Local Dynamical Entropies and Their Applications in Number Theory

The concept of entropy of a dynamical system, which was introduced by Ya.,G. Sinai and A.,N. Kolmogorov, describes the degree of randomness/complexity of its trajectories. The construction of the entropy theory of dynamical systems is an important step in the study of their ergodic properties, but it is, in a sense, an averaged characterization. Complexity/chaoticity of separate trajectories varies extremely strongly---from the trivial behavior in the case of periodic trajectories to the highly involved behavior for “generic” trajectories of the same chaotic systems. We propose a new approach to the construction of local dynamical entropies, which are applicable to characterizing the complexity of individual trajectories, in turn, enabling us to close this gap. Moreover, this approach can be applied not only to trajectories but also to arbitrary sequences. In particular, we employ it to estimate the complexity of various concepts in number theory (prime numbers, quadratic residues, etc.).