Optimally topologically transitive orbits in discrete dynamical systems

Every orbit of a rigid rotation of a circle by a fixed irrational angle is dense. However, the apparent uniformity of the distribution of iterates after a finite number of iterations appears strikingly different for various choices of a rotation angle. Motivated by this observation, we introduce a scalar function on the orbits of a discrete dynamical system defined on a bounded metric space, called the linear limit density, which we interpret as a measure of an orbit's approach to density. Utilizing the three-distance theorem, we compute the exact value of the linear limit density of orbits of rigid rotations by irrational rotation angles with period-1 continued fraction expansions. We further show that any discrete dynamical system defined by an orientation-preserving diffeomorphism of the circle has an orbit with a larger linear limit density than any orbit of the rigid rotation by the golden number. Bernoulli shift maps acting on sequences over a finite alphabet provide another illustrative class of dynamical systems with dense orbits. Our study of the efficiency of an orbit's approach to density leads us to demonstrate the existence of a class of infinite sequences with finite linear limit density constructed by recursively extending finite de Bruijn sequences.