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.

THE NON-SPECTRAL PROPERTY OF A CLASS OF PLANAR SELF-SIMILAR MEASURES

Fractals ◽  
2020 ◽  
Vol 28 (05) ◽  
pp. 2050091
Author(s):  
YANG-YANG XU ◽  
JING-CHENG LIU
Keyword(s):  
Spectral Property ◽  
The Self ◽  
Real Matrix ◽  
Exponential Functions ◽  
Similar Measure ◽  
Digit Set ◽  
Matrix Formula ◽  
Self Similar ◽  
Orthogonal Set

Let the self-similar measure [Formula: see text] be generated by an expanding real matrix [Formula: see text] and a digit set [Formula: see text] in space [Formula: see text]. In this paper, we only consider [Formula: see text] and the case [Formula: see text] is similar. We show that there exists an infinite orthogonal set of exponential functions in [Formula: see text] if and only if [Formula: see text] for some [Formula: see text] with [Formula: see text]. Furthermore, for the cases that [Formula: see text] does not admit any infinite orthogonal set of exponential functions, the exact cardinality of orthogonal exponential functions in [Formula: see text] is given.

ORTHOGONAL EXPONENTIAL FUNCTIONS OF SELF-AFFINE MEASURES IN ℝn

Fractals ◽  
2019 ◽  
Vol 27 (03) ◽  
pp. 1950029 ◽  
Author(s):  
ZHI-MIN WANG ◽  
XIN-HAN DONG ◽  
ZHI-YONG WANG
Keyword(s):  
Positive Integer ◽  
The Self ◽  
Real Matrix ◽  
Exponential Functions ◽  
Digit Set ◽  
Matrix Formula ◽  
Orthogonal Set

For a positive integer [Formula: see text], let [Formula: see text]. Let the self-affine measure [Formula: see text] be generated by an expanding real matrix [Formula: see text] and a finite digit set [Formula: see text], where [Formula: see text] with [Formula: see text] and [Formula: see text] is the [Formula: see text]th column of the [Formula: see text] identical matrix [Formula: see text], [Formula: see text]. In this paper, we prove that [Formula: see text] contains an infinite orthogonal set of exponential functions if and only if there exists [Formula: see text] such that [Formula: see text] for some [Formula: see text] with [Formula: see text] and [Formula: see text].

Spectral property of the planar self-affine measures with three-element digit sets

2020 ◽  
Vol 32 (3) ◽  
pp. 673-681
Author(s):  
Ming-Liang Chen ◽  
Jing-Cheng Liu ◽  
Juan Su
Keyword(s):  
Spectral Property ◽  
Necessary Conditions ◽  
The Self ◽  
Real Matrix ◽  
Exponential Functions ◽  
Digit Set ◽  
Sufficient And Necessary Conditions ◽  
Orthogonal Set

AbstractLet the self-affine measure {\mu_{M,D}} be generated by an expanding real matrix {M=\operatorname{diag}(\rho_{1}^{-1},\rho_{2}^{-1})} and an integer digit set {D=\{(0,0)^{t},(\alpha_{1},\alpha_{2})^{t},(\beta_{1},\beta_{2})^{t}\}} with {\alpha_{1}\beta_{2}-\alpha_{2}\beta_{1}\neq 0}. In this paper, the sufficient and necessary conditions for {L^{2}(\mu_{M,D})} to contain an infinite orthogonal set of exponential functions are given.

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.

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.

THE CARDINALITY OF ORTHOGONAL EXPONENTIAL FUNCTIONS ON THE SPATIAL SIERPINSKI GASKET

Fractals ◽  
2019 ◽  
Vol 27 (04) ◽  
pp. 1950056 ◽  
Author(s):  
JIA ZHENG ◽  
JING-CHENG LIU ◽  
MING-LIANG CHEN
Keyword(s):  
Sierpinski Gasket ◽  
Diagonal Matrix ◽  
The Self ◽  
The Other ◽  
Exponential Functions ◽  
Sierpiński Gasket ◽  
Digit Set ◽  
Orthogonal Exponentials ◽  
Matrix Formula

For the self-affine measures [Formula: see text] generated by a diagonal matrix [Formula: see text] with entries [Formula: see text] and the digit set [Formula: see text], Li showed that there exists an infinite orthogonal exponential functions set in [Formula: see text] if and only if at least two of the three numbers [Formula: see text] are even, and conjectured that there exist at most four mutually orthogonal exponential functions in [Formula: see text] for other cases [J.-L. Li, Non-spectrality of self-affine measures on the spatial Sierpinski gasket, J. Math. Anal. Appl. 432 (2015) 1005–1017]. This conjecture was disproved by Wang and Li through constructing a class of the eight-element orthogonal exponential functions [Q. Wang and J.-L. Li, There are eight-element orthogonal exponentials on the spatial Sierpinski gasket, Math. Nachr. 292 (2019) 211–226]. In this paper, we will show that there are any number of orthogonal exponential functions in [Formula: see text] if two of the three numbers [Formula: see text] are different odd and the other is even.

