Beyond Hamiltonian and symplectic dynamics, conformal dynamics studies diffeomorphisms and flows that rescale the symplectic form by a factor different from one. This framework includes, for instance, Hamiltonian systems with a friction term. While certain tools from conservative and symplectic dynamics remain available in this setting, new dissipative phenomena emerge, such as the presence of attractors. In this course, we will study conformal dynamics on both classical symplectic manifolds and on more general locally symplectic manifolds. We will provide several examples. In the symplectic setting, we will study attractors and the isotropy property of certain invariant submanifolds. In the locally symplectic setting, we will investigate the relations between the recurrence properties, the shape of the orbits, and the exponential divergence rate of the linearized dynamics (namely the 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.