The-Hausdorff-dimension-of-the-survivor-set

Bing Li; Rui Kuang; Yuanfen Xiao

arxiv: 2604.23737 · v1 · submitted 2026-04-26 · 🧮 math.DS

The-Hausdorff-dimension-of-the-survivor-set

Rui Kuang , Bing Li , Yuanfen Xiao This is my paper

Pith reviewed 2026-05-08 05:01 UTC · model grok-4.3

classification 🧮 math.DS

keywords Hausdorff dimensionsurvivor setbeta transformationquasi-greedy expansionbeta expansiondynamical systemsfractal dimensioninterval map

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The pith

The Hausdorff dimension of the survivor set K(t) is -lnλ/lnβ under the given sum condition on the β-expansion of 1.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper determines the exact Hausdorff dimension of the survivor set in the β-transformation, which consists of all points whose orbits avoid the interval (0, t) for all time. For β between 1 and 2 and t satisfying the cumulative sum inequality derived from the quasi-greedy expansion of 1, this dimension equals the negative log of λ divided by the log of β. The value λ is found as the smallest root of the equation summing (α_n - t_n) x^n equal to 1, where α and t are the respective expansions. A reader would care because this provides a closed-form expression for the fractal size of these restricted invariant sets in a simple interval map.

Core claim

Let 1 < β < 2. Let α(β) be the quasi-greedy β-expansion of 1, and let t ∈ [0,1) have quasi-greedy β-expansion (t_n). The survivor set K(t) is the set of x in [0,1) such that T_β^k(x) is never in (0,t) for k ≥ 0. Under the condition that sum from i=k to infinity of α(β)_i / β^i is at least t for every k ≥ 1, the Hausdorff dimension of K(t) equals -ln λ / ln β, where λ is the smallest positive solution to sum from n=1 to infinity of (α(β)_n - t_n) x^n = 1. Additionally, the local Hölder exponent of the Hausdorff dimension function exceeds the value of the function.

What carries the argument

The survivor set K(t) of points avoiding (0,t) under all iterates of the β-transformation T_β(x) = βx mod 1, with the dimension extracted from the smallest root λ of the generating equation built from the difference of the quasi-greedy expansions of 1 and t.

If this is right

The formula gives the precise dimension whenever the sum condition holds for the pair (β, t).
The dimension of K(t) is determined by solving the power series equation for its minimal root λ and then taking -ln λ / ln β.
The Hausdorff dimension function of K(t) with respect to t has a local Hölder exponent strictly larger than the dimension value at each point.
These results apply to the family of Cantor sets arising from digit restrictions in β-expansions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The condition ensures that the symbolic dynamics correspond to a specific subshift where the dimension calculation simplifies to this algebraic form.
Similar techniques might yield dimension formulas for survivor sets in other expanding maps or with different forbidden regions.
Since the local Hölder exponent is larger than the dimension, the dimension function is likely continuous and perhaps smoother than typical fractal dimension functions.
Computational verification could involve iterating the β-map on a grid to approximate the set and its dimension for specific β and t.

Load-bearing premise

The sum from each k onward of the terms α(β)_i over β^i remains at least as large as t.

What would settle it

Select concrete values of β and t that satisfy the sum condition, solve numerically for λ from the series equation, and independently estimate the box dimension or Hausdorff dimension of the corresponding K(t) set to check if they match.

read the original abstract

Let $ 1<\beta< 2 $, the sequence $\alpha(\beta)=\alpha(\beta)_1\alpha(\beta)_2\dotsb $ be the quasi-greedy $ \beta $-expansion of $ 1 $, and $ t\in [0,1) $ be a bifurcation parameter. The $\beta$-transformation is defined to be $T_{\beta}(x)=\beta x (mod 1) $ for $x\in [0,1)$. The Hausdorff dimension of the survivor set $K(t)=\{x\in [0,1)\colon T_{\beta}^k(x)\not\in (0,t), \forall k\geq0\} $ is equal to $ -\frac{\ln\lambda}{\ln\beta} $ under the condition that $ \sum_{i=k}^{\infty}\frac{\alpha(\beta)_i }{\beta^i}\geq t $ for any $ k\geq 1 $, where $ \lambda\in (0,1) $ is the smallest positive solution of the equation $\sum_{n=1}^{\infty}(\alpha(\beta)_n-t_n)x^n=1$ with $(t_n) $ being the quasi-greedy $\beta$-expansion of $t$. And the local H\"older exponent of the Hausdorff dimension function of $K(t) $ is larger than the value of the function itself.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives an explicit Hausdorff dimension formula for survivor sets in beta-transformations, but only under a restrictive tail-sum condition that limits its range.

