Approximate Identities and Lagrangian Poincar\'e Recurrence

TL;DR

This paper explores the existence of maps that approximate the identity under various norms and discusses the Lagrangian Poincaré recurrence conjecture within Hamiltonian and smooth diffeomorphism dynamics.

Contribution

It connects three problems: approximation of identity maps in different norms and the Lagrangian Poincaré recurrence conjecture, providing new insights into their interrelations.

Findings

Existence of maps approximating the identity in various norms.

Properties of such maps in Hamiltonian and smooth diffeomorphisms.

Discussion on the Lagrangian Poincaré recurrence conjecture.

Abstract

In this note we discuss three interconnected problems about dynamics of Hamiltonian or, more generally, just smooth diffeomorphisms. The first two concern the existence and properties of the maps whose iterations approximate the identity map with respect to some norm, e.g., $C^1$- or $C^0$-norm for general diffeomorphisms and the $\gamma$-norm in the Hamiltonian case, and the third problem is the Lagrangian Poincar\'e recurrence conjecture.