We consider a symbolic
dynamical system (X,σ) on a countable state space. We introduce a kind of
topological entropy for such systems, denoted h∗, which coincides with usual
topological entropy when X is compact. We use a pictorial approach, to classify a
graph Γ (or a chain) as transient, null recurrent, or positive recurrent. We show that
given 0 ≤ α ≤ β ≤∞, there is a chain whose h∗ entropy is β and where Gurevic
entropy is α. We compute the topological entropies of some classes of chains,
including larger chains built up from smaller ones by a new operation which we call
the Cartesian sum.
