Overview
This research introduces the concepts of flat translated chains of contactomorphisms and periodic flat translated chains for finite sequences of contactomorphisms. The study extends a theorem by Viterbo (1992) concerning the multiplicity of periodic points in compactly supported Hamiltonian diffeomorphisms of $\mathbb{R}^{2n}$ to these newly defined chain types. Specifically, the findings indicate that non-trivial, compactly supported contactomorphisms that are contact isotopic to the identity exhibit an infinite number of geometrically distinct flat translated chains within specific mathematical spaces.
Research Context
The work builds upon Viterbo's (1992) theorem regarding the multiplicity of periodic points for compactly supported Hamiltonian diffeomorphisms on $\mathbb{R}^{2n}$. This prior theorem addressed characteristics of Hamiltonian systems, a class of dynamical systems fundamental in classical mechanics and symplectic geometry. The current research aims to extend similar multiplicity properties to the realm of contactomorphisms, which are transformations preserving a contact structure on odd-dimensional manifolds. The specific domains considered are $\mathbb{R}^{2n+1}$ and $\mathbb{R}^{2n} \times S^1$, equipped with a standard contact form.
Approach
The methodology involves defining two new mathematical constructs: flat translated chains of contactomorphisms and periodic flat translated chains of finite sequences of contactomorphisms. Once these notions are established, the core approach is to demonstrate their existence and multiplicity properties under specific conditions for contactomorphisms. The conditions include the contactomorphism being non-trivial, compactly supported, and contact isotopic to the identity. The analysis is conducted with respect to the standard contact form in the interior of the support.
Findings
- The research introduces the notions of flat translated chains of contactomorphisms.
- The research introduces the notions of periodic flat translated chains of finite sequences of contactomorphisms.
- It is shown 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.
- These infinitely many geometrically distinct flat translated chains exist in the interior of the support with respect to the standard contact form.
- For non-negative contactomorphisms, the growth rate of such translated chains is at least linear.
- Similar statements are made for periodic flat translated chains of finite sequences of contactomorphisms.