arXiv subjects

Viktor L. Ginzburg

Publications and source records attributed to Viktor L. Ginzburg.

At least 19 recordsLinked to original sources

arXiv2026

Closed Orbits of Dynamically Convex Reeb Flows: Towards the HZ- and Multiplicity Conjectures

We study the multiplicity problem for prime closed orbits of dynamically convex Reeb flows on the boundary of a star-shaped domain in $\mathbb{R}^{2n}$. The first of our two main results asserts that such a flow has at least $n$ prime closed Reeb orbits, improving the previously known lower bound by a factor of two and settling a long-standing open question. The second main theorem is that when, in addition, the domain is centrally symmetric and the Reeb flow is non-degenerate, the flow has either exactly $n$ or infinitely many prime closed orbits. This is a higher-dimensional contact variant of Franks' celebrated $2$-or-infinity theorem and, viewed from the symplectic dynamics perspective, settles a particular case of the contact Hofer-Zehnder conjecture. The proofs are based on several auxiliary results of independent interest on the structure of the filtered symplectic homology and the properties of closed orbits.

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arXiv2026

From Barcode Entropy to Metric Entropy

We establish a connection between barcode entropy and metric entropy. Namely, we show that the barcode entropy bounds the metric entropy from below for a measure from a specific class of invariant measures associated with a pair of Lagrangian or Legendrian submanifolds, which we call pseudo-chord measures. This inequality refines, via the variational principle, the previously known upper bound of barcode entropy by topological entropy.

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arXiv2026

Filtered Symplectic Homology and Closed Reeb Orbits

We further explore connections between the symplectic homology persistence module and the properties of closed Reeb orbits for star-shaped domains in higher dimensions. Our first result is that the sequence of $S^1$-equivariant spectral invariants over a field of positive characteristic is bounded from above, in contrast with the case of characteristic zero. We also prove that the dimension of the filtered symplectic homology is bounded as a function of the action whenever the flow is a pseudo-rotation, i.e., it has finitely many prime closed orbits. Finally, we show that a non-degenerate Reeb flow has infinitely many prime closed orbits whenever it has one closed orbit with negative mean index.

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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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arXiv2025

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.

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arXiv2025

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.

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arXiv2025

Barcode growth for toric-integrable Hamiltonian systems

We continue investigating the connection between the dynamics of a Hamiltonian system and the barcode growth of the associated Floer or symplectic homology persistence module, focusing now on completely integrable systems. We show that for convex/concave or real analytic toric domains and convex/concave or real analytic completely integrable Hamiltonians on closed toric manifolds the barcode has polynomial growth with degree (i.e., slow barcode entropy) not exceeding half of the dimension. This slow polynomial growth contrasts with exponential growth for many systems with sufficiently non-trivial dynamics. We also touch upon the barcode growth function as an invariant of the interior of the domain and use it to distinguish some open domains.

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arXiv2024

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.

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arXiv2024

Topological entropy of Hamiltonian diffeomorphisms: a persistence homology and Floer theory perspective

We study topological entropy of compactly supported Hamiltonian diffeomorphisms from a perspective of persistence homology and Floer theory. We introduce barcode entropy, a Floer-theoretic invariant of a Hamiltonian diffeomorphism, measuring exponential growth under iterations of the number of not-too-short bars in the barcode of the Floer complex. We prove that the barcode entropy is bounded from above by the topological entropy and, conversely, that the barcode entropy is bounded from below by the topological entropy of any hyperbolic invariant set, e.g., a hyperbolic horseshoe. As a consequence, we conclude that for Hamiltonian diffeomorphisms of surfaces the barcode entropy is equal to the topological entropy.

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arXiv2024

On the growth of the Floer barcode

This paper is a follow up to the authors' recent work on barcode entropy. We study the growth of the barcode of the Floer complex for the iterates of a compactly supported Hamiltonian diffeomorphism. In particular, we introduce sequential barcode entropy which has properties similar to barcode entropy, bounds it from above and is more sensitive to the barcode growth. We prove that in dimension two the sequential barcode entropy equals the topological entropy and hence equals the ordinary barcode entropy. We also study the behavior of the $γ$-norm under iterations. We show that the $γ$-norm of the iterates is separated from zero when the map has sufficiently many hyperbolic periodic points and, as a consequence, it is separated from zero $C^\infty$-generically in dimension two. We also touch upon properties of the barcode entropy of pseudo-rotations and, more generally, $γ$-almost periodic maps.

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arXiv2024

On the generic behavior of the spectral norm

The main result of the paper is that for any closed symplectic manifold the spectral norm of the iterates of a Hamiltonian diffeomorphism is locally uniformly bounded away from zero $C^\infty$-generically.

