# Differentiability of the Evolution Map and Mackey Continuity

## TL;DR

This paper establishes that Mackey k-continuity ensures the differentiability of the evolution map in Milnor's infinite dimensional Lie groups, removing previous continuity assumptions and linking regularity to Mackey continuity.

## Contribution

It introduces Mackey k-continuity as a key condition for differentiability of the evolution map, advancing the understanding of regularity in infinite dimensional Lie groups.

## Key findings

- Mackey k-continuity implies differentiability of the evolution map.
- Regularity and Mackey continuity are equivalent for Lie groups with Fréchet Lie algebras.
- The strong Trotter property is analyzed under Mackey continuity.

## Abstract

We solve the differentiability problem for the evolution map in Milnor's infinite dimensional setting. We first show that the evolution map of each $C^k$-semiregular Lie group $G$ (for $k\in \mathbb{N}\sqcup\{\mathrm{lip},\infty\}$) admits a particular kind of sequentially continuity $-$ called Mackey k-continuity. We then prove that this continuity property is strong enough to ensure differentiability of the evolution map. In particular, this drops any continuity presumptions made in this context so far. Remarkably, Mackey k-continuity arises directly from the regularity problem itself, which makes it particular among the continuity conditions traditionally considered. As an application of the introduced notions, we discuss the strong Trotter property in the sequentially-, and the Mackey continuous context. We furthermore conclude that if the Lie algebra of $G$ is a Fr\'{e}chet space, then $G$ is $C^k$-semiregular (for $k\in \mathbb{N}\sqcup\{\infty\}$) if and only if $G$ is $C^k$-regular.