Möbius Transformations in the Second