Narrow Results

Invariant quantities of the classical motion of an ideal incompressible fluid in a two-dimensional bounded domain are used to construct a family {Π_α}_α of probability measures of the Gibbs form, which are invariant under the flow. The Gibbs exponent H is given by the renormalized energy. These measures are supported by the space of configuration Γ, i.e. the fluid vorticity is concentrated in a finite number of distinct points. Properties of a deterministic vortex dynamics having Π_α as invariant measure are investigated; in particular Markov uniqueness is proven. The classical (pre-)Dirichlet form associated to Π_α is also introduced and analyzed.

The goal of this work is to characterize the fine structures of fractals that are in the frontier between order and chaos or whose dynamics are chaotic.