Inner Functions and Laminations
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866917657336348672 |
|---|---|
| author | Ivrii, Oleg Urbański, Mariusz |
| author_facet | Ivrii, Oleg Urbański, Mariusz |
| contents | In this paper, we study orbit counting problems for inner functions using geodesic and horocyclic flows on Riemann surface laminations. For a one component inner function of finite Lyapunov exponent with $F(0) = 0$, other than $z \to z^d$, we show that the number of pre-images of a point $z \in \mathbb{D} \setminus \{ 0\}$ that lie in a ball of hyperbolic radius $R$ centered at the origin satisfies $$ \mathcal{N}(z, R) \, \sim \, \frac{1}{2} \log \frac{1}{|z|} \cdot \frac{1}{\int_{\partial \mathbb{D}} \log |F'| dm}, \quad \text{as }R \to \infty. $$ For a general inner function of finite Lyapunov exponent, we show that the above formula holds up to a Cesàro average. Our main insight is that iteration along almost every inverse orbit is asymptotically linear. We also prove analogues of these results for parabolic inner functions of infinite height. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_02878 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Inner Functions and Laminations Ivrii, Oleg Urbański, Mariusz Dynamical Systems Complex Variables 30J05, 37D40, 37A25 In this paper, we study orbit counting problems for inner functions using geodesic and horocyclic flows on Riemann surface laminations. For a one component inner function of finite Lyapunov exponent with $F(0) = 0$, other than $z \to z^d$, we show that the number of pre-images of a point $z \in \mathbb{D} \setminus \{ 0\}$ that lie in a ball of hyperbolic radius $R$ centered at the origin satisfies $$ \mathcal{N}(z, R) \, \sim \, \frac{1}{2} \log \frac{1}{|z|} \cdot \frac{1}{\int_{\partial \mathbb{D}} \log |F'| dm}, \quad \text{as }R \to \infty. $$ For a general inner function of finite Lyapunov exponent, we show that the above formula holds up to a Cesàro average. Our main insight is that iteration along almost every inverse orbit is asymptotically linear. We also prove analogues of these results for parabolic inner functions of infinite height. |
| title | Inner Functions and Laminations |
| topic | Dynamical Systems Complex Variables 30J05, 37D40, 37A25 |
| url | https://arxiv.org/abs/2405.02878 |