{"title":"Pseudometric spaces: From minimality to maximality in the groups of combinatorial self-similarities","authors":"Viktoriia Bilet, Oleksiy Dovgoshey","doi":"10.1515/agms-2023-0103","DOIUrl":null,"url":null,"abstract":"The group of combinatorial self-similarities of a pseudometric space <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0103_eq_001.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>X</m:mi> <m:mo>,</m:mo> <m:mi>d</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>\\left(X,d)</jats:tex-math> </jats:alternatives> </jats:inline-formula> is the maximal subgroup of the symmetric group <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0103_eq_002.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi mathvariant=\"normal\">Sym</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>X</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>{\\rm{Sym}}\\left(X)</jats:tex-math> </jats:alternatives> </jats:inline-formula> whose elements preserve the four-point equality <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0103_eq_003.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>d</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>y</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mi>d</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>u</m:mi> <m:mo>,</m:mo> <m:mi>v</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>d\\left(x,y)=d\\left(u,v)</jats:tex-math> </jats:alternatives> </jats:inline-formula>. Let us denote by <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0103_eq_004.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi mathvariant=\"script\">ℐP</m:mi> </m:math> <jats:tex-math>{\\mathcal{ {\\mathcal I} P}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> the class of all pseudometric spaces <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0103_eq_005.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>X</m:mi> <m:mo>,</m:mo> <m:mi>d</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>\\left(X,d)</jats:tex-math> </jats:alternatives> </jats:inline-formula> for which every combinatorial self-similarity <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0103_eq_006.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi mathvariant=\"normal\">Φ</m:mi> <m:mo>:</m:mo> <m:mi>X</m:mi> <m:mo>→</m:mo> <m:mi>X</m:mi> </m:math> <jats:tex-math>\\Phi :X\\to X</jats:tex-math> </jats:alternatives> </jats:inline-formula> satisfies the equality <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0103_eq_007.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>d</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi mathvariant=\"normal\">Φ</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> </m:math> <jats:tex-math>d\\left(x,\\Phi \\left(x))=0,</jats:tex-math> </jats:alternatives> </jats:inline-formula> but all permutations of metric reflection of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0103_eq_008.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>X</m:mi> <m:mo>,</m:mo> <m:mi>d</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>\\left(X,d)</jats:tex-math> </jats:alternatives> </jats:inline-formula> are combinatorial self-similarities of this reflection. The structure of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_agms-2023-0103_eq_009.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi mathvariant=\"script\">ℐP</m:mi> </m:math> <jats:tex-math>{\\mathcal{ {\\mathcal I} P}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-spaces is fully described.","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":"168-169 1","pages":""},"PeriodicalIF":0.9000,"publicationDate":"2024-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Analysis and Geometry in Metric Spaces","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1515/agms-2023-0103","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
The group of combinatorial self-similarities of a pseudometric space (X,d)\left(X,d) is the maximal subgroup of the symmetric group Sym(X){\rm{Sym}}\left(X) whose elements preserve the four-point equality d(x,y)=d(u,v)d\left(x,y)=d\left(u,v). Let us denote by ℐP{\mathcal{ {\mathcal I} P}} the class of all pseudometric spaces (X,d)\left(X,d) for which every combinatorial self-similarity Φ:X→X\Phi :X\to X satisfies the equality d(x,Φ(x))=0,d\left(x,\Phi \left(x))=0, but all permutations of metric reflection of (X,d)\left(X,d) are combinatorial self-similarities of this reflection. The structure of ℐP{\mathcal{ {\mathcal I} P}}-spaces is fully described.
伪几何空间 ( X , d ) \left(X,d)的组合自相似性群是对称群 Sym ( X ) {\rm{Sym}}\left(X) 的最大子群,其元素保持四点相等 d ( x , y ) = d ( u , v ) d\left(x,y)=d\left(u,v) 。让我们用 ℐP {\mathcal{ {\mathcal I} P}} 表示所有伪几何空间 ( X , d ) 的类 \left(X,d),其中每个组合自相似性 Φ : X → X \Phi :X\to X 满足等式 d ( x , Φ ( x ) ) = 0 , d\left(x,\Phi \left(x))=0,但是 ( X , d ) \left(X,d)的度量反射的所有排列都是这种反射的组合自相似性。对ℐP {\mathcal{ {\mathcal I} P}} 的结构进行了全面描述。 -空间的结构得到了充分描述。
期刊介绍:
Analysis and Geometry in Metric Spaces is an open access electronic journal that publishes cutting-edge research on analytical and geometrical problems in metric spaces and applications. We strive to present a forum where all aspects of these problems can be discussed.
AGMS is devoted to the publication of results on these and related topics:
Geometric inequalities in metric spaces,
Geometric measure theory and variational problems in metric spaces,
Analytic and geometric problems in metric measure spaces, probability spaces, and manifolds with density,
Analytic and geometric problems in sub-riemannian manifolds, Carnot groups, and pseudo-hermitian manifolds.
Geometric control theory,
Curvature in metric and length spaces,
Geometric group theory,
Harmonic Analysis. Potential theory,
Mass transportation problems,
Quasiconformal and quasiregular mappings. Quasiconformal geometry,
PDEs associated to analytic and geometric problems in metric spaces.