Beyond Hamiltonian and symplectic Dynamics, the conformal dynamics study dynamics that multiply a local symplectic form by a factor that is different from one. This includes for example Hamiltonian systems with a friction term. Even if some tools coming from conservative and symplectic dynamics can be used in this setting, some dissipative phenomena appear, as attractors. In this course, we will study the conformal dynamics of both classical symplectic manifolds and their extensions, the locally symplectic manifolds. In particular, in the symplectic setting, we introduce and study the natural generalization of Birkhoff attractor in higher dimension. In the locally symplectic setting, we will explain which relation exists between recurrence properties, shape of the orbits and exponential rate of the linearized dynamics (as Lyapunov exponents).

The mini-course will focus on connections and applications of symplectic topological and Morse theoretic methods to dynamics of Hamiltonian systems, including the multiplicity problem for periodic orbits and also other aspects of dynamics. In particular, it will discuss applications of Floer theory or variants of Morse theory to topological entropy bounds and constructions of invariant sets and measures with specific properties.

We will give an introduction to embedded contact homology, periodic Floer homology, and related tools using holomorphic curves in four dimensions. We will show how these can be applied to questions about the dynamics of Reeb vector fields in three dimensions and area-preserving surface diffeomorphisms.

We first define the spectral metric on the space of Lagrangians and Hamiltonians and define the notions of \(\gamma\)-support and of \(\gamma\)-coisotropic sets. We first prove some basic results about such objects and illustrate them on non-trivial examples. We then show how this completion can be used for dynamical questions, for example for generalizing the Birkhoff attractor of dissipative twist maps of the annulus to higher dimensional settings, for studying Hamiltonian flows for Hamiltonians with singularities, and for providing a symplectic approach to the study of Hamilton-Jacobi equations.