Space–time complexity in Hamiltonian dynamics

New notions of the complexity function C(e;t,s) and entropy function S(e;t,s) are introduced to describe systems with nonzero or zero Lyapunov exponents or systems that exhibit strong intermittent behavior with “flights,” trappings, weak mixing, etc. The important part of the new notions is the first appearance of e-separation of initially close trajectories. The complexity function is similar to the propagator p(t0,x0;t,x) with a replacement of x by the natural lengths s of trajectories, and its introduction does not assume of the space–time independence in the process of evolution of the system. A special stress is done on the choice of variables and the replacement t→η=ln t, s→ξ=ln s makes it possible to consider time-algebraic and space-algebraic complexity and some mixed cases. It is shown that for typical cases the entropy function S(e;ξ,η) possesses invariants (α,β) that describe the fractal dimensions of the space–time structures of trajectories. The invariants (α,β) can be linked to the transport...