Let Ψ = {ψn}n≥1 be an iterated function system (IFS) on [0, 1] with attractor J. Associated with each x ∈ J, there is a sequence {ωn(x)}n≥1 consisting of integers, called the digit sequence of x, such that<Equation ID="Equ1"> <EquationNumber>(1)</EquationNumber> <EquationSource Format="TEX">$x=\lim_{n\rightarrow\infty}\psi_{\omega_{1}(x)}\circ\cdots\circ \psi_{\omega_{n}(x)}.$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mi>x</mi> <mo>=</mo> <munder> <mo form="prefix" movablelimits="true">lim</mo> <mrow> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msub> <mi>ψ</mi> <mrow> <msub> <mi>ω</mi> <mrow> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo>∘</mo> <mo>⋯</mo> <mo>∘</mo> <msub> <mi>ψ</mi> <mrow> <msub> <mi>ω</mi> <mrow> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo>.</mo> </math></EquationSource> </Equation>We revisit the Borel-Bernstein theorem in a d-decaying Gauss-like IFS, and completely characterize the metrical properties of the set<Equation ID="Equa"> <EquationSource Format="TEX">$E(\Phi)=\big\{x\in J\colon \omega_{n}(x)\geq \Phi(n)\, \text{for infinitely many}\, n\in \mathbb{N}\big\},$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mi>E</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo maxsize="1.2em" minsize="1.2em">{</mo> </mrow> <mi>x</mi> <mo>∈</mo> <mi>J</mi> <mo>:</mo> <msub> <mi>ω</mi> <mrow> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mtext>for infinitely many</mtext> <mspace width="thinmathspace" /> <mi>n</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">N</mi> </mrow> <mrow> <mo maxsize="1.2em" minsize="1.2em">}</mo> </mrow> <mo>,</mo> </math></EquationSource> </Equation>where Φ: ℕ → ℝ is a positive function.

Growth rate of digits in a Gauss-like IFS

期刊论文
发表日期：

2025年12月1日

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Saisai Shi,
Bo Tan,
Qinglong Zhou

摘要

Let Ψ = {ψ_n}_n≥1 be an iterated function system (IFS) on [0, 1] with attractor J. Associated with each x ∈ J, there is a sequence {ω_n(x)}_n≥1 consisting of integers, called the digit sequence of x, such that

(1) $x=\lim_{n\rightarrow\infty}\psi_{\omega_{1}(x)}\circ\cdots\circ \psi_{\omega_{n}(x)}.$ $x = lim_{n \to \infty} ψ_{ω_{1} (x)} \circ \dots \circ ψ_{ω_{n} (x)} .$

We revisit the Borel-Bernstein theorem in a d-decaying Gauss-like IFS, and completely characterize the metrical properties of the set

$E(\Phi)=\big\{x\in J\colon \omega_{n}(x)\geq \Phi(n)\, \text{for infinitely many}\, n\in \mathbb{N}\big\},$ $E (Φ) = {x \in J : ω_{n} (x) \geq Φ (n) for infinitely many n \in N},$

where Φ: ℕ → ℝ is a positive function.