scholarly journals Topology of planar self-affine tiles with collinear digit set

2020 ◽  
Author(s):  
Shigeki Akiyama ◽  
Benoît Loridant ◽  
Jörg Thuswaldner
Keyword(s):  
Digit Set

On the spectrality of self-affine measures with four digits on ℝ2

2021 ◽  
pp. 2150004
Author(s):  
Ming-Liang Chen ◽  
Zhi-Hui Yan
Keyword(s):  
Spectral Property ◽  
Orthonormal Basis ◽  
Zero Set ◽  
Real Matrix ◽  
Exponential Functions ◽  
Bernoulli Convolution ◽  
Digit Set ◽  
Similar Characterization ◽  
Equivalent Characterization ◽  
Matrix Formula

In this paper, we study the spectral property of the self-affine measure [Formula: see text] generated by an expanding real matrix [Formula: see text] and the four-element digit set [Formula: see text]. We show that [Formula: see text] is a spectral measure, i.e. there exists a discrete set [Formula: see text] such that the collection of exponential functions [Formula: see text] forms an orthonormal basis for [Formula: see text], if and only if [Formula: see text] for some [Formula: see text]. A similar characterization for Bernoulli convolution is provided by Dai [X.-R. Dai, When does a Bernoulli convolution admit a spectrum? Adv. Math. 231(3) (2012) 1681–1693], over which [Formula: see text]. Furthermore, we provide an equivalent characterization for the maximal bi-zero set of [Formula: see text] by extending the concept of tree-mapping in [X.-R. Dai, X.-G. He and C. K. Lai, Spectral property of Cantor measures with consecutive digits, Adv. Math. 242 (2013) 187–208]. We also extend these results to the more general self-affine measures.

NON-SPECTRALITY OF THE PLANAR SELF-AFFINE MEASURES WITH FOUR-ELEMENT DIGIT SETS

Fractals ◽  
2019 ◽  
Vol 27 (07) ◽  
pp. 1950115 ◽  
Author(s):  
JUAN SU ◽  
YAO LIU ◽  
JING-CHENG LIU
Keyword(s):  
Exponential Functions ◽  
Integer Matrix ◽  
Digit Set ◽  
Number Formula ◽  
Matrix Formula ◽  
Orthogonal Set

In this paper, we consider the non-spectrality of the planar self-affine measures [Formula: see text] generated by an expanding integer matrix [Formula: see text] and a four-element digit set [Formula: see text] We show that [Formula: see text] contains an infinite orthogonal set of exponential functions if and only if [Formula: see text]. Moreover, if [Formula: see text], then there exist at most [Formula: see text] mutually orthogonal exponential functions in [Formula: see text], and the number [Formula: see text] is the best.

Spectrality of planar self-affine measures with two-element digit set

2011 ◽  
Vol 55 (3) ◽  
pp. 593-605 ◽  
Author(s):  
JianLin Li ◽  
ZhiYing Wen
Keyword(s):  
Digit Set

scholarly journals Connectedness of number theoretical tilings

2005 ◽  
Vol Vol. 7 ◽  
Author(s):  
Shigeki Akiyama ◽  
Nertila Gjini
Keyword(s):  
Complete Classification ◽  
Necessary And Sufficient Condition ◽  
Digit Set ◽  
Number Systems ◽  
Arcwise Connected ◽  
International Audience ◽  
Consecutive Integers ◽  
Pisot Unit ◽  
Necessary And Sufficient

International audience Let T=T(A,D) be a self-affine tile in \reals^n defined by an integral expanding matrix A and a digit set D. In connection with canonical number systems, we study connectedness of T when D corresponds to the set of consecutive integers \0,1,..., |det(A)|-1\. It is shown that in \reals^3 and \reals^4, for any integral expanding matrix A, T(A,D) is connected. We also study the connectedness of Pisot dual tilings which play an important role in the study of β -expansion, substitution and symbolic dynamical system. It is shown that each tile generated by a Pisot unit of degree 3 is arcwise connected. This is naturally expected since the digit set consists of consecutive integers as above. However surprisingly, we found families of disconnected Pisot dual tiles of degree 4. Also we give a simple necessary and sufficient condition for the connectedness of the Pisot dual tiles of degree 4. As a byproduct, a complete classification of the β -expansion of 1 for quartic Pisot units is given.

ORTHOGONAL EXPONENTIAL FUNCTIONS OF THE PLANAR SELF-AFFINE MEASURES WITH FOUR DIGITS

Fractals ◽  
2020 ◽  
Vol 28 (01) ◽  
pp. 2050016
Author(s):  
JUAN SU ◽  
ZHI-YONG WANG ◽  
MING-LIANG CHEN
Keyword(s):  
Complete Characterization ◽  
The Self ◽  
Exponential Functions ◽  
Digit Set ◽  
Matrix Formula ◽  
Orthogonal Set

For the self-affine measure [Formula: see text] generated by an expanding matrix [Formula: see text] and an integer digit set [Formula: see text] with [Formula: see text], Su et al. proved that if [Formula: see text], then [Formula: see text] contains an infinite orthogonal set of exponential functions if and only if [Formula: see text] [J. Su, Y. Liu and J. C. Liu, Non-spectrality of the planar self-affine measures with four-element digit sets, Fractals (2019), https://doi.org/10.1142/S0218348X19501159 ]. In this paper, we show that the above conclusion also holds for [Formula: see text]. So, a complete characterization of [Formula: see text] containing an infinite orthogonal set of exponential functions is given.

Two-step digit-set-restricted modified signed-digit addition–subtraction algorithm and its optoelectronic implementation

1999 ◽  
Vol 38 (26) ◽  
pp. 5621 ◽  
Author(s):  
Feng Qian ◽  
Guoqiang Li ◽  
Hao Ruan ◽  
Hongmei Jing ◽  
Liren Liu
Keyword(s):  
Digit Set
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