Dimension of attractors associated to autonomous and non-autonomous dynamical systems - Moura, Rafael de Oliveira (2025)
This thesis explores the dimensional properties of attractors in dynamical systems, offering theoretical advancements and applications within both autonomous and non-autonomous settings. A comprehensive framework is presented to analyze evolution processes, pullback attractors, and uniform attractors, providing tools for the study of asymptotic behaviors in partial and ordinary differential equations. Novel contributions include improved dimension estimates for uniform attractors, extensions of Mañés Embedding Theorem, and the development of dimension reduction techniques for finite-dimensional Hilbert spaces. Central results include a generalization of box-counting dimension estimates for uniform attractors, removing the assumption of finite-dimensional symbol spaces. Applications to parabolic semilinear equations demonstrate the practical relevance of these findings, establishing that systems with infinite-dimensional symbol spaces can still exhibit finite-dimensional attractors. Further, the differences between box-counting dimension and inertial manifold theories are explored, revealing complementary approaches to embedding attractors in finite-dimensional spaces with varying regularity. This work opens avenues for bridging the box-counting and spectral approaches, refining dimension estimates, and enhancing our understanding of the geometric and spectral structure of attractors in dynamical systems. These findings provide a deeper mathematical foundation for applications in science and engineering, extending the reach of dynamical systems theory.