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Spectra of Cantor measures
The Cantor measures μ q , b with 2 = q , b / q ∈ Z are few of known singular continuous measures that admits a spectrum. The spectra of Cantor measures μ q , b are not unique and their structure is not well-studied. It has been observed that maximal orthogonal sets of the Cantor measure μ q , b have...
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| Published in: | Mathematische annalen 2016-12, Vol.366 (3-4), p.1621-1647 |
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| Format: | Article |
| Language: | English |
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| cited_by | cdi_FETCH-LOGICAL-c316t-3214c8b6403caf146e6cef3c0021c2dbe43c742b912ff8f205a073b7613dad483 |
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| cites | cdi_FETCH-LOGICAL-c316t-3214c8b6403caf146e6cef3c0021c2dbe43c742b912ff8f205a073b7613dad483 |
| container_end_page | 1647 |
| container_issue | 3-4 |
| container_start_page | 1621 |
| container_title | Mathematische annalen |
| container_volume | 366 |
| creator | Dai, Xin-Rong |
| description | The Cantor measures
μ
q
,
b
with
2
=
q
,
b
/
q
∈
Z
are few of known singular continuous measures that admits a spectrum. The spectra of Cantor measures
μ
q
,
b
are not unique and their structure is not well-studied. It has been observed that maximal orthogonal sets of the Cantor measure
μ
q
,
b
have tree structure and they can be built by selecting maximal tree appropriately. A challenging problem is when a maximal orthogonal set becomes a spectrum. In this paper, we introduce a measurement on the tree structure associated with a maximal orthogonal set
Λ
of the Cantor measure
μ
q
,
b
, and we use boundedness and linear increment of that measurement to justify whether
Λ
is a spectrum or not. As applications of our justification, we provide a characterization for the expanding set
K
Λ
of a spectrum
Λ
to be a spectrum again. Furthermore, we construct a spectrum
Λ
such that for some integer
K
, the shrinking set
Λ
/
K
is a maximal orthogonal set but not a spectrum. |
| doi_str_mv | 10.1007/s00208-016-1374-5 |
| format | article |
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μ
q
,
b
with
2
=
q
,
b
/
q
∈
Z
are few of known singular continuous measures that admits a spectrum. The spectra of Cantor measures
μ
q
,
b
are not unique and their structure is not well-studied. It has been observed that maximal orthogonal sets of the Cantor measure
μ
q
,
b
have tree structure and they can be built by selecting maximal tree appropriately. A challenging problem is when a maximal orthogonal set becomes a spectrum. In this paper, we introduce a measurement on the tree structure associated with a maximal orthogonal set
Λ
of the Cantor measure
μ
q
,
b
, and we use boundedness and linear increment of that measurement to justify whether
Λ
is a spectrum or not. As applications of our justification, we provide a characterization for the expanding set
K
Λ
of a spectrum
Λ
to be a spectrum again. Furthermore, we construct a spectrum
Λ
such that for some integer
K
, the shrinking set
Λ
/
K
is a maximal orthogonal set but not a spectrum.</description><identifier>ISSN: 0025-5831</identifier><identifier>EISSN: 1432-1807</identifier><identifier>DOI: 10.1007/s00208-016-1374-5</identifier><language>eng</language><publisher>Berlin/Heidelberg: Springer Berlin Heidelberg</publisher><subject>Mathematics ; Mathematics and Statistics</subject><ispartof>Mathematische annalen, 2016-12, Vol.366 (3-4), p.1621-1647</ispartof><rights>Springer-Verlag Berlin Heidelberg 2016</rights><rights>Copyright Springer Science & Business Media 2016</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c316t-3214c8b6403caf146e6cef3c0021c2dbe43c742b912ff8f205a073b7613dad483</citedby><cites>FETCH-LOGICAL-c316t-3214c8b6403caf146e6cef3c0021c2dbe43c742b912ff8f205a073b7613dad483</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>314,780,784,27923,27924</link.rule.ids></links><search><creatorcontrib>Dai, Xin-Rong</creatorcontrib><title>Spectra of Cantor measures</title><title>Mathematische annalen</title><addtitle>Math. Ann</addtitle><description>The Cantor measures
