On behaviour of holomorphically contractible systems under non-monotonic sequences of sets

The new results concerning the continuity of holomorphically contractible systems treated as set functions with respect to non-monotonic sequences of sets are given. In particular, continuity properties of Kobayashi and Carath\'eodory pseudodistances, as well as Lempert and Green functions with respect to sequences of domains converging in Hausdorff metric are delivered.


Introduction
It is known that both Carathéodory and Kobayashi pseudodistances depend continuously on increasing and decreasing sequences of domains (in the latter case, adding some regularity assumptions on limiting domain; cf. [3] and references therein). The pseudodistances mentioned above are particular examples of wider class of holomorphically contractible systems, i.e. systems of functions D running through all domains in all C n 's, such that d D is forced to be p, the hyperbolic distance on D, the unit disc on the plane and all holomorphic mappings are contractions with respect to the system (d D ) (cf. Definition 2.1). The question about the behaviour of holomorphically contractible systems under not necessarily monotonic sequences of sets seems to be natural and important. In the present note, inspired by [1], we shall give a very general result stating the continuity of holomorphically contractible systems under the sequences of domains convergent with respect to Hausdorff distance (for two nonempty bounded sets A, B it is defined as where for a set S and a positive number ε, the set S (ε) := s∈S B(s, ε) is the ε-envelope of S; B(x, r) denotes the open Euclidean ball of center x and radius r). Namely, our main result reads as follows:  In particular, we get the results in this spirit for Carathéodory and Kobayashi pseudodistances as well as for Green and Lempert functions (cf. Corollaries 2.3, 2.4, and 2.6). We believe they are interesting in their own.
In [1] all the results are settled in the context of complex Banach spaces, yet under strong assumption about the convexity of the approximating domains together with the limiting one. Our results are free from this restrictive assumption.
In Section 2 we give the formal definition of holomorphically contractible system and both list and prove the corollaries form Theorem 1.

Holomorphically contractible systems
Let us start with the precise definition of holomorphically contractible system.
where D runs over all domains in C n with arbitrary n, is called a holomorphically contractible system if the following two conditions are satisfied: ( Remark 2.2. If in the above definition we replace p by m, the Möbius distance on D, then we speak of m-contractible system. This distinction is however somewhat artificial, since having (d D ), a holomorphically contractible system, we may define d * D := tanh d D and then the operator sending (d D ) to (d * D ) is a bijection between the class of contractible systems and the class of m-contractible systems (see [2], Section 4.1).
The most important examples of holomorphically contractible systems are the following: (1) Carathéodory pseudodistance: (2) Lempert function: (3) Kobayashi pseudodistance:  Proof. The proof goes along the same lines as the proof of Corollary 2.3.
In the case of Green function, things go a little bit more complicated. Let us see the details.  Assume that for each compact K ⊂ D there exists an n 0 ∈ N such that for any n ≥ n 0 , K ⊂ D n . Then for any z, w ∈ D lim n→∞ g D n (z, w) = g D (z, w).
Proof. By [4] we know that the Green function is continuous with respect to increasing sequences of domains. Therefore, (I n ) n∈N may be chosen as some exhausting sequence of smoothly bounded strictly pseudoconvex relatively compact open subsets of D. Also, using results of [5], it is clear that the good candidate for the "exterior" sequence is (E n ) n∈N := (D n ) n∈N .

Proof of Theorem 1.1
Proof of Theorem 1.1. There exists an m 1 ∈ N such that for m ≥ m 1 we have We may choose the smallest possible such an m 1 . In what follows, we shall construct two sequences of sets, (L n ) n∈N , (U n ) n∈N , such that L n ⊂ L n+1 , n ∈ N, ∞ n=1 L n = D, U n+1 ⊂⊂ U n , n ∈ N, ∞ n=1 U n = D and L n ⊂ D m 1 +n−1 ⊂ U n , n ∈ N.
Then for n large enough, z, w ∈ L n and Finally, letting n → ∞ and using the assumptions concerning continuity of system (d D ) with respect to monotonic sequences of domains (I n ) n∈N , (E n ) n∈N , we reach the conclusion of Theorem 1.1. Let us pass to the construction. Let L 1 := I 1 , U 1 := E 1 . We proceed as follows: Choose the smallest m 2 ∈ N such that for any m ≥ m 2 we have There are two cases to be considered: Case 1. m 2 ∈ {m 1 , m 1 + 1}. Then We define L 2 = . . . = L s := I 1 , L s+1 := I 2 . Further, as U 2 we choose a domain relatively compact in E 1 , containing in its interior E 2 ∪ m 1 +s−1 l=m 1 +1 D l . Inductively, for k = 2, . . . , s a domain U k is chosen as a domain relatively compact in U k−1 , containing in its interior E 2 ∪ m 1 +s−1 l=m 1 +k−1 D l . Finally, we put U s+1 := E 2 . Suppose we have constructed domains L 1 ⊂ . . . ⊂ L r and U 1 ⊂⊂ . . . ⊂⊂ U r such that L j ⊂ D m 1 +j−1 ⊂ U j , j = 1, . . . , r and L r = I M , U r = E M , m 1 + r − 1 = m M with some M ∈ N We choose the smallest m M+1 ∈ N with Similarly as before, there are two cases to be considered: