On the arithmetic difference of middle Cantor sets

Mehdi Pourbarat

doi:10.3934/dcds.2018186

Article Contents

2018, Volume 38, Issue 9: 4259-4278. Doi: 10.3934/dcds.2018186

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On the arithmetic difference of middle Cantor sets

Mehdi Pourbarat

Dep. Math., Shahid Beheshti Univ., Tehran, Iran

Received: May 2017

Published: September 2018

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Abstract

We determine all triples $(α, β, λ)$ such that $C_α- λ C_β $ forms a closed interval, where $C_α$ and $C_β$ are middle Cantor sets. This follows from a new recurrence type result for certain renormalization operators. We also consider the affine Cantor sets $K$ and $K'$ defined by two increasing maps which the product of their thicknesses is bigger than one. Then we construct a recurrent set for their renormalization operators. This leads us to characterize all $λ$ that $K- λ K' $ is a closed interval.

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Mathematics Subject Classification: 28A78, 28A80, 58F14.

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Figure 1. Blue region and curves determine all $\alpha, \beta$ that $C_\alpha+C_\beta= [0, ~2] $. For instant, the curves $r_{3, 1}, r_{2, 1}, r_{3, 2}, r_{4, 3}, r_{5, 4}$ are selected, that we characterized them by the functions $\beta=\alpha ^\frac{3}{1}, \alpha ^\frac{2}{1}, \alpha ^\frac{3}{2}, \alpha ^\frac{4}{3}, \alpha ^\frac{5}{4}$, respectively. The yellow Curves are the graph of these functions from downside to upside which have been drawn by Maple program

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Figure 2. Gray region illustrates the recurrent set $R$

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Figure 3. For middle Cantor sets, the sets $E, F, G, \Delta_1, \Delta_2$ and intervals $I, J$ are illustrated. Triangles $\Delta_1$ and $\Delta_2$ are non-empty and both project to the horizontal interval $I$

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Figure 4. The graph of map $T$ is illustrated

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Figure 5. The functions $g_0$ and $g_k$ are illustrated

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