Integral self-affine sets with positive Lebesgue measures

Deng, Guotai; He, Xing‐Gang

doi:10.1007/s00013-007-2266-4

Cited by 2 publications

(4 citation statements)

References 10 publications

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“…It is well known [17] that the tile T has integer Lebesgue measure. The question how to find this measure is studied in [17,12,5], and finally answered in [9]. A related question how to find the measure of intersection T ∩ (T + a) for a ∈ Z n is studied in [9,7].…”

Section: Applications and Examplesmentioning

confidence: 99%

Graph-directed systems and self-similar measures on limit spaces of self-similar groups

Bondarenko

Kravchenko

2011

Advances in Mathematics

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Let G be a group and φ : H → G be a contracting homomorphism from a subgroup H < G of finite index. V. Nekrashevych [25] associated with the pair (G, φ) the limit dynamical system (J G , s) and the limit G-space X G together with the covering ∪ g∈G T · g by the tile T. We develop the theory of self-similar measures m on these limit spaces. It is shown that (J G , s, m) is conjugated to the one-sided Bernoulli shift. Using sofic subshifts we prove that the tile T has integer measure and we give an algorithmic way to compute it. In addition we give an algorithm to find the measure of the intersection of tiles T ∩ (T · g) for g ∈ G. We present applications to the invariant measures for the rational functions on the Riemann sphere and to the evaluation of the Lebesgue measure of integral self-affine tiles.

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Section: Applications and Examplesmentioning

confidence: 99%

Graph-directed systems and self-similar measures on limit spaces of self-similar groups

Bondarenko

Kravchenko

2011

Advances in Mathematics

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“…Hence these two problems are reduced to the question of how to find the measure of the intersection F (1) v 1 ∩ F (2) v 2 , where each set F (i) v i is defined in some finite graph Γ (i) = (V (i) , E (i) ) with its vertex v i . One can construct a new finite graph Γ (sometimes called the labeled product of graphs Γ (i) ) with the set of vertices V (1) × V (2) , where we put an edge (u 1 , u 2 ) x → (w 1 , w 2 ) for every edges u 1…”

Section: Corollary 2 Every Self-affine Set Has Rational Lebesgue Measmentioning

confidence: 99%

“…The last case was treated by Gabardo and Yu [3] and in more general settings by Bondarenko and Kravchenko [1]. The positivity of the Lebesgue measure of self-affine sets was also studied in [8,6,2].…”

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confidence: 99%

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On Lebesgue Measure of Integral Self-Affine Sets

Bondarenko

Kravchenko

2010

Discrete Comput Geom

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Let A be an expanding integer n × n matrix and D be a finite subset of Z n . The self-affine set T = T (A, D) is the unique compact set satisfying the equality A(T ) = d∈D (T + d). We present an effective algorithm to compute the Lebesgue measure of the self-affine set T , the measure of the intersection T ∩ (T + u) for u ∈ Z n , and the measure of the intersection of self-affine setsLet A be an expanding integer n × n matrix, where expanding means that every eigenvalue has modulus greater than 1, and let D be a finite subset of Z n . There exists a unique nonempty compact set T = T (A, D) ⊂ R n , called (integral) self-affine set, satisfying A(T ) = d∈D (T + d). It can be given explicitly byThe self-affine set T with |D| = | det A| and of positive Lebesgue measure is called a self-affine tile. Self-affine tiles were intensively studied for the last two decades in the context of self-replicating tilings, radix systems, Haar-type wavelets, etc. R.V. Kravchenko was partially supported by NSF grant 0503688. I.V. Bondarenko ( ) National Taras Shevchenko University of Kiev, Kiev, Ukraine

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Integral self-affine sets with positive Lebesgue measures

Cited by 2 publications

References 10 publications

Graph-directed systems and self-similar measures on limit spaces of self-similar groups

Graph-directed systems and self-similar measures on limit spaces of self-similar groups

On Lebesgue Measure of Integral Self-Affine Sets

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