μ
q
,
b
with
2
=
q
,
b
/
q
∈
Z
are few of known singular continuous measures that admits a spectrum. The spectra of Cantor measures
μ
q
,
b
are not unique and their structure is not well-studied. It has been observed that maximal orthogonal sets of the Cantor measure
μ
q
,
b
have tree structure and they can be built by selecting maximal tree appropriately. A challenging problem is when a maximal orthogonal set becomes a spectrum. In this paper, we introduce a measurement on the tree structure associated with a maximal orthogonal set
Λ
of the Cantor measure
μ
q
,
b
, and we use boundedness and linear increment of that measurement to justify whether
Λ
is a spectrum or not. As applications of our justification, we provide a characterization for the expanding set
K
Λ
of a spectrum
Λ
to be a spectrum again. Furthermore, we construct a spectrum
Λ
such that for some integer
K
, the shrinking set
Λ
/
K
is a maximal orthogonal set but not a spectrum.</description><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><issn>0025-5831</issn><issn>1432-1807</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2016</creationdate><recordtype>article</recordtype><recordid>eNp1kE1LxDAURYMoWEd_gK4KrqPv5XuWUnQUBlyo65CmiTg4bU3ahf_eDHXhxtVbvHvuhUPIJcINAujbDMDAUEBFkWtB5RGpUHBG0YA-JlV5SyoNx1NylvMOADiArMjVyxj8lFw9xLpx_TSkeh9cnlPI5-Qkus8cLn7virw93L82j3T7vHlq7rbUc1QT5QyFN60SwL2LKFRQPkTuyyJ61rVBcK8Fa9fIYjSRgXSgeasV8s51wvAVuV56xzR8zSFPdjfMqS-TFo0BI2GtdEnhkvJpyDmFaMf0sXfp2yLYgwK7KLBFgT0osLIwbGFyyfbvIf1p_hf6ATVgW68</recordid><startdate>20161201</startdate><enddate>20161201</enddate><creator>Dai, Xin-Rong</creator><general>Springer Berlin Heidelberg</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>20161201</creationdate><title>Spectra of Cantor measures</title><author>Dai, Xin-Rong</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c316t-3214c8b6403caf146e6cef3c0021c2dbe43c742b912ff8f205a073b7613dad483</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2016</creationdate><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Dai, Xin-Rong</creatorcontrib><collection>CrossRef</collection><jtitle>Mathematische annalen</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Dai, Xin-Rong</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Spectra of Cantor measures</atitle><jtitle>Mathematische annalen</jtitle><stitle>Math. Ann</stitle><date>2016-12-01</date><risdate>2016</risdate><volume>366</volume><issue>3-4</issue><spage>1621</spage><epage>1647</epage><pages>1621-1647</pages><issn>0025-5831</issn><eissn>1432-1807</eissn><abstract>The Cantor measures
μ
q
,
b
with
2
=
q
,
b
/
q
∈
Z
are few of known singular continuous measures that admits a spectrum. The spectra of Cantor measures
μ
q
,
b
are not unique and their structure is not well-studied. It has been observed that maximal orthogonal sets of the Cantor measure
μ
q
,
b
have tree structure and they can be built by selecting maximal tree appropriately. A challenging problem is when a maximal orthogonal set becomes a spectrum. In this paper, we introduce a measurement on the tree structure associated with a maximal orthogonal set
Λ
of the Cantor measure
μ
q
,
b
, and we use boundedness and linear increment of that measurement to justify whether
Λ
is a spectrum or not. As applications of our justification, we provide a characterization for the expanding set
K
Λ
of a spectrum
Λ
to be a spectrum again. Furthermore, we construct a spectrum
Λ
such that for some integer
K
, the shrinking set
Λ
/
K
is a maximal orthogonal set but not a spectrum.</abstract><cop>Berlin/Heidelberg</cop><pub>Springer Berlin Heidelberg</pub><doi>10.1007/s00208-016-1374-5</doi><tpages>27</tpages></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0025-5831 |
| ispartof | Mathematische annalen, 2016-12, Vol.366 (3-4), p.1621-1647 |
| issn | 0025-5831 1432-1807 |
| language | eng |
| recordid | cdi_proquest_journals_1880850967 |
| source | Springer Nature |
| subjects | Mathematics Mathematics and Statistics |
| title | Spectra of Cantor measures |
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