What does it mean for an orbit to "accumulate"?

Question

For example, from Beardon's "A Primer On Riemann Surfaces",

...Show that g is a homeomorphism of D onto itself, and that no orbit accumulates in D.

I'm simply looking for a definition of the word accumulates; I haven't encountered it yet and he presents no formal meaning previously.

Answer 1

Very roughly, it is saying something like that the orbit should not have a subsequence with a Limit Point. That wiki page contains various definitions related to accumulation points in a topological space. The specific one given is:

"A point x ∈ X is a cluster point or accumulation point of a sequence
(xn)n ∈ N if, for every neighbourhood V of x, there are infinitely many 
natural numbers n such that xn ∈ V."