Abstract
If is a continuous map of a compact metric space
,
and if
is a sequence of positive reals converging to 0, we investigate the properties of the set
. We show that
is a dense
subset of
for every
when x is a recurrent point, even though
can be disjoint with the orbit of x for some
. Under the assumption that f has an invariant non-atomic Borel probability measure
, we prove results to the effect that (i) there is a uniform upper limit to the speed with which the orbit of each x can approach y for
-almost every
, (ii) if
is ergodic with full support and if
is the set of points having dense orbits, then for
-almost every
and for every
there is a uniform upper limit to the speed with which the orbit of x can approach y. Next, using
as a useful tool in proofs, we establish the following. If f is totally transitive and X is infinite, then there is a dense subset
which is a countable union of Cantor sets such that
and
for any two distinct
and any two distinct
. If f is a transitive map enjoying a certain type of continuity in the backward direction, then f has a residual set of points with dense backward orbits.