Abstract
Let Δ \Delta be the unit disc of C \mathbb C and let f , g ∈ H o l ( Δ , Δ ) f,g \in \mathrm {Hol}(\Delta ,\Delta ) be such that f ∘ g = g ∘ f f \circ g = g \circ f . For A > 1 A>1 , let F i x A ( f ) := { p ∈ ∂ Δ ∣ lim r → 1 f ( r p ) = p , lim r → 1 | f ′ ( r p ) | ≤ A } \mathrm {Fix}_A (f):=\{p \in \partial \Delta \mid \lim _{r \to 1}f(rp)=p, \lim _{r \to 1}|f’(rp)|\leq A \} . We study the behavior of g g on F i x A ( f ) \mathrm {Fix}_A (f) . In particular, we prove that g ( F i x A ( f ) ) ⊆ F i x A ( f ) g(\mathrm {Fix}_A (f))\subseteq \mathrm {Fix}_A (f) . As a consequence, besides conditions for F i x A ( f ) ∩ F i x A ( g ) ≠ ∅ \mathrm {Fix}_A(f) \cap \mathrm {Fix}_A(g) \neq \emptyset , we prove a conjecture of C. Cowen in case f f and g g are univalent mappings.
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