CONTRACTIVE CURVES

We discuss the dynamics of the correspondences associated to those plane curves whose local sections contract the Poincaré metric in a hyperbolic planar domain


Introduction.
We consider certain 1-dimensional, holomorphic correspondences of hyperbolic type, which we call "contractive curves."These are curves whose local sections contract the Poincaré metric of a hyperbolic planar domain.The model for our analysis is given by the hyperbolic Julia sets.
This work has been motivated by the work in Ochs' recent paper [5].In the first part, we adapt to our purposes the Schwarz lemma for correspondences that appears in Minda's paper [3].In the second part, we discuss the dynamics of a contractive curve, and the properties of the associated attractor.Such curves usually do not have global sections that contract the hyperbolic metric.Nevertheless, the associated dynamical systems have much in common with the iterated function systems of hyperbolic type.The basis for our discussion is the paper [2] by Barnsley and Demko.

Hyperbolic metric. The hyperbolic metric (infinitesimal length-element) on the unit disk
Let U ⊂ C be a planar domain (open and connected subset of C), and assume that the complement C \ U contains at least two points, so that U is a hyperbolic Riemann surface; the universal covering of U is (biholomorphic to) ∆.The density ρ U of the hyperbolic metric in U , ds U (z) = ρ U (z)|dz|, is defined as follows.Fix an unramified covering ∆ φ → U .Given z ∈ U, choose any t ∈ φ −1 (z), and define ρ U (z) = 2/(1 − |t| 2 )|φ (t)|.The definition of ρ U (z) does not depend on the choice of ∆ φ → U , nor on the choice of t ∈ φ −1 (z).Note that ρ U is positive and real-analytic.
The hyperbolic distance s U in U is obtained by integrating the infinitesimal lengthelement ds U .The metric space (U , s U ) is complete.
Holomorphic maps do not expand the hyperbolic metric; given any holomorphic map U f → V of hyperbolic domains, f * ds V ≤ ds U , that is, ρ V (f )|f | ≤ ρ U .Equality at some point of U implies that U f → V is an unramified covering; in this case, equality holds everywhere in U .(2.1)

Hausdorff distance. Given two subsets
Denote by (X, d) := ( X , d ) the metric space of nonempty compact subsets of X.
If (X, d) is a complete metric space, then (X, d) is a complete metric space.Therefore, when U is a hyperbolic domain, (U, s U ) is a complete metric space.

Stolz subdomains.
Given a subdomain V of a hyperbolic planar domain Roughly speaking, if V ⊂ U are planar domains with piecewise C 1 -smooth boundaries, then V is Stolz in U if and only if the boundaries ∂U and ∂V have no tangency at any common boundary point.

Proper curves.
By a curve in a given holomorphic manifold, we always mean a closed analytic subset of pure dimension one.

Schwarz lemma of Nehari and Minda.
Using Ahlfors' lemma, Nehari [4] and then Minda [3] prove a Schwarz-Pick lemma for multivalent functions.We formulate and prove their result, in a more convenient form.Proposition 3.4.Let U and V be hyperbolic planar domains.
Clearly, ρ B is real-analytic, and 0 positive functions in a planar domain, and let W g → (0, ∞) denote their geometric mean, g = [ i ρ i ] 1/m .The inequality between the arithmetic mean and the geometric mean of finitely many positive numbers implies that if K := max i (K ρ i ) < 0, then K g ≤ K.
If x ≠ p(B) is in a small neighborhood of p(B), then B is given over a small neighborhood W of x by the union of graphs of m(B) biholomorphisms The restriction to W of ρ B is the geometric mean of the functions ρ i .The conformal invariance of the curvature implies that In conclusion, ρ is ultra-hyperbolic in U .Ahlfors' lemma implies that ρ ≤ ρ U , and the proof is finished.
and the Schwarz-Pick lemma implies, for all z ∈ ∆, It is amusing to see this directly, as follows.Rewrite the inequality as Using the conformal invariance of the quantity (1−|z| 2 )|g (z)|, we may assume that z = 0 and g(0) ≠ 0. We need to show that |g /g|(0) ≤ u n (|g(0)|).Note two properties of u n : (i) and the inequality (3.2) is shown.
Let U , V be planar domains, and To show that such v 0 exists, we use an analytic continuation argument.
Any local branch of C at (x 0 ,y 0 ) has a normalization of form t (x, y) → U be the analytic extension of the section-germ (f 0 ,x n ).Then Since p is proper, we may assume (extracting a subsequence), that The proof is complete.Remark 4.2.The proof of Corollary 3.7 shows that contractive curves have the following coherence property: there is a constant k = k(U, V ) > 1 such that, for all ((x, y), u) ∈ C × U , there exists v ∈ V with (u, v) ∈ C and ks U (y, v) ≤ s U (x, u).
The hyperbolic metric is locally equivalent to the Euclidean metric , for all K ∈ U .We call A C the attractor associated to the contractive curve C.

