Answer is given in:  Density of Beckward Paths of the Julia set of Semigroup, SJM, Vol.(22)(2014), 77-85., G.R. Chacon, R. Coluci & D. D'Angeli))

That it is possible introduce a measure on the set of infinite paths of the tree by using the Lebesgue measure on   [0, 1]  we can see considering two simple quadratic functions.

                                    f(z) = z2 – 1    and     g(z) = z2

whose Julia sets (by Gaston Julia)  are known.  These are described in  Figure  1.   (Basilica - the Julia set is the boundary of the gray domain) by the numerical experiment  (using Matlab and Maple). it is possible to find the whole Julia set of the semigroup in the following way:

We consider the backward orbits of any compositions of the inverses of  functions f and g. Each of this function has two inverse images:

                                    ho(z)   = √1 + z,   h1(z)   = −√1 + z,

                                     h2 (z) = √z,          h3(z)  = −√z,

where ho and  h1 are the inverse images of  f  and   h2 and  h3 of  g.   

Consider sequences of  length  218 of elements of

                                      H = { ho, h1, h2, h3 }

 generated in each  step by two random  choices, the choice between the functions  f  and  g and  between their two pre-images.

After several experiments, the picture obtained is very similar to the one obtained  by considering the whole backward  orbit.

Moreover, it is clear that not every infinite path in the tree gives rise to an approximation of the Julia set of the semigroup. Thus, this allows us to conjecture that almost every infinite path of the backward tree is dense in the Julia set.

Amendment to previous answers:

For two considered functions

                                   f (z) = z2 - 1  and   g(z) = z2 

each of which has two inverse functions (see above, my previous answer) the Julia sets are presented in Figure 2 in new attachment, while the approximation of the Julia set of the semigroup   < f, g >   is can be found in Figure 3 in attachment.