In this paper, the concept of an
inverse arc function is introduced. An inverse arc function f is a function such that
for each arc L in the range of f, there exists an arc L1 in the domain of
f such that f(L1) = L. It is proved that a dendrite D is the continuous
image of an inverse arc function f with domain an arc L if and only if D has
only a finite number of endpoints. Other results are obtained telling what
dendrites can be ranges of continuous inverse arc functions having dendrites as
domains.
The dimension raising ability of a continuous inverse arc function whose domain is
a dendrite is questioned. It is proved that if D is a dendrite with only a countable
number of endpoints, then there does not exist a continuous inverse arc function f
with domain D such that dimf(D) ≧ 2. If a dendrite D has uncountably many
endpoints, then the question is left unanswered.
