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

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 $\mathbb{R}^{2n}$. 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.