On flat translated chains of contactomorphisms of ℝ²ⁿ⁺¹ and ℝ²ⁿ × S¹

2026/07/13 by Marco Mazzucchelli, Sheila Sandon · 1 voice
Mathematics · #math.SG

paper · pdf

Abstract

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 ℝ²ⁿ. More precisely, we show that every non-trivial compactly supported contactomorphism of either ℝ²ⁿ⁺¹ or ℝ²ⁿ × S¹ 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.

Citations

• Non-orderability and the contact Hofer norm
• Contact non-squeezing at large scale via generating functions
• Contactomorphisms of the sphere without translated points
• Morse estimates for translated points on unit tangent bundles
• Maximum principles in symplectic homology
• Givental's non-linear Maslov index on lens spaces
• The Hofer norm of a contactomorphism
• Non-squeezing property of contact balls
• Orderability and the Weinstein Conjecture
• Translated points and Rabinowitz Floer homology
• A Morse estimate for translated points of contactomorphisms of spheres and projective spaces
• On iterated translated points for contactomorphisms of R²ⁿ⁺¹ and R²ⁿ x S¹
• An integer valued bi-invariant metric on the group of contactomorphisms of R²n x S¹
• Contact Homology, Capacity and Non-Squeezing in R²n x S¹ via Generating Functions
• Partially ordered groups and geometry of contact transformations
• Critical points of quasi-functions and generating families of Legendrian manifolds

Discussions

• Marco Mazzucchelli, Sheila Sandon: On flat translated chains of contactomorphisms of $\mathbb{R}^{2n+1}$ and $\mathbb{R}^{2n} \times S^1$ https://arxiv.org/abs/2607.11757 https://arxiv.org/pdf/2607.11 [bsky, 0 points, 0 comments]

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• Geodesics of norms on the contactomorphisms group of ℝ²ⁿ × S¹
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• On non-contractible hyperbolic periodic orbits and periodic points of symplectomorphisms