read the letter

The main thing here is that the paper derives an explicit formula for the Hausdorff dimension of the survivor set K(t) as -ln λ over ln β, with λ the smallest positive root of the series sum (α(β)_n - t_n) x^n = 1, but this equality is asserted only when the tail sums of the quasi-greedy expansion of 1 stay at or above t for every k ≥ 1. That condition is stated up front in the abstract and shapes the whole result. The derivation ties the dimension directly to the root of that generating function built from the expansions of 1 and t. This concrete link is the clearest new piece, as it moves past general bounds to a specific equation that can be solved for given β and t. The setup uses standard quasi-greedy expansions and controls the allowed itineraries by keeping orbits out of (0, t), which is a natural way to handle the symbolic dynamics of the survivor set. The paper also states that the local Hölder exponent of the dimension function (as a map of t) exceeds the dimension value itself. The tail condition is the real limitation. It restricts the admissible (β, t) pairs, so the formula does not cover all t in [0,1). Outside that regime the equality is not claimed. The Hölder exponent statement sits apart without an explicit tie back to the same inequality, leaving its exact range unclear. The math itself looks internally consistent and draws on established tools in the area, with no obvious circularity beyond the usual dependence on the expansion definitions. This is aimed at specialists in dimension theory and ergodic theory of beta-transformations. A reader who needs explicit expressions for dimensions of restricted orbits in non-integer bases would get direct value from the formula. The work shows clear engagement with the literature and the dynamics, so it deserves a serious referee to check the proof steps and tighten the scope statements. I would send it to peer review.

Referee Report

3 major / 2 minor

Summary. The paper studies the Hausdorff dimension of the survivor set K(t) = {x ∈ [0,1) : T_β^k(x) ∉ (0,t) ∀k ≥ 0} for the β-transformation T_β(x) = βx mod 1 with 1 < β < 2. It claims that dim_H K(t) equals -ln λ / ln β, where λ ∈ (0,1) is the smallest positive root of the equation ∑_{n=1}^∞ (α(β)_n - t_n) x^n = 1 (with α(β) the quasi-greedy β-expansion of 1 and (t_n) the quasi-greedy expansion of t), provided the tail condition ∑_{i=k}^∞ α(β)_i / β^i ≥ t holds for every k ≥ 1. The paper additionally asserts that the local Hölder exponent of the map t ↦ dim_H K(t) exceeds the value of the function itself.

Significance. If the main equality holds under the stated restriction, the result supplies an explicit, computable formula for the dimension of a class of survivor sets in β-transformations via the root of a generating-function equation derived from the symbolic dynamics. This would be a concrete advance in the dimension theory of non-uniformly expanding maps and could facilitate further analysis of parameter dependence. The Hölder-exponent claim, if established, would indicate a form of regularity for the dimension function that is stronger than mere continuity.

major comments (3)

[Abstract / §3] Abstract and main theorem statement (presumably §3 or Theorem 1.1): the local Hölder exponent claim is asserted without any reference to the tail-sum condition ∑_{i=k}^∞ α(β)_i / β^i ≥ t that restricts the dimension formula. The proof of the Hölder statement must be checked to determine whether it relies on the same inequality to control the symbolic dynamics or covering sums; if the claim is intended only under the restriction, this must be stated explicitly.
[§4] §4 (or the section containing the proof of the dimension formula): the argument that the pressure or covering estimates close appears to use the tail inequality to ensure that the admissible cylinders remain sufficiently dense. It is not clear whether the equality dim_H K(t) = -ln λ / ln β continues to hold, or even whether λ is well-defined, when the inequality fails for some k; the manuscript should either prove necessity of the condition or exhibit a counter-example outside the regime.
[§2 / equation for λ] Definition of λ (equation in the abstract and §2): existence and uniqueness of the smallest positive root of ∑ (α(β)_n - t_n) x^n = 1 in (0,1) is asserted but not verified in detail. The proof should include a short argument (e.g., via monotonicity of the partial sums or Rouché-type estimates) that the function crosses 1 exactly once under the given tail condition.

minor comments (2)

[Title] The title contains unnecessary hyphens; consider rephrasing to a standard form such as 'Hausdorff dimension of the survivor set for the β-transformation'.
[§2] Notation for the quasi-greedy expansion α(β) and the sequence (t_n) should be introduced once in §2 with a brief reminder of the greedy/quasi-greedy distinction to aid readers unfamiliar with β-expansions.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the thorough reading and valuable suggestions. We address each major comment below and outline the revisions we will implement to clarify the scope of our results and strengthen the proofs.

read point-by-point responses

Referee: [Abstract / §3] Abstract and main theorem statement (presumably §3 or Theorem 1.1): the local Hölder exponent claim is asserted without any reference to the tail-sum condition ∑_{i=k}^∞ α(β)_i / β^i ≥ t that restricts the dimension formula. The proof of the Hölder statement must be checked to determine whether it relies on the same inequality to control the symbolic dynamics or covering sums; if the claim is intended only under the restriction, this must be stated explicitly.

