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Realizing rotation numbers on annular continua
An annular continuum is a compact connected set K which separates a closed annulus A into exactly two connected components, one containing each boundary component. The topology of such continua can be very intricate (for instance, non-locally connected). We adapt a result proved by Handel in the cas...
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| Published in: | Mathematische Zeitschrift 2017-02, Vol.285 (1-2), p.549-564 |
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| Language: | English |
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| container_title | Mathematische Zeitschrift |
| container_volume | 285 |
| creator | Koropecki, Andres |
| description | An annular continuum is a compact connected set
K
which separates a closed annulus
A
into exactly two connected components, one containing each boundary component. The topology of such continua can be very intricate (for instance, non-locally connected). We adapt a result proved by Handel in the case where
K
=
A
, showing that if
K
is an invariant annular continuum of a homeomorphism of
A
isotopic to the identity, then the rotation set in
K
is closed. Moreover, every element of the rotation set is realized by an ergodic measure supported in
K
(and by a periodic orbit if the rotation number is rational) and most elements are realized by a compact invariant set. Our second result shows that if the continuum
K
is minimal with the property of being annular (what we call a
circloid
), then every rational number between the extrema of the rotation set in
K
is realized by a periodic orbit in
K
. As a consequence, the rotation set is a closed interval, and every number in this interval (rational or not) is realized by an orbit (moreover, by an ergodic measure) in
K
. This improves a previous result of Barge and Gillette. |
| doi_str_mv | 10.1007/s00209-016-1720-z |
| format | article |
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K
which separates a closed annulus
A
into exactly two connected components, one containing each boundary component. The topology of such continua can be very intricate (for instance, non-locally connected). We adapt a result proved by Handel in the case where
K
=
A
, showing that if
K
is an invariant annular continuum of a homeomorphism of
A
isotopic to the identity, then the rotation set in
K
is closed. Moreover, every element of the rotation set is realized by an ergodic measure supported in
K
(and by a periodic orbit if the rotation number is rational) and most elements are realized by a compact invariant set. Our second result shows that if the continuum
K
is minimal with the property of being annular (what we call a
circloid
), then every rational number between the extrema of the rotation set in
K
is realized by a periodic orbit in
K
. As a consequence, the rotation set is a closed interval, and every number in this interval (rational or not) is realized by an orbit (moreover, by an ergodic measure) in
K
. This improves a previous result of Barge and Gillette.</description><identifier>ISSN: 0025-5874</identifier><identifier>EISSN: 1432-1823</identifier><identifier>DOI: 10.1007/s00209-016-1720-z</identifier><language>eng</language><publisher>Berlin/Heidelberg: Springer Berlin Heidelberg</publisher><subject>Invariants ; Mathematics ; Mathematics and Statistics</subject><ispartof>Mathematische Zeitschrift, 2017-02, Vol.285 (1-2), p.549-564</ispartof><rights>Springer-Verlag Berlin Heidelberg 2016</rights><rights>Copyright Springer Science & Business Media 2017</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c316t-ad0cc79322dde2701c50a6a13b17fcccde772b40606cd2af2c60e7d313eade73</citedby><cites>FETCH-LOGICAL-c316t-ad0cc79322dde2701c50a6a13b17fcccde772b40606cd2af2c60e7d313eade73</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>314,776,780,27903,27904</link.rule.ids></links><search><creatorcontrib>Koropecki, Andres</creatorcontrib><title>Realizing rotation numbers on annular continua</title><title>Mathematische Zeitschrift</title><addtitle>Math. Z</addtitle><description>An annular continuum is a compact connected set
K
which separates a closed annulus
A
into exactly two connected components, one containing each boundary component. The topology of such continua can be very intricate (for instance, non-locally connected). We adapt a result proved by Handel in the case where