Orbits and limit sets
for some r ≥ 1.We denote by Fix(C) the set of fixed points of C, and by Per(C) the set of periodic points of C.
A weak orbit of a point x ∈ U is a sequence of points x r ∈ r C (x), r ≥ 0. An orbit of x is a sequence of points x r ∈ U , r ≥ 0, with x 0 = x and (x r ,x r +1 ) ∈ C for all r ≥ 0. A (weak) suborbit of x is a subsequence of a (weak) orbit.
The limit set A wo (x) is the set of points u ∈ U such that some weak orbit of x converges to u; A so (x) is the set of points u ∈ U such that some suborbit of x converges to u.Similarly, define the limit sets A wso (x) and A o (x).
The total orbit X + of X ⊂ U is the union of orbits of the points of X.
Corollary 4.4.If C ⊂ U × V is a contractive curve, then, for all points x ∈ U , Proof.Given arbitrary points x ∈ U and a ∈ A C , lim r s U ( r C (x), A C ) = 0, hence lim r s U ( r C (x), a) = 0. We get a sequence x r ∈ r C (x) with lim r x r = a, so that a ∈ A wo (x).Therefore, A C ⊂ A wo (x).Fix u ∈ A wso (x), and let (x r j ) j be a weak suborbit of x that converges to u.Since lim j s U ( r j C (x), A C ) = 0, we get lim j s U (x r j ,A C ) = 0, and then Clearly, A wo (x) ⊂ A wso (x), and we obtain A C = A wo (x) = A wso (x).Let ( j ) j be a sequence that decreases to 0. Since a ∈ A wo (x), there exists Repeating this procedure, we get a suborbit x j of x with s U (a, x j ) ≤ j , hence lim j x j = a.It follows that A C ⊂ A so (x).Since A so (x) ⊂ A wso (x) = A C , we get A so (x) = A C .
Fix an orbit (x r ) r of x that converges to u.Since C is closed, (x r ,x r +1 ) ∈ C, and lim r (x r ,x r +1 ) = (u, u), we get (u, u) ∈ C. Therefore, A o (x) ⊂ Fix(C).

Periodic points.
We prove the density in A C of the periodic points of a contractive curve C without singular branches.
Then, for all x ∈ A C and all branches B of C at x, B has a section defined on ∆ U (x, C ).In particular, for all x ∈ A C and all y ∈ C (x), Therefore, f (∆ U (x, )) ⊂ ∆ U (x, ), and f must have a fixed point z ∈ ∆ U (x, ).Then z ∈ r C (z), so that z ∈ Per(C).