Orthogonal exponentials of self-affine measures on ℝn

2020 ◽  
Vol 31 (08) ◽  
pp. 2050063
Author(s):  
Juan Su ◽  
Ming-Liang Chen
Keyword(s):  
The Self ◽  
Exponential Functions ◽  
Digit Set ◽  
Orthogonal Exponentials ◽  
Matrix Formula ◽  
Finite Integer ◽  
Orthogonal Set

Let the self-affine measure [Formula: see text] be generated by an expanding matrix [Formula: see text] and a finite integer digit set [Formula: see text], where [Formula: see text] with [Formula: see text] and [Formula: see text]. In this paper, we show that if [Formula: see text] for an integer [Formula: see text], then [Formula: see text] admits an infinite orthogonal set of exponential functions if and only if there exists [Formula: see text] such that [Formula: see text] for some [Formula: see text] with [Formula: see text] and [Formula: see text].

SPECTRAL PROPERTY OF CERTAIN MORAN MEASURES WITH THREE-ELEMENT DIGIT SETS

Fractals ◽  
2019 ◽  
Vol 27 (04) ◽  
pp. 1950068 ◽  
Author(s):  
XIU-QUN FU ◽  
XIN-HAN DONG ◽  
ZONG-SHENG LIU ◽  
ZHI-YONG WANG
Keyword(s):  
Spectral Property ◽  
Orthonormal Basis ◽  
Probability Measures ◽  
Finite Support ◽  
Equal Distribution ◽  
Infinite Convolution

Let [Formula: see text],[Formula: see text][Formula: see text],[Formula: see text][Formula: see text], satisfy [Formula: see text]. Let [Formula: see text] be the infinite convolution of probability measures with finite support and equal distribution. In this paper, we show that if [Formula: see text], then there exists a discrete set [Formula: see text] such that [Formula: see text] is an orthonormal basis for [Formula: see text].

Non-spectrality of self-affine measures on the three-dimensional Sierpinski gasket

2019 ◽  
Vol 31 (6) ◽  
pp. 1447-1455 ◽  
Author(s):  
Zheng-Yi Lu ◽  
Xin-Han Dong ◽  
Peng-Fei Zhang
Keyword(s):  
Sierpinski Gasket ◽  
Three Dimensional ◽  
Diagonal Matrix ◽  
Exponential Functions ◽  
Sierpiński Gasket ◽  
Digit Set ◽  
Orthogonal Set

AbstractLet {\mu_{M,D}} be a self-affine measure generated by an expanding diagonal matrix {M\in M_{3}(\mathbb{R})} with entries {\rho_{1},\rho_{2},\rho_{3}} and the digit set {D=\{(0,0,0)^{t},(1,0,0)^{t},(0,1,0)^{t},(0,0,1)^{t}\}}. In this paper, we prove that for any {\rho_{1},\rho_{2},\rho_{3}\in(1,\infty)}, if {\rho_{1},\rho_{2},\rho_{3}\in\{\pm x^{\frac{1}{r}}:x\in\mathbb{Q}^{+},r\in% \mathbb{Z}^{+}\}}, then {L^{2}(\mu_{M,D})} contains an infinite orthogonal set of exponential functions if and only if there exist two numbers of {\rho_{1},\rho_{2},\rho_{3}} that are in the set {\{\pm(\frac{p}{q})^{\frac{1}{r}}:p\in 2\mathbb{Z}^{+},q\in 2\mathbb{Z}^{+}-1% \text{ and }r\in\mathbb{Z}^{+}\}}. In particular, if {\rho_{1},\rho_{2},\rho_{3}\in\{\frac{p}{q}:p,q\in 2\mathbb{Z}+1\}}, then there exist at most 4 mutually orthogonal exponential functions in {L^{2}(\mu_{M,D})}, and the number 4 is the best possible.

scholarly journals Irrational rotation factors for conservative torus homeomorphisms

2016 ◽  
Vol 37 (5) ◽  
pp. 1537-1546 ◽  
Author(s):  
T. JÄGER ◽  
F. TAL
Keyword(s):  
Irrational Rotation ◽  
One Dimensional ◽  
Similar Characterization ◽  
Equivalent Characterization ◽  
Area Preserving ◽  
Rational Direction ◽  
Uniformly Bounded

We provide an equivalent characterization for the existence of one-dimensional irrational rotation factors of conservative torus homeomorphisms that are not eventually annular. It states that an area-preserving non-annular torus homeomorphism $f$ is semiconjugate to an irrational rotation $R_{\unicode[STIX]{x1D6FC}}$ of the circle if and only if there exists a well-defined speed of rotation in some rational direction on the torus, and the deviations from the constant rotation in this direction are uniformly bounded. By means of a counterexample, we also demonstrate that a similar characterization does not hold for eventually annular torus homeomorphisms.

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