Linear fractional mappings: invariant sets, semigroups and commutativity

Abstract

We study commutativity and embeddability (into continuous semigroups) properties of linear fractional self-mappings of the open unit disk in the complex plane. The common thread in our approach is the classical notion of the Koenigs function which we use in each of the three possible cases (dilation, hyperbolic and parabolic). Since we are interested in a classical subject, the paper is written in the style of a survey, in order to make it accessible to a wider audience. Therefore it contains, in addition to our new results, an exposition of most relevant facts. (2000). 30C45, 47H20. Mathematics Subject Classification

Holomorphic self-maps of the disk intertwining two linear fractional maps

“…The fact that the commutation equation f • ϕ = ϕ • f implies that f is an LFT whenever ϕ is such, discarding the exceptional cases mentioned earlier, was also obtained in [15,Theorem 2] by using completely different methods.…”

“…The same idea appears in the second part, which refers to the elliptic case. Our method also yields a criterion somewhat different from the analytic condition presented in [15], [16], and [25, Proposition 5.9.5], as well as from the geometric criterion given most recently in [8,Proposition 3.4].…”

“…A relationship with the so-called polymorphic functions [19] should also be mentioned. Several partial results regarding the commutation with LFT's can be found in the recent paper [15].…”

Convergence characteristics of one-parameter continuous semigroups

“…Suppose that S = {F t } t≥0 is a semigroup of self-mappings of such that F(z) = F 1 (z) = az+b cz+d with two different fixed points z 1 = F (z 1 ) and z 2 = F (z 2 ), and assume that 0 < F (z 1 ) < 1. Then the following assertions hold (see [20]):…”

“…Theorem 3.7 (see [20]) Let S = {F t } t≥0 be a semigroup with F 1 (z) = az+b cz+d , F 1 (1) = 1, and let F 1 (1) be a positive real number. Then Note that λ = e −β , where β is the angular derivative of the generator f of S at 1 (see [11]).…”