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{"title":"群作用下移位映射的动力学性质","authors":"Zhan-Huai Ji","doi":"10.1155/2022/5969042","DOIUrl":null,"url":null,"abstract":"<jats:p>Firstly, we introduced the concept of <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M1\">\n <mi>G</mi>\n <mo>‐</mo>\n </math>\n </jats:inline-formula>Lipschitz tracking property, <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M2\">\n <mi>G</mi>\n <mo>‐</mo>\n </math>\n </jats:inline-formula>asymptotic average tracking property, and <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M3\">\n <mi>G</mi>\n <mo>‐</mo>\n </math>\n </jats:inline-formula>periodic tracking property. Secondly, we studied their dynamical properties and topological structure and obtained the following conclusions: (1) let <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M4\">\n <mfenced open=\"(\" close=\")\">\n <mrow>\n <mi>X</mi>\n <mo>,</mo>\n <mi>d</mi>\n </mrow>\n </mfenced>\n </math>\n </jats:inline-formula> be compact metric <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M5\">\n <mi>G</mi>\n <mo>‐</mo>\n </math>\n </jats:inline-formula>space and the metric <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M6\">\n <mi>d</mi>\n </math>\n </jats:inline-formula> be invariant to <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M7\">\n <mi>G</mi>\n </math>\n </jats:inline-formula>. Then, <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M8\">\n <mi>σ</mi>\n </math>\n </jats:inline-formula> has <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M9\">\n <mover accent=\"true\">\n <mi>G</mi>\n <mo stretchy=\"true\">¯</mo>\n </mover>\n <mo>‐</mo>\n </math>\n </jats:inline-formula>asymptotic average tracking property; (2) let <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M10\">\n <mfenced open=\"(\" close=\")\">\n <mrow>\n <mi>X</mi>\n <mo>,</mo>\n <mi>d</mi>\n </mrow>\n </mfenced>\n </math>\n </jats:inline-formula> be compact metric <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M11\">\n <mi>G</mi>\n <mo>‐</mo>\n </math>\n </jats:inline-formula>space and the metric <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M12\">\n <mi>d</mi>\n </math>\n </jats:inline-formula> be invariant to <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M13\">\n <mi>G</mi>\n </math>\n </jats:inline-formula>. Then, <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M14\">\n <mi>σ</mi>\n </math>\n </jats:inline-formula> has <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M15\">\n <mover accent=\"true\">\n <mi>G</mi>\n <mo stretchy=\"true\">¯</mo>\n </mover>\n <mo>‐</mo>\n </math>\n </jats:inline-formula>Lipschitz tracking property; (3) let <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M16\">\n <mfenced open=\"(\" close=\")\">\n <mrow>\n <mi>X</mi>\n <mo>,</mo>\n <mi>d</mi>\n </mrow>\n </mfenced>\n </math>\n </jats:inline-formula> be compact metric <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M17\">\n <mi>G</mi>\n <mo>‐</mo>\n </math>\n </jats:inline-formula>space and the metric <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M18\">\n <mi>d</mi>\n </math>\n </jats:inline-formula> be invariant to <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M19\">\n <mi>G</mi>\n </math>\n </jats:inline-formula>. Then, <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M20\">\n <mi>σ</mi>\n </math>\n </jats:inline-formula> has <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M21\">\n <mover accent=\"true\">\n <mi>G</mi>\n <mo stretchy=\"true\">¯</mo>\n </mover>\n <mo>‐</mo>\n </math>\n </jats:inline-formula>periodic tracking property. The above results make up for the lack of theory of <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M22\">\n <mi>G</mi>\n <mo>‐</mo>\n </math>\n </jats:inline-formula>Lipschitz tracking property, <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M23\">\n <mi>G</mi>\n <mo>‐</mo>\n </math>\n </jats:inline-formula>asymptotic average tracking property, and <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M24\">\n <mi>G</mi>\n <mo>‐</mo>\n </math>\n </jats:inline-formula>periodic tracking property in infinite product space under group action.</jats:p>","PeriodicalId":49111,"journal":{"name":"Advances in Mathematical Physics","volume":" ","pages":""},"PeriodicalIF":1.0000,"publicationDate":"2022-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Dynamical Property of the Shift Map under Group Action\",\"authors\":\"Zhan-Huai Ji\",\"doi\":\"10.1155/2022/5969042\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<jats:p>Firstly, we introduced the concept of <jats:inline-formula>\\n <math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\" id=\\\"M1\\\">\\n <mi>G</mi>\\n <mo>‐</mo>\\n </math>\\n </jats:inline-formula>Lipschitz tracking property, <jats:inline-formula>\\n <math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\" id=\\\"M2\\\">\\n <mi>G</mi>\\n <mo>‐</mo>\\n </math>\\n </jats:inline-formula>asymptotic average tracking property, and <jats:inline-formula>\\n <math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\" id=\\\"M3\\\">\\n <mi>G</mi>\\n <mo>‐</mo>\\n </math>\\n </jats:inline-formula>periodic tracking property. Secondly, we studied their dynamical properties and topological structure and obtained the following conclusions: (1) let <jats:inline-formula>\\n <math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\" id=\\\"M4\\\">\\n <mfenced open=\\\"(\\\" close=\\\")\\\">\\n <mrow>\\n <mi>X</mi>\\n <mo>,</mo>\\n <mi>d</mi>\\n </mrow>\\n </mfenced>\\n </math>\\n </jats:inline-formula> be compact metric <jats:inline-formula>\\n <math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\" id=\\\"M5\\\">\\n <mi>G</mi>\\n <mo>‐</mo>\\n </math>\\n </jats:inline-formula>space and the metric <jats:inline-formula>\\n <math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\" id=\\\"M6\\\">\\n <mi>d</mi>\\n </math>\\n </jats:inline-formula> be invariant to <jats:inline-formula>\\n <math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\" id=\\\"M7\\\">\\n <mi>G</mi>\\n </math>\\n </jats:inline-formula>. Then, <jats:inline-formula>\\n <math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\" id=\\\"M8\\\">\\n <mi>σ</mi>\\n </math>\\n </jats:inline-formula> has <jats:inline-formula>\\n <math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\" id=\\\"M9\\\">\\n <mover accent=\\\"true\\\">\\n <mi>G</mi>\\n <mo stretchy=\\\"true\\\">¯</mo>\\n </mover>\\n <mo>‐</mo>\\n </math>\\n </jats:inline-formula>asymptotic average tracking property; (2) let <jats:inline-formula>\\n <math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\" id=\\\"M10\\\">\\n <mfenced open=\\\"(\\\" close=\\\")\\\">\\n <mrow>\\n <mi>X</mi>\\n <mo>,</mo>\\n <mi>d</mi>\\n </mrow>\\n </mfenced>\\n </math>\\n </jats:inline-formula> be compact metric <jats:inline-formula>\\n <math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\" id=\\\"M11\\\">\\n <mi>G</mi>\\n <mo>‐</mo>\\n </math>\\n </jats:inline-formula>space and the metric <jats:inline-formula>\\n <math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\" id=\\\"M12\\\">\\n <mi>d</mi>\\n </math>\\n </jats:inline-formula> be invariant to <jats:inline-formula>\\n <math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\" id=\\\"M13\\\">\\n <mi>G</mi>\\n </math>\\n </jats:inline-formula>. Then, <jats:inline-formula>\\n <math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\" id=\\\"M14\\\">\\n <mi>σ</mi>\\n </math>\\n </jats:inline-formula> has <jats:inline-formula>\\n <math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\" id=\\\"M15\\\">\\n <mover accent=\\\"true\\\">\\n <mi>G</mi>\\n <mo stretchy=\\\"true\\\">¯</mo>\\n </mover>\\n <mo>‐</mo>\\n </math>\\n </jats:inline-formula>Lipschitz tracking property; (3) let <jats:inline-formula>\\n <math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\" id=\\\"M16\\\">\\n <mfenced open=\\\"(\\\" close=\\\")\\\">\\n <mrow>\\n <mi>X</mi>\\n <mo>,</mo>\\n <mi>d</mi>\\n </mrow>\\n </mfenced>\\n </math>\\n </jats:inline-formula> be compact metric <jats:inline-formula>\\n <math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\" id=\\\"M17\\\">\\n <mi>G</mi>\\n <mo>‐</mo>\\n </math>\\n </jats:inline-formula>space and the metric <jats:inline-formula>\\n <math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\" id=\\\"M18\\\">\\n <mi>d</mi>\\n </math>\\n </jats:inline-formula> be invariant to <jats:inline-formula>\\n <math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\" id=\\\"M19\\\">\\n <mi>G</mi>\\n </math>\\n </jats:inline-formula>. Then, <jats:inline-formula>\\n <math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\" id=\\\"M20\\\">\\n <mi>σ</mi>\\n </math>\\n </jats:inline-formula> has <jats:inline-formula>\\n <math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\" id=\\\"M21\\\">\\n <mover accent=\\\"true\\\">\\n <mi>G</mi>\\n <mo stretchy=\\\"true\\\">¯</mo>\\n </mover>\\n <mo>‐</mo>\\n </math>\\n </jats:inline-formula>periodic tracking property. The above results make up for the lack of theory of <jats:inline-formula>\\n <math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\" id=\\\"M22\\\">\\n <mi>G</mi>\\n <mo>‐</mo>\\n </math>\\n </jats:inline-formula>Lipschitz tracking property, <jats:inline-formula>\\n <math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\" id=\\\"M23\\\">\\n <mi>G</mi>\\n <mo>‐</mo>\\n </math>\\n </jats:inline-formula>asymptotic average tracking property, and <jats:inline-formula>\\n <math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\" id=\\\"M24\\\">\\n <mi>G</mi>\\n <mo>‐</mo>\\n </math>\\n </jats:inline-formula>periodic tracking property in infinite product space under group action.</jats:p>\",\"PeriodicalId\":49111,\"journal\":{\"name\":\"Advances in Mathematical Physics\",\"volume\":\" \",\"pages\":\"\"},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2022-12-22\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Advances in Mathematical Physics\",\"FirstCategoryId\":\"101\",\"ListUrlMain\":\"https://doi.org/10.1155/2022/5969042\",\"RegionNum\":4,\"RegionCategory\":\"物理与天体物理\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"PHYSICS, MATHEMATICAL\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Advances in Mathematical Physics","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.1155/2022/5969042","RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
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Dynamical Property of the Shift Map under Group Action
Firstly, we introduced the concept of
G
‐
Lipschitz tracking property,
G
‐
asymptotic average tracking property, and
G
‐
periodic tracking property. Secondly, we studied their dynamical properties and topological structure and obtained the following conclusions: (1) let
X
,
d
be compact metric
G
‐
space and the metric
d
be invariant to
G
. Then,
σ
has
G
¯
‐
asymptotic average tracking property; (2) let
X
,
d
be compact metric
G
‐
space and the metric
d
be invariant to
G
. Then,
σ
has
G
¯
‐
Lipschitz tracking property; (3) let
X
,
d
be compact metric
G
‐
space and the metric
d
be invariant to
G
. Then,
σ
has
G
¯
‐
periodic tracking property. The above results make up for the lack of theory of
G
‐
Lipschitz tracking property,
G
‐
asymptotic average tracking property, and
G
‐
periodic tracking property in infinite product space under group action.