A note on forward iteration of inner functions
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arXiv
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866917002064429056 |
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| author | Ferreira, Gustavo Rodrigues |
| author_facet | Ferreira, Gustavo Rodrigues |
| contents | A well-known problem in holomorphic dynamics is to obtain Denjoy--Wolff-type results for compositions of self-maps of the unit disc. Here, we tackle the particular case of inner functions: if $f_n:\mathbb{D}\to\mathbb{D}$ are inner functions fixing the origin, we show that a limit function of $f_n\circ\cdots\circ f_1$ is either constant or an inner function. For the special case of Blaschke products, we prove a similar result and show, furthermore, that imposing certain conditions on the speed of convergence guarantees $L^1$ convergence of the boundary extensions. We give a counterexample showing that, without these extra conditions, the boundary extensions may diverge at all points of $\partial\mathbb{D}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2206_00374 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A note on forward iteration of inner functions Ferreira, Gustavo Rodrigues Complex Variables Dynamical Systems A well-known problem in holomorphic dynamics is to obtain Denjoy--Wolff-type results for compositions of self-maps of the unit disc. Here, we tackle the particular case of inner functions: if $f_n:\mathbb{D}\to\mathbb{D}$ are inner functions fixing the origin, we show that a limit function of $f_n\circ\cdots\circ f_1$ is either constant or an inner function. For the special case of Blaschke products, we prove a similar result and show, furthermore, that imposing certain conditions on the speed of convergence guarantees $L^1$ convergence of the boundary extensions. We give a counterexample showing that, without these extra conditions, the boundary extensions may diverge at all points of $\partial\mathbb{D}$. |
| title | A note on forward iteration of inner functions |
| topic | Complex Variables Dynamical Systems |
| url | https://arxiv.org/abs/2206.00374 |