Dimension Theory of Stationary Fractal Measures

Descripción del proyecto

La geometría del azar

Los fractales son patrones intrincados que se observan en las costas, las nubes e incluso en los mercados financieros. Tienen una forma de dimensión que no se ajusta a los números enteros. Durante décadas, los matemáticos han predicho lo densas que deberían ser estas formas, basándose en su aleatoriedad subyacente. Sin embargo, demostrar esta llamada «dimensión esperada» es un reto porque los fractales se solapan o existen en espacios de alta dimensión. El equipo del proyecto DIM-FRACTAL, financiado por el CEI, pretende resolver este problema combinando herramientas de la combinatoria aditiva y la teoría de matrices aleatorias. Los investigadores están creando un método unificado para casos en los que las técnicas estándar no funcionan. Demostrarán que los fractales alcanzan toda su complejidad geométrica. Resolver este problema ofrecería nuevas formas de entender los sistemas no lineales y caóticos que conforman el mundo físico.

Objetivo

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.