Dimension Theory of Stationary Fractal Measures

Descrizione del progetto

La geometria della casualità

I frattali sono modelli complessi che si osservano nelle linee costiere, nelle nuvole e persino nei mercati finanziari. Hanno una dimensione che non si adatta perfettamente ai numeri interi. Per decenni, i matematici hanno previsto la densità che queste forme dovrebbero avere, basandosi sulla loro casualità intrinseca. Tuttavia, dimostrare questa cosiddetta «dimensione attesa» è complesso, poiché i frattali si sovrappongono o esistono in spazi ad alta dimensionalità. Il progetto DIM-FRACTAL, finanziato dal CER, intende affrontare questa problematica combinando strumenti della combinatoria additiva e della teoria delle matrici casuali. I ricercatori stanno sviluppando un approccio unificato per i casi in cui le tecniche standard risultano inadeguate. Si dimostrerà che i frattali raggiungono la loro piena complessità geometrica. La risoluzione di questo problema offrirebbe nuovi modi di comprendere i sistemi non lineari e caotici che modellano il mondo fisico.

Obiettivo

We consider the dimension of stationary fractal measures. Our examples include self-affine, self-similar, and Furstenberg measures, which are among the most fundamental and studied fractal objects. The dimension of such a measure has a natural upper bound defined in terms of entropy, Lyapunov exponents, and the dimension of the ambient space. Equality to the upper bound is expected in the absence of obvious algebraic obstructions. In very simple situations, this equality is easy to demonstrate and was known to hold long ago. In more complicated cases, it has been shown to hold almost surely under a natural randomization of the parameters.

On the other hand, proving the expected equality under the minimal conjectured assumptions, or even under some mild algebraic conditions, is an extremely challenging problem. In recent years, there has been considerable progress in this direction. However, due to major obstacles in current methods, a satisfactory solution is still a long way off. These obstacles mainly stem from high dimensionality of the ambient space, and from lack of separation in the associated semigroup.

The primary goal of the proposed research is to make substantial progress in verifying the equality by addressing the aforementioned obstacles through the use of new techniques and ideas. Our suggested arguments include methods from additive combinatorics, ergodic theory, and the theory of random products of matrices.

To address difficulties caused by lack of separation, we suggest extending the theory developed for Bernoulli convolutions, and its partial generalization to the setup of three maps, to systems consisting of more maps. To tackle high dimensionality difficulties, we propose an approach requiring the factorization of the Furstenberg boundary maps. By implementing these strategies, we expect our research to play a crucial role in forming a more unified and complete dimension theory of stationary fractal measures.