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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2604.23737 |
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| _version_ | 1866911623370768384 |
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| author | Kuang, Rui Li, Bing Xiao, Yuanfen |
| author_facet | Kuang, Rui Li, Bing Xiao, Yuanfen |
| contents | Let $ 1<β< 2 $, the sequence $α(β)=α(β)_1α(β)_2\dotsb $ be the quasi-greedy $ β$-expansion of $ 1 $, and $ t\in [0,1) $ be a bifurcation parameter. The $β$-transformation is defined to be $T_β(x)=βx (mod 1) $ for $x\in [0,1)$. The Hausdorff dimension of the survivor set $K(t)=\{x\in [0,1)\colon T_β^k(x)\not\in (0,t), \forall k\geq0\} $ is equal to $ -\frac{\lnλ}{\lnβ} $ under the condition that $ \sum_{i=k}^{\infty}\frac{α(β)_i }{β^i}\geq t $ for any $ k\geq 1 $, where $ λ\in (0,1) $ is the smallest positive solution of the equation $\sum_{n=1}^{\infty}(α(β)_n-t_n)x^n=1$ with $(t_n) $ being the quasi-greedy $β$-expansion of $t$. And the local Hölder exponent of the Hausdorff dimension function of $K(t) $ is larger than the value of the function itself. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_23737 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The-Hausdorff-dimension-of-the-survivor-set Kuang, Rui Li, Bing Xiao, Yuanfen Dynamical Systems 37B40, 37B10 Let $ 1<β< 2 $, the sequence $α(β)=α(β)_1α(β)_2\dotsb $ be the quasi-greedy $ β$-expansion of $ 1 $, and $ t\in [0,1) $ be a bifurcation parameter. The $β$-transformation is defined to be $T_β(x)=βx (mod 1) $ for $x\in [0,1)$. The Hausdorff dimension of the survivor set $K(t)=\{x\in [0,1)\colon T_β^k(x)\not\in (0,t), \forall k\geq0\} $ is equal to $ -\frac{\lnλ}{\lnβ} $ under the condition that $ \sum_{i=k}^{\infty}\frac{α(β)_i }{β^i}\geq t $ for any $ k\geq 1 $, where $ λ\in (0,1) $ is the smallest positive solution of the equation $\sum_{n=1}^{\infty}(α(β)_n-t_n)x^n=1$ with $(t_n) $ being the quasi-greedy $β$-expansion of $t$. And the local Hölder exponent of the Hausdorff dimension function of $K(t) $ is larger than the value of the function itself. |
| title | The-Hausdorff-dimension-of-the-survivor-set |
| topic | Dynamical Systems 37B40, 37B10 |
| url | https://arxiv.org/abs/2604.23737 |