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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2409.14428 |
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| _version_ | 1866909712524509184 |
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| author | Dajani, Karma Huang, Yan |
| author_facet | Dajani, Karma Huang, Yan |
| contents | Given $β>1$ and $α\in[0,1)$, let $T_{β, α}(x)=βx+α\pmod 1$. Then under the map $T_{β,α}$ each $x\in[0,1]$ has an \emph{intermediate $β$-expansion} of the form $x=\sum_{i=1}^\infty\frac{c_i-α}{β^i}$ {with each $c_i\in\{0,1,\ldots,\lf β+α\rf\}$}. In this paper we study the approximation properties of $T_{β,α}$ by considering the expected value $M_β(α)$ of the \emph{normalized errors} $(θ_{β,α}^n(x))_{n\geq 1}$, where $$θ_{β,α}^n(x):=β^n\left(x-\sum_{i=1}^n\frac{c_i-α}{β^i}\right),\quad n\in\mathbb{N}.$$ We prove that $M_β(\cdot)$ is continuous on $[0,1)$. As a result, $\mathcal{M_β}:=\{M_β(α):α\in[0,1)\}$ is a closed interval. In particular, if $β$ is a multinacci number, the map $T_{β,α}$ has matching for Lebesgue almost every $α\in[0,1)$, and then $M_β(\cdot)$ is locally linear almost everywhere on $[0,1)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_14428 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Approximation properties of the intermediate $β$-expansions Dajani, Karma Huang, Yan Dynamical Systems Given $β>1$ and $α\in[0,1)$, let $T_{β, α}(x)=βx+α\pmod 1$. Then under the map $T_{β,α}$ each $x\in[0,1]$ has an \emph{intermediate $β$-expansion} of the form $x=\sum_{i=1}^\infty\frac{c_i-α}{β^i}$ {with each $c_i\in\{0,1,\ldots,\lf β+α\rf\}$}. In this paper we study the approximation properties of $T_{β,α}$ by considering the expected value $M_β(α)$ of the \emph{normalized errors} $(θ_{β,α}^n(x))_{n\geq 1}$, where $$θ_{β,α}^n(x):=β^n\left(x-\sum_{i=1}^n\frac{c_i-α}{β^i}\right),\quad n\in\mathbb{N}.$$ We prove that $M_β(\cdot)$ is continuous on $[0,1)$. As a result, $\mathcal{M_β}:=\{M_β(α):α\in[0,1)\}$ is a closed interval. In particular, if $β$ is a multinacci number, the map $T_{β,α}$ has matching for Lebesgue almost every $α\in[0,1)$, and then $M_β(\cdot)$ is locally linear almost everywhere on $[0,1)$. |
| title | Approximation properties of the intermediate $β$-expansions |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2409.14428 |