Survivor Set of Gauss Map with a Hole at $1$
Abstract
Let T be the Gauss map. For each α∈ [0,1], let E(α) be the survivor set consisting of all x∈ [0,1] whose orbit {Tk(x):k≥ 0} never enters the interval (α,1]. We prove that the set-valued function α↦ E(α) is increasing and locally constant almost everywhere. Moreover, we explicitly characterize all intervals where the function α↦ E(α) remains constant. We also describe the points at which the map α↦ E(α) is not locally constant. Further, we obtain the Hausdorff dimension of E(α) for all α∈ [0,1]. We prove that the function α↦ dimH(E(α)) forms a devil's staircase on (0,1], but is not continuous at $0$. This discontinuity implies that the result of Raith [Acta Math. Univ. Comenian. 63 (1994), 39-53] fails for piecewise monotone interval maps with infinitely many branches. Finally, we give an application quantifying the set of points sharing identical α-continued fraction expansion and regular continued fraction expansion.
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BibTeX
@misc{liu2026survivor,
title = {{Survivor Set of Gauss Map with a Hole at $1$}},
author = {Cheng Liu and Lingmin Liao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.14534},
eprint = {2610.14534},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.14534},
}
RIS
TY - UNPB
TI - Survivor Set of Gauss Map with a Hole at $1$
AU - Cheng Liu
AU - Lingmin Liao
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2610.14534
UR - https://arxiv.org/abs/2610.14534
AB - Let T be the Gauss map. For each α∈ [0,1], let E(α) be the survivor set consisting of all x∈ [0,1] whose orbit {T^k(x):k≥ 0} never enters the interval (α,1]. We prove that the set-valued function α↦ E(α) is increasing and locally constant almost everywhere. Moreover, we explicitly characterize all intervals where the function α↦ E(α) remains constant. We also describe the points at which the map α↦ E(α) is not locally constant. Further, we obtain the Hausdorff dimension of E(α) for all α∈ [0,1]. We prove that the function α↦ dim_H(E(α)) forms a devil's staircase on (0,1], but is not continuous at $0$. This discontinuity implies that the result of Raith [Acta Math. Univ. Comenian. 63 (1994), 39-53] fails for piecewise monotone interval maps with infinitely many branches. Finally, we give an application quantifying the set of points sharing identical α-continued fraction expansion and regular continued fraction expansion.
ER -