# Generalized Hannay-Berry Connections on Foliated Manifolds and Applications

## TL;DR

This paper generalizes the averaging method for Poisson connections on foliated manifolds with symmetry, extending Hannay-Berry connections, and explores their applications in Hamiltonian systems of adiabatic type.

## Contribution

It introduces a generalized framework for Hannay-Berry connections on foliated manifolds, broadening their applicability in geometric and Hamiltonian system analysis.

## Key findings

- Extended the averaging method for Poisson connections
- Generalized Hannay-Berry connections to foliated manifolds
- Applications in normal form theory for Hamiltonian systems

## Abstract

In this paper, we discuss some aspects of the averaging method for Poisson connections on foliated manifolds with symmetry generalizing the previous results on the Hannay-Berry connections on fibrations due to \cite{Mn-88,MaMoRa-90} which play an important role in the normal form theory for Hamiltonian systems of adiabatic type.