Canonical models on strongly convex domains via the squeezing function. (English) Zbl 1464.32025
Summary: We prove that if a holomorphic self-map \(f:\Omega\rightarrow\Omega\) of a bounded strongly convex domain \(\Omega\subset\mathbb{C}^q\) with smooth boundary is hyperbolic then it admits a natural semi-conjugacy with a hyperbolic automorphism of a possibly lower dimensional ball \(\mathbb{B}^k\). We also obtain the dual result for a holomorphic self-map \(f:\Omega\rightarrow\Omega\) with a boundary repelling fixed point. Both results are obtained by rescaling the dynamics of \(f\) via the squeezing function.
MSC:
| 32H50 | Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables |
| 32A40 | Boundary behavior of holomorphic functions of several complex variables |
| 32T15 | Strongly pseudoconvex domains |
| 37F99 | Dynamical systems over complex numbers |
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