows of closed surfaces

We prove that a $C^2$-generic Riemannian metric on a closed surface has either an elliptic closed geodesic or an Anosov geodesic flow. As a consequence, we prove the $C^2$-stability conjecture for Riemannian geodesic flows of closed surfaces: a $C^2$-structurally stable Riemannian geodesic flow of a closed surface is Anosov. In order to prove these statements, we establish a general result that may be of independent interest and provides sufficient conditions for a Reeb flow of a closed 3-manifold to be Anosov.

math.DS↗

arXiv2021

Higher systolic inequalities for 3-dimensional contact manifolds

A contact form is called Besse when the associated Reeb flow is periodic. We prove that Besse contact forms on closed connected 3-manifolds are the local maximizers of suitable higher systolic ratios. Our result extends earlier ones for Zoll contact forms, that is, contact forms whose Reeb flow defines a free circle action.

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arXiv2020

On the structure of Besse convex contact spheres

We consider convex contact spheres $Y$ all of whose Reeb orbits are closed. Any such $Y$ admits a stratification by the periods of closed Reeb orbits. We show that $Y$ "resembles" a contact ellipsoid: any stratum of $Y$ is an integral homology sphere, and the sequence of Ekeland-Hofer spectral invariants of $Y$ coincides with the full sequence of action values, each one repeated according to its multiplicity.

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