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arXiv2022

Lower semi-continuity of Lagrangian volume

We study lower semi-continuity properties of the volume, i.e., the surface area, of a closed Lagrangian manifold with respect to the Hofer- and $γ$-distance on a class of monotone Lagrangian submanifolds Hamiltonian isotopic to each other. We prove that volume is $γ$-lower semi-continuous in two cases. In the first one the volume form comes from a Kähler metric with a large group of Hamiltonian isometries, but there are no additional constraints on the Lagrangian submanifold. The second one is when the volume is taken with respect to any compatible metric, but the Lagrangian submanifold must be a torus. As a consequence, in both cases, the volume is Hofer lower semi-continuous.

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arXiv2021

Pseudo-rotations vs. Rotations

Continuing the study of Hamiltonian pseudo-rotations of projective spaces, we focus on the conjecture that the fixed-point data set (the actions and the linearized flows at one-periodic orbits) of a pseudo-rotation exactly matches that data for a suitable unique true rotation even though the two maps can have very different dynamics. We prove this conjecture in several instances, e.g., for strongly non-degenerate pseudo-rotations of ${{\mathbb C}{\mathbb P}}^2$ with some notable exceptions, which we call ghost pseudo-rotations. The existence of ghost pseudo-rotations is a completely open question. The conjecture is closely related to the properties of the action and index spectra of pseudo-rotations, and ghost pseudo-rotations, if they exist, satisfy all known restrictions on the fixed-point data for pseudo-rotations but these data are distinctly different from the data for any rotation. The main new ingredient of the proofs is purely combinatorial and of independent interest. This is the index divisibility theorem connecting the divisibility properties of the Conley--Zehnder index sequence for the iterates of a map with the properties of its spectrum.

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arXiv2020

Dynamical convexity and closed orbits on symmetric spheres

The main theme of this paper is the dynamics of Reeb flows with symmetries on the standard contact sphere. We introduce the notion of strong dynamical convexity for contact forms invariant under a group action, supporting the standard contact structure, and prove that in dimension $2n+1$ any such contact form satisfying a condition slightly weaker than strong dynamical convexity has at least $n+1$ simple closed Reeb orbits. For contact forms with antipodal symmetry, we prove that strong dynamical convexity is a consequence of ordinary convexity. In dimension five or greater, we construct examples of antipodally symmetric dynamically convex contact forms which are not strongly dynamically convex, and thus not contactomorphic to convex ones via a contactomorphism commuting with the antipodal map. Finally, we relax this condition on the contactomorphism furnishing a condition that has non-empty $C^1$-interior.

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arXiv2020

Another look at the Hofer-Zehnder conjecture

We give a different and simpler proof of a slightly modified (and weaker) variant of a recent theorem of Shelukhin extending Franks' "two-or-infinitely-many" theorem to Hamiltonian diffeomorphisms in higher dimensions and establishing a sufficiently general case of the Hofer-Zehnder conjecture. A few ingredients of our proof are common with Shelukhin's original argument, the key of which is Seidel's equivariant pair-of-pants product, but the new proof highlights a different aspect of the periodic orbit dynamics of Hamiltonian diffeomorphisms.

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arXiv2020

On the iterated Hamiltonian Floer homology

The focus of the paper is the behavior under iterations of the filtered and local Floer homology of a Hamiltonian on a symplectically aspherical manifold. The Floer homology of an iterated Hamiltonian comes with a natural cyclic group action. In the filtered case, we show that the supertrace of a generator of this action is equal to the Euler characteristic of the homology of the un-iterated Hamiltonian. For the local homology the supertrace is the Lefschetz index of the fixed point. We also prove an analog of the classical Smith inequality for the iterated local homology and the equivariant versions of these results.

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arXiv2020

On the spectral characterization of Besse and Zoll Reeb flows

A closed contact manifold is called Besse when all its Reeb orbits are closed, and Zoll when they have the same minimal period. In this paper, we provide a characterization of Besse contact forms for convex contact spheres and Riemannian unit tangent bundles in terms of $S^1$-equivariant spectral invariants. Furthermore, for restricted contact type hypersurfaces of symplectic Euclidean spaces, we give a sufficient condition for the Besse property via the Ekeland-Hofer capacities.

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arXiv2020

From pseudo-rotations to holomorphic curves via quantum Steenrod squares

In the context of symplectic dynamics, pseudo-rotations are Hamiltonian diffeomorphisms with finite and minimal possible number of periodic orbits. These maps are of interest in both dynamics and symplectic topology. We show that a closed, monotone symplectic manifold, which admits a non-degenerate pseudo-rotation, must have a deformed quantum Steenrod square of the top degree element, and hence non-trivial holomorphic spheres. This result (partially) generalizes a recent work by Shelukhin and complements the results by the authors on non-vanishing Gromov-Witten invariants of manifolds admitting pseudo-rotations.

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