K
=
A
, showing that if
K
is an invariant annular continuum of a homeomorphism of
A
isotopic to the identity, then the rotation set in
K
is closed. Moreover, every element of the rotation set is realized by an ergodic measure supported in
K
(and by a periodic orbit if the rotation number is rational) and most elements are realized by a compact invariant set. Our second result shows that if the continuum
K
is minimal with the property of being annular (what we call a
circloid
), then every rational number between the extrema of the rotation set in
K
is realized by a periodic orbit in
K
. As a consequence, the rotation set is a closed interval, and every number in this interval (rational or not) is realized by an orbit (moreover, by an ergodic measure) in
K
. This improves a previous result of Barge and Gillette.</description><subject>Invariants</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><issn>0025-5874</issn><issn>1432-1823</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2017</creationdate><recordtype>article</recordtype><recordid>eNp1kMtKxDAUhoMoWEcfwF3BdcZzkjZplzJ4gwFBZh_SNB06dJIxaRf26c1QF25cnR_-y4GPkHuENQLIxwjAoKaAgqJkQOcLkmHBGcWK8UuSJbukZSWLa3IT4wEgmbLIyPrT6qGfe7fPgx_12HuXu-nY2BDzJLVz06BDbrwbezfpW3LV6SHau9-7IruX593mjW4_Xt83T1tqOIqR6haMkTVnrG0tk4CmBC008gZlZ4xprZSsKUCAMC3THTMCrGw5cquTx1fkYZk9Bf812Tiqg5-CSx8VVhXIWlRFmVK4pEzwMQbbqVPojzp8KwR1pqIWKipRUWcqak4dtnRiyrq9DX-W_y39ANzkZVs</recordid><startdate>20170201</startdate><enddate>20170201</enddate><creator>Koropecki, Andres</creator><general>Springer Berlin Heidelberg</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>20170201</creationdate><title>Realizing rotation numbers on annular continua</title><author>Koropecki, Andres</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c316t-ad0cc79322dde2701c50a6a13b17fcccde772b40606cd2af2c60e7d313eade73</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2017</creationdate><topic>Invariants</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Koropecki, Andres</creatorcontrib><collection>CrossRef</collection><jtitle>Mathematische Zeitschrift</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Koropecki, Andres</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Realizing rotation numbers on annular continua</atitle><jtitle>Mathematische Zeitschrift</jtitle><stitle>Math. Z</stitle><date>2017-02-01</date><risdate>2017</risdate><volume>285</volume><issue>1-2</issue><spage>549</spage><epage>564</epage><pages>549-564</pages><issn>0025-5874</issn><eissn>1432-1823</eissn><abstract>An annular continuum is a compact connected set
K
which separates a closed annulus
A
into exactly two connected components, one containing each boundary component. The topology of such continua can be very intricate (for instance, non-locally connected). We adapt a result proved by Handel in the case where
K
=
A
, showing that if
K
is an invariant annular continuum of a homeomorphism of
A
isotopic to the identity, then the rotation set in
K
is closed. Moreover, every element of the rotation set is realized by an ergodic measure supported in
K
(and by a periodic orbit if the rotation number is rational) and most elements are realized by a compact invariant set. Our second result shows that if the continuum
K
is minimal with the property of being annular (what we call a
circloid
), then every rational number between the extrema of the rotation set in
K
is realized by a periodic orbit in
K
. As a consequence, the rotation set is a closed interval, and every number in this interval (rational or not) is realized by an orbit (moreover, by an ergodic measure) in
K
. This improves a previous result of Barge and Gillette.</abstract><cop>Berlin/Heidelberg</cop><pub>Springer Berlin Heidelberg</pub><doi>10.1007/s00209-016-1720-z</doi><tpages>16</tpages></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0025-5874 |
| ispartof | Mathematische Zeitschrift, 2017-02, Vol.285 (1-2), p.549-564 |
| issn | 0025-5874 1432-1823 |
| language | eng |
| recordid | cdi_proquest_journals_1880796845 |
| source | Springer Nature |
| subjects | Invariants Mathematics Mathematics and Statistics |
| title | Realizing rotation numbers on annular continua |
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