Shrinking targets and recurrent behaviour for forward compositions of inner functions
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909207151771648 |
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| author | Benini, Anna Miriam Evdoridou, Vasiliki Fagella, Núria Rippon, Philip J. Stallard, Gwyneth M. |
| author_facet | Benini, Anna Miriam Evdoridou, Vasiliki Fagella, Núria Rippon, Philip J. Stallard, Gwyneth M. |
| contents | We prove sharp results about recurrent behaviour of orbits of forward compositions of inner functions, inspired by fundamental results about iterates of inner functions, and give examples to illustrate behaviours that cannot occur in the simpler case of iteration.
A result of Fernández, Melián and Pestana gives a precise version of the classical Poincaré recurrence theorem for iterates of the boundary extension of an inner function that fixes~0. We generalise this to forward composition sequences $F_n=f_n\circ \dots\circ f_1,$ $n\in \mathbb{N},$ where $f_n$ are inner functions that fix~0, giving conditions on the contraction of $(F_n)$ so that the radial boundary extension $F_n$ hits any shrinking target of arcs $(I_n)$ of a given size.
Next, Aaronson, and also Doering and Mañé, gave a remarkable dichotomy for iterates of any inner function, showing that the behaviour of the boundary extension is of two entirely different types, depending on the size of the sequence $(|f^n(0)|)$. In earlier work, we showed that one part of this dichotomy holds in the non-autonomous setting of forward compositions.
It turns out that this dichotomy is closely related to the result of Fernández, Melián and Pestana, and here we show that a version of the second part of the dichotomy holds in the non-autonomous setting provided we impose a condition on the contraction of $(F_n)$ in relation to the size of the sequence $(|F_n(0)|)$. The techniques we use include a strong version of the second Borel--Cantelli lemma and strong mixing results of Pommerenke for contracting sequences of inner functions. We give examples to show that the contraction conditions that we need to impose in the non-autonomous setting are best possible. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_11866 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Shrinking targets and recurrent behaviour for forward compositions of inner functions Benini, Anna Miriam Evdoridou, Vasiliki Fagella, Núria Rippon, Philip J. Stallard, Gwyneth M. Dynamical Systems 37D05, 37A25, 30D05, 37F10, 28D05, 37F99 We prove sharp results about recurrent behaviour of orbits of forward compositions of inner functions, inspired by fundamental results about iterates of inner functions, and give examples to illustrate behaviours that cannot occur in the simpler case of iteration. A result of Fernández, Melián and Pestana gives a precise version of the classical Poincaré recurrence theorem for iterates of the boundary extension of an inner function that fixes~0. We generalise this to forward composition sequences $F_n=f_n\circ \dots\circ f_1,$ $n\in \mathbb{N},$ where $f_n$ are inner functions that fix~0, giving conditions on the contraction of $(F_n)$ so that the radial boundary extension $F_n$ hits any shrinking target of arcs $(I_n)$ of a given size. Next, Aaronson, and also Doering and Mañé, gave a remarkable dichotomy for iterates of any inner function, showing that the behaviour of the boundary extension is of two entirely different types, depending on the size of the sequence $(|f^n(0)|)$. In earlier work, we showed that one part of this dichotomy holds in the non-autonomous setting of forward compositions. It turns out that this dichotomy is closely related to the result of Fernández, Melián and Pestana, and here we show that a version of the second part of the dichotomy holds in the non-autonomous setting provided we impose a condition on the contraction of $(F_n)$ in relation to the size of the sequence $(|F_n(0)|)$. The techniques we use include a strong version of the second Borel--Cantelli lemma and strong mixing results of Pommerenke for contracting sequences of inner functions. We give examples to show that the contraction conditions that we need to impose in the non-autonomous setting are best possible. |
| title | Shrinking targets and recurrent behaviour for forward compositions of inner functions |
| topic | Dynamical Systems 37D05, 37A25, 30D05, 37F10, 28D05, 37F99 |
| url | https://arxiv.org/abs/2405.11866 |