Quantitative views of recurrence and proximality

In a topological dynamical system with a dense set of recurrent points, we investigate whether there are 'plenty' of points whose recurrence is 'fast'. Depending upon how we make our query precise, we get affirmative as well as negative answers. We carry out a similar study about proximal pairs; that is, for 'most' proximal pairs of points, how fast the distance between the corresponding terms in the two orbits can go to zero. For instance, we show that if f:[0, 1] → [0, 1] is a continuous map having a periodic point whose period is not a power of 2, then for every function φ:N→N, there is an uncountable scrambled set S ⊂ [0, 1] for f satisfying the extra property that lim infn→∞ φ(n)|fn(x)-fn(y)|=0 for all x, y in S. We also provide characterizations of weak mixing and mixing for a topological dynamical system in terms of proximality of orbits to arbitrary sequences in the phase space