Belinschi and Nica introduced a composition semigroup of maps on the set of probability measures. Using this semigroup, they introduced a free divisibility indicator, from which one can know quantitatively if a measure is freely infinitely divisible or not.In the first half of the paper, we further investigate this indicator: we calculate how the indicator changes with respect to free and Boolean powers; we prove that free and Boolean 1/2-stable laws have free divisibility indicators equal to infinity; we derive an upper bound of the indicator in terms of Jacobi parameters. This upper bound is achieved only by free Meixner distributions. We also prove Bożejko's conjecture which says the Boolean powers µ ⊎ , t ∈ [0, 1], of a probability measure µ are freely infinitely divisible if the measure µ is freely infinitely divisible.In the other half of this paper, we introduce an analogous composition semigroup for multiplicative convolutions and define free divisibility indicators for these convolutions. Moreover, we prove that a probability measure on the unit circle is freely infinitely divisible relative to the free multiplicative convolution if and only if the indicator is not less than one. We also prove how the multiplicative divisibility indicator changes under free and Boolean powers and then the multiplicative analogue of Bożejko's conjecture. We include an appendix, where the Cauchy distributions and point measures are shown to be the only fixed points of the Boolean-to-free Bercovici-Pata bijection.

We completely determine the free infinite divisibility for the Boolean stable law which is parametrized by a stability index α and an asymmetry coefficient ρ. We prove that the Boolean stable law is freely infinitely divisible if and only if one of the following conditions holds: 0 < α ≤ 1 2 ; 1 2 < α ≤ 2 3 and 2 − 1 α ≤ ρ ≤ 1 α − 1; α = 1, ρ = 1 2. Positive Boolean stable laws corresponding to ρ = 1 and α ≤ 1 2 have completely monotonic densities and they are both freely and classically infinitely divisible. We also show that continuous Boolean convolutions of positive Boolean stable laws with different stability indices are also freely and classically infinitely divisible. Boolean stable laws, free stable laws and continuous Boolean convolutions of positive Boolean stable laws are non-trivial examples whose free divisibility indicators are infinity. We also find that the free multiplicative convolution of Boolean stable laws is again a Boolean stable law.