Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2504.02135 |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866914538168778752 |
|---|---|
| author | Tryniecki, Rafał Urbański, Mariusz Zdunik, Anna |
| author_facet | Tryniecki, Rafał Urbański, Mariusz Zdunik, Anna |
| contents | Let $G(x):=\{1/x\}$ be the Gauss map. By $g_n(x)=\frac{1}{x+n}$ we denote its continuous/real analytic inverse branches. We define iterated function system (IFS) $G_n$ by limiting the collection of functions $g_k$, $k\in\mathbb N$, to the first $n$ elements, meaning that $G_n = \{g_k \}_{k=1}^n$. We are interested in the asymptotics of the Hausdorff measure of the limit set $J_n$ i. e. set consisting of irrational elements of $[0,1]$ having continued fraction expansion with entries at most $n$. In the first part of the paper, we deal with the piecewise-linear analogue of the Gauss map and resulting IFSs. We prove that \[ \lim \limits_{n \to \infty } \frac{1-H_n(J_n)}{1-h_n} \cdot \frac{1}{\ln n} = 1, \] where $J_n$ is the limit set of the piecewise-linear analogue of $G_n$, $h_n$ is its Hausdorff dimension and $H_n$ is the value of $h_n$-dimensional Hausdorff measure of the set $J_n$, $H_n:=H_{h_n}(J_n)$.
In the second part, we focus on the IFS generated by the first $n$ branches of Gauss map and prove, as our main result, that $$ \lim_{n\to\infty} \frac{1-H_n}{(1-h_n)\ln n}= 1 $$ and equivalently, due to Hensley's result, $$ \lim_{n\to\infty} \frac{n(1-H_n)}{\ln n}= \frac{6}{π^2}, $$ where $J_n$ is the limit set of the system $G_n$, i.e. the set consisting of irrational numbers in $[0,1]$ that continued fraction expansion with entries not exceeding $n$. Similarly as for the piecewise linear map, $h_n$ is the Hausdorff dimension of $J_n$ and $H_n$ is the value of $h_n$-dimensional Hausdorff measure of the set $J_n$, $H_n:=H_{h_n}(J_n)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_02135 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Asymptotics of the Hausdorff measure for the Gauss map and its linearized analogue Tryniecki, Rafał Urbański, Mariusz Zdunik, Anna Dynamical Systems 37E05 Let $G(x):=\{1/x\}$ be the Gauss map. By $g_n(x)=\frac{1}{x+n}$ we denote its continuous/real analytic inverse branches. We define iterated function system (IFS) $G_n$ by limiting the collection of functions $g_k$, $k\in\mathbb N$, to the first $n$ elements, meaning that $G_n = \{g_k \}_{k=1}^n$. We are interested in the asymptotics of the Hausdorff measure of the limit set $J_n$ i. e. set consisting of irrational elements of $[0,1]$ having continued fraction expansion with entries at most $n$. In the first part of the paper, we deal with the piecewise-linear analogue of the Gauss map and resulting IFSs. We prove that \[ \lim \limits_{n \to \infty } \frac{1-H_n(J_n)}{1-h_n} \cdot \frac{1}{\ln n} = 1, \] where $J_n$ is the limit set of the piecewise-linear analogue of $G_n$, $h_n$ is its Hausdorff dimension and $H_n$ is the value of $h_n$-dimensional Hausdorff measure of the set $J_n$, $H_n:=H_{h_n}(J_n)$. In the second part, we focus on the IFS generated by the first $n$ branches of Gauss map and prove, as our main result, that $$ \lim_{n\to\infty} \frac{1-H_n}{(1-h_n)\ln n}= 1 $$ and equivalently, due to Hensley's result, $$ \lim_{n\to\infty} \frac{n(1-H_n)}{\ln n}= \frac{6}{π^2}, $$ where $J_n$ is the limit set of the system $G_n$, i.e. the set consisting of irrational numbers in $[0,1]$ that continued fraction expansion with entries not exceeding $n$. Similarly as for the piecewise linear map, $h_n$ is the Hausdorff dimension of $J_n$ and $H_n$ is the value of $h_n$-dimensional Hausdorff measure of the set $J_n$, $H_n:=H_{h_n}(J_n)$. |
| title | Asymptotics of the Hausdorff measure for the Gauss map and its linearized analogue |
| topic | Dynamical Systems 37E05 |
| url | https://arxiv.org/abs/2504.02135 |