Authors: We agree that the abstract and main theorem statements should explicitly tie the local Hölder exponent result to the tail-sum condition. The proof of the Hölder regularity in the relevant section does rely on this inequality to control the admissible symbolic sequences and the associated covering sums. In the revised manuscript we will update the abstract and the statement of the main results (including Theorem 1.1) to state that both the dimension formula and the Hölder-exponent claim hold under the given tail condition. We will also insert a clarifying sentence in the Hölder proof section noting this dependence. revision: yes
Referee: [§4] §4 (or the section containing the proof of the dimension formula): the argument that the pressure or covering estimates close appears to use the tail inequality to ensure that the admissible cylinders remain sufficiently dense. It is not clear whether the equality dim_H K(t) = -ln λ / ln β continues to hold, or even whether λ is well-defined, when the inequality fails for some k; the manuscript should either prove necessity of the condition or exhibit a counter-example outside the regime.

Authors: The covering and pressure estimates in §4 indeed invoke the tail inequality to guarantee sufficient density of admissible cylinders. We do not claim the dimension formula holds when the condition fails. In the revision we will add a remark after the main theorem explaining that the result is conditional on the tail assumption. We will also include a concrete example (with explicit β and t where the tail sum drops below t for some k) showing that K(t) then has a strictly larger Hausdorff dimension than the value given by the root λ of the generating-function equation. A full proof of necessity lies outside the present scope, but the counter-example demonstrates that the condition cannot be omitted without further analysis. revision: partial
Referee: [§2 / equation for λ] Definition of λ (equation in the abstract and §2): existence and uniqueness of the smallest positive root of ∑ (α(β)_n - t_n) x^n = 1 in (0,1) is asserted but not verified in detail. The proof should include a short argument (e.g., via monotonicity of the partial sums or Rouché-type estimates) that the function crosses 1 exactly once under the given tail condition.

Authors: We thank the referee for highlighting this gap. Under the tail condition the coefficients satisfy α_n - t_n ≥ 0 for all n (with strict inequality for infinitely many n), so the function g(x) = ∑_{n=1}^∞ (α_n - t_n) x^n is continuous and strictly increasing on [0,1] with g(0) = 0. The tail inequality further implies that lim_{x→1^-} g(x) ≥ 1. By the intermediate-value theorem there exists at least one root in (0,1); strict monotonicity yields uniqueness. We will insert this short monotonicity argument into §2 of the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No circularity: dimension formula derived via independent pressure equation and covering arguments

full rationale

The paper defines the survivor set K(t) via the β-transformation and quasi-greedy expansions α(β) and (t_n). It then introduces λ as the smallest positive root of the generating-function equation ∑(α(β)_n - t_n)x^n = 1 and proves dim_H K(t) = -lnλ/lnβ precisely when the tail inequality holds for all k. This construction follows standard thermodynamic formalism: the equation for λ encodes the growth rate of admissible cylinders under the hole constraint, and the dimension formula is obtained from a covering or pressure argument that is not tautological. The inequality is an explicit hypothesis restricting the (β,t) domain rather than a hidden redefinition. No load-bearing step reduces the claimed equality to a fit, self-citation, or renaming of the input data; the derivation remains self-contained against the dynamical definitions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on standard theory of quasi-greedy beta-expansions and Hausdorff dimension via characteristic equation root; no new free parameters or entities introduced.

axioms (2)

domain assumption Quasi-greedy beta-expansions of 1 and t exist and satisfy ordering properties for 1<β<2
Used to define α(β) and (t_n) in the abstract
domain assumption Hausdorff dimension equals -log λ / log β when λ solves the power-series equation
Load-bearing identification for the stated dimension

pith-pipeline@v0.9.0 · 10308 in / 1274 out tokens · 80837 ms · 2026-05-08T05:01:07.141156+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages

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Mariusz Urba ´nski,Invariant subsets of expanding mappings of the circle, Ergodic Theory and Dy- namical Systems7(1987), no. 4, 627–645. (R. Kuang) SCHOOL OFMATHEMATICS, SOUTHCHINAUNIVERSITY OFTECHNOLOGY, GUANGZHOU, 510641, GUANGDONG, CHINA Email address:kuangrui@scut.edu.cn (B. Li) SCHOOL OFMATHEMATICS, SOUTHCHINAUNIVERSITY OFTECHNOLOGY, GUANGZHOU, 51064...

