Tools from dynamical systems, especially from ergodic theory and thermodynamic formalism, are extremely useful to compute and estimate various dimensions. This includes dimensions of limit sets of geometric constructions (the standard Cantor set being most famous example), which a priori, are not related to dynamical systems.
It has been known for many years that one can construct dynamical systems exhibiting various pathological behaviors by requiring a key dynamical characteristic to have special number theoretic properties (such as requiring a rotation number to admit abnormally fast rational approximations). In a similar spirit, one can sometimes construct pathological examples of dimension in dynamical systems using ideas in number theory.
In this short survey we attempt to present some of the important ideas, models, tools and results in the interplay of dimension theory and dynamical systems, as well as to give a glimpse of applications in disparate areas of mathematics.