Sixth Dynamic Day
DAI Day 6

Abstract: A point mass moves inside a convex planar domain under the gravitational attraction of a fixed positive mass. Upon reaching the boundary, the particle is reflected back into the domain according to the geometrical optics law: the angle of incidence equals the angle of reflection. It is known that this type of dynamical system is integrable (for all energy levels) when the domain is an ellipse and the attracting mass is placed at one of its foci or, more generally, when the domain is composed of arcs of confocal quadrics, again with the attracting mass at a focus.
In this talk we will discuss the following rigidity phenomenon: among analytic, centrally symmetric, compact convex domains, the only ones for which the corresponding Kepler billiard is analytically integrable (for sufficiently large energies) are circles with the attracting mass placed at their centers and ellipses, with the attracting mass at a focus. Our approach is based on an explicit construction of a symbolic dynamics for the system.
If time permits, I will discuss some connections between this construction and the presence of a non-rational invariant curve of rotation number 1/2 for the standard billiard map and how to build billiard tables with exotic 1/2-invariant curves.
Joint work with Vivina Barutello, Irene De Blasi and Susanna Terracini.

Abstract: The climate is a highly complex and heterogeneous nonequilibrium system under the continuous action of forcing and dissipation. Its main energy source is the spatially inhomogeneous absorption of incoming solar radiation, which triggers a rich set of instabilities and feedback mechanisms across a wide range of temporal and spatial scales, driving the redistribution of heat and water mass. This process continues until the system reaches an approximate steady state lying on a high-dimensional attractor.
In this talk, we will describe climate simulations that exhibit multiple attractors—ranging from a snowball state to a hot, ice-free state—in both simplified configurations, such as aquaplanets, and more realistic setups with continental distributions. We will characterise these attractors, their basin boundaries, tipping points, and the associated bifurcation diagram as external forcing is varied, using methods and diagnostics drawn from dynamical systems theory, such as instantaneous dimension and persistence. We will also discuss novel topological techniques that can be used to identify characteristic climate dynamics within each attractor.
Joint works with C. Ragon, L. Moinat, D. Sciamarella.

Abstract: This talk introduces a dissipative variant of symplectic billiards within strictly convex planar domains. In this setting, the associated billiard map is no longer conservative and therefore admits a compact invariant set –known as the Birkhoff attractor. The complexity of this attractor depends on both the dissipation rate and the geometry of the billiard table. We will discuss the qualitative features of the Birkhoff attractor in two distinct regimes, determined by the strength of dissipation and the shape of the domain.
Based on joint work with L. Baracco, O. Bernardi, and A. Florio.

Abstract: I will present recent advances on the spectral method using transfer operators applied to partially hyperbolic systems, exploring the consequences in terms of the statistical properties such as ergodicity, mixing and limit theorems.
Based on joint works with C. Liverani and with K. Fernando.