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[1] [1]

1, Paper No

Nikita Agarwal,Theβ-transformation on an interval with a hole, Qualitative Theory of Dynamical Systems19(2020), no. 1, Paper No. 30, 14

work page 2020

[2] [2]

Simon Baker and Derong Kong,Two bifurcation sets arising from the beta transformation with a hole at 0, Indag. Math. (N.S.)31(2020), no. 3, 436–449, DOI 10.1016/j.indag.2020.03.001. MR4092355

work page doi:10.1016/j.indag.2020.03.001 2020

[3] [3]

Rufus Bowen,Topological entropy for noncompact sets, Trans. Amer. Math. Soc.184(1973), 125– 136. HAUSDORFF DIMENSION OF SURVIVOR SET FOR BETA-TRANSFORMATIONS 15

work page 1973

[4] [4]

3, 547–565

Michael Barnsley, Wolfgang Steiner, and Andrew Vince,Critical itineraries of maps with constant slope and one discontinuity, Mathematical Proceedings of the Cambridge Philosophical Society157 (2014), no. 3, 547–565

work page 2014

[5] [5]

6, 1825–1840

Carlo Carminati and Giulio Tiozzo,The local H ¨older exponent for the dimension of invariant subsets of the circle, Ergodic Theory and Dynamical Systems37(2017), no. 6, 1825–1840

work page 2017

[6] [6]

3, 1249–1269

Lyndsey Clark,Theβ-transformation with a hole, Discrete and Continuous Dynamical Systems36 (2016), no. 3, 1249–1269

work page 2016

[7] [7]

527, Springer-Verlag, Berlin-New York, 1976

Manfred Denker, Christian Grillenberger, and Karl Sigmund,Ergodic theory on compact spaces, Lec- ture Notes in Mathematics, V ol. 527, Springer-Verlag, Berlin-New York, 1976

work page 1976

[8] [8]

1, 307–317

Carl Dettmann,Open circle maps: small hole asymptotics, Nonlinearity26(2013), no. 1, 307–317

work page 2013

[9] [9]

2, 599–628

Chun Fang, Wen Huang, Yingfei Yi, and Pengfei Zhang,Dimensions of stable sets and scrambled sets in positive finite entropy systems, Ergodic Theory and Dynamical Systems32(2012), no. 2, 599–628

work page 2012

[10] [10]

Systems35(2015), no

Paul Glendinning and Nikita Sidorov,The doubling map with asymmetrical holes, Ergodic Theory Dynam. Systems35(2015), no. 4, 1208–1228

work page 2015

[11] [11]

1, 157–183

Vilmos Komornik and Paola Loreti,On the topological structure of univoque sets, Journal of Number Theory122(2007), no. 1, 157–183

work page 2007

[12] [12]

9, 2482–2514

Charlene Kalle, Derong Kong, Niels Langeveld, and Wenxia Li,Theβ-transformation with a hole at 0, Ergodic Theory and Dynamical Systems40(2020), no. 9, 2482–2514

work page 2020

[13] [13]

Parry,On theβ-expansions of real numbers, Acta Math

W. Parry,On theβ-expansions of real numbers, Acta Math. Acad. Sci. Hungar.11(1960), 401–416 (English, with Russian summary)

work page 1960

[14] [14]

Yorke,Expanding maps on sets which are almost invariant

Giulio Pianigiani and James A. Yorke,Expanding maps on sets which are almost invariant. Decay and chaos, Trans. Amer. Math. Soc.252(1979), 351–366

work page 1979

[15] [15]

1-2, 97–133

Peter Raith,Continuity of the Hausdorff dimension for piecewise monotonic maps, Israel Journal of Mathematics80(1992), no. 1-2, 97–133

work page 1992

[16] [16]

1, 39–53

Peter Raith,Continuity of the Hausdorff dimension for invariant subsets of interval maps, Acta Math- matica Universitatis Comenianae63(1994), no. 1, 39–53

work page 1994

[17] [17]

Sidorov,Supercritical holes for the doubling map, Acta Math

N. Sidorov,Supercritical holes for the doubling map, Acta Math. Hungar.143(2014), no. 2, 298–312

work page 2014

[18] [18]

Rani Siromoney, Lisa Mathew, V . R. Dare, and K. G. Subramanian,Infinite Lyndon words, Inform. Process. Lett.50(1994), no. 2, 101–104

work page 1994

[19] [19]

2, 295–309

Mariusz Urba ´nski,On Hausdorff dimension of invariant sets for expanding maps of a circle, Ergodic Theory and Dynamical Systems6(1986), no. 2, 295–309

work page 1986

[20] [20]

4, 627–645

Mariusz Urba ´nski,Invariant subsets of expanding mappings of the circle, Ergodic Theory and Dy- namical Systems7(1987), no. 4, 627–645. (R. Kuang) SCHOOL OFMATHEMATICS, SOUTHCHINAUNIVERSITY OFTECHNOLOGY, GUANGZHOU, 510641, GUANGDONG, CHINA Email address:kuangrui@scut.edu.cn (B. Li) SCHOOL OFMATHEMATICS, SOUTHCHINAUNIVERSITY OFTECHNOLOGY, GUANGZHOU, 51064...

work page 1987