An overview of the dimension theory of dynamical systems (Q2778784)

This paper is a short survey on the dimension theory of dynamical systems. The dimension of a set is a subtle characteristic which measures the geometric complexity of the set at arbitrarily fine scale. There are many notions of dimensions and for general sets there is no reason why any of these should coincide. Through a series of examples, the authors aim at giving an overview of the deep connections between dynamical systems and dimension theory. Rather than presenting a point of view with fool proofs, the survey has the goal to summarize and compare the main ideas and results in dimension theory providing a road map (necessarily omitting details) of a fraction of the literature. The presented examples are particurlarly enlighting and compensate for the sketchy presentation. NEWLINENEWLINENEWLINEThe paper is divided into three main parts: 1) the role of dimension theory in dynamical systems; 2) dimension theory of low-dimensional dynamical systems; 3) dimension theory of higher dimensional dynamical systems.NEWLINENEWLINENEWLINEIn Part 1, the main notions of dimension are introduced, and a short section is devoted to the illustration of Pesin's theory of dimension-like characteristics. This theory is a general theory which unifies many notions of dimension along with many fundamental quantities in dynamical systems such as entropies, pressure, etc. Special attention also goes to the different methods to estimate Hausdorff dimension from below. Several examples are discussed in Section 1.3. E.g. in Section 1.3.4 results for the dyadic expansion and continued fraction expansion of numbers are recast as dimension results for dynamical systems and attacked using tools from dynamics. NEWLINENEWLINENEWLINEPart 2 starts with the example of the affine horseshoe in \(\mathbb{R}^2\). Many peculiarities of the theory of dimension in low-dimensional dynamical systems are discussed and the concept of dimension of measures is introduced. The ideas behind the proof of Young's dimension formula and Manning and McClusky's dimension formula are given. NEWLINENEWLINENEWLINESection 2.4 is then devoted to multifractal analysis and its attempt to understand the fine structure of the level sets of the fundamental asymptotic quantities in ergodic theory. NEWLINENEWLINENEWLINEIn Part 3, some key examples illustrate what is known about the dimension theory of higher-dimensional dynamical systems. By the example of the affine horseshoe in \(\mathbb{R}^3\) the authors point out the difficulties on the way for a general theory of the Hausdorff dimension of invariant sets. (They also discuss the example of solenoids.) NEWLINENEWLINENEWLINEThe last part is devoted to the study of the dimension of invariant measures which is a more tractable and fruitful tool in the case of higher dimensional dynamical system than the dimension of invariant sets. To keep the exposition simple, the attention is restricted to ergodic measures. The connection between Lyapunov exponents and measure-theoretic entropy via dimensions of measures is established.NEWLINENEWLINEFor the entire collection see [Zbl 0973.00044].