Continuity. Let
Corollary 4.7.Let (Ꮿ(U , V ), s ) be the space of contractive curves in U × V , and denote by  Consequently, .12) so the limit is independent of the divisor.Let K = r r C (supp(D)).Then K is compact in U , and C (K) ⊂ K. Given φ 1 and φ 2 in Ꮿ(K), Φ 1 ( r D) and Φ 2 ( r D) are well defined.Moreover, for all r ≥ 0, |Φ 1 ( r D) − Φ 2 ( r D)| ≤ φ 1 − φ 2 K .The subspace of Ꮿ(K) formed by the functions that admit a Lipschitz extension to U is dense in Ꮿ(K).It follows that, for arbitrary φ ∈ Ꮿ(U), lim r Φ( r D) exists in C. To see that this limit is independent of the divisor, take A, B ∈ Sym(U ), put K = r r C (supp(A + B)), and approximate φ in Ꮿ(K) with functions that admit Lipschitz extensions to U.
Proposition 4.11.There exists a (unique) probability measure µ C on A C with lim r →∞ Φ( r D) = A C φdµ C for all φ ∈ Ꮿ(U) and all D ∈ Sym(U).
By definition, the measure µ C describes the frequency with which Borel sets are visited by the total orbit of any point.
where U is the union of the three disjoint disks of radius 0.1 and centers 0, 1, 2. Thus, A can be viewed as the attractor associated to the mixed iteration of f and g on U .But the point 2 is not periodic.Of course, no domain containing the points 0 and 1 can be f -invariant.Then C is contractive in a suitable product of annuli centered at 0. Figure 5.1b shows the corresponding attractor, for C = (y 5 = 29i(x − 1) 2 ).
and define the Hausdorff distance d (K, H) := max sup k∈K d(k, H), sup h∈H d(K, h) .
be a normalization map of the (possibly reducible) plane curve C. Recall that C is the curve of local branches of C. Let C ν → Č be the dual curve of C, that is, the curve of tangents of local branches of C. Then Č ⊂ ᏸ, where ᏸ C 2 is the space of nonvertical lines in C 2 .Let ᏸ σ → C be the holomorphic map that associates to a line its slope.The map C s → C, B s(B), is holomorphic since it factorizes as C ν → Č ⊂ ᏸ σ → C. Given x ∈ U , let C(x) be the set of local branches B of C with p(B) = x.Since the projection C p → U is proper, C(x) is finite.Define the function U ρ → [0, ∞), ρ(x) := max B∈C(x) ρ V (q(B))•|s(B)|.By definition, ρ V (f )|f | ≤ ρ for every local section f of C. Clearly, ρ ∈ C(U).We show that ρ is ultra-hyperbolic.Let B be a local branch of C with s(B) ≠ 0. Let m(B) denote the multiplicity of B. Since B has nonvertical tangent, m(B) equals the local degree of B p → U. Since B has nonhorizontal tangent, m(B) equals the local degree of q : B → V .We can define, in a punctured neighborhood of p(B), the function denote the discrete set of branch points of C p → U .Choose x ∈ D 0 \ B(p) and u ∈ U \ B(p).Let y = f 0 (x).Let D ⊂ U \ B(p) be a simply connected domain with {x, u} ⊂ D. The section-germ (f 0 ,x) can be analytically extended to a section D f → U .If we put v = s(u), we have s V (y, v) = s V (f (x), f (u)) ≤ s U (x, u).Now, let D 0 \B(p) x n → x 0 and U \B(p) u n → u 0 .Let D n ⊂ U \B(p) be a simply connected domain with {x n ,u n } ⊂ D n , and let D n fn

4. Attractor associated to a contractive curve 4 . 1 .
Contractive curves.Given V ⊂ U , let V i → U denote the inclusion map.Given a proper curve C ⊂ U × V , U C → U denotes the self-map C = i C .Definition 4.1.A contractive curve is a proper curve C ⊂ U × V ,where U is a hyperbolic planar domain and V is a Stolz subdomain of U.

Corollary 4 . 6 .
If a contractive curve C ⊂ U × V has no singular branches, then A C = Per(C).Proof.Clearly, Per(C) ⊂ A C .To prove the other inclusion, note that, under the assumption that C has no singular branches, every local branch B of C has a section defined in a hyperbolic disk ∆ U (p(c), ), for some > 0. Let (B) be the supremum of such .The function C B (B) ∈ (0, ∞] is lower semicontinuous.Let C := min{ (B) :

4. 6 .
Balanced measure of Barnsley-Demko.Sym a (U) denotes the quotient of U a by the action of the symmetric group S a .In other words, Sym a (U) is the space of effective divisors on U of degree a. Define a complete distance δ on Sym a (U) as follows.Given A = a j=1 (x j ) and B = a j=1 (u j ), δ(A, B) := (1/a) min σ ∈Sa ( a j=1 s U (x j ,u σ (j) )).Clearly, the map (U , s U ) I → (Sym a (U ), δ), where I(x) = a(x), is an isometric embedding.Let Sym(U ) := ⊕ a Sym a (U ).Let V i U be a Stolz subdomain of a hyperbolic planar domain.Recall that k = 1/ Lip(i) > 1.Let C ⊂ U × V be a contractive curve, with projections C p → U and C q → V .Let d be the topological degree of p. Define Sym a (U) → Sym da (U), (D) := i * q * p * D, and Sym a (U ) γ → [0, ∞), γ(D) := δ(dD, D).

Definition 3.2. Let
U and V be planar domains.A curve C ⊂ U ×V is called proper if and only if the following two conditions are satisfied: