# KAM, $\alpha$-Gevrey regularity and the $\alpha$-Bruno-R\"ussmann condition

## TL;DR

This paper establishes a new invariant torus theorem for $ extalpha$-Gevrey Hamiltonian systems under an arithmetic condition called the $ extalpha$-Bruno-R"ussmann condition, extending classical results and providing direct proofs without complex extensions or smoothing.

## Contribution

It introduces the $ extalpha$-Bruno-R"ussmann condition for $ extalpha$-Gevrey systems and proves an invariant torus theorem using a direct approach, avoiding complex analysis and smoothing techniques.

## Key findings

- Proves invariant tori exist under the $ extalpha$-Bruno-R"ussmann condition.
- Shows the destruction of tori if a weaker condition is not met.
- Develops new functional estimates in the Gevrey class.

## Abstract

We prove a new invariant torus theorem, for $\alpha$-Gevrey smooth Hamiltonian systems, under an arithmetic assumption which we call the $\alpha$-Bruno-R\"ussmann condition, and which reduces to the classical Bruno-R\"ussmann condition in the analytic category. Our proof is direct in the sense that, for analytic Hamiltonians, we avoid the use of complex extensions and, for non-analytic Hamiltonians, we do not use analytic approximation nor smoothing operators. Following Bessi, we also show that if a slightly weaker arithmetic condition is not satisfied, the invariant torus may be destroyed. Crucial to this work are new functional estimates in the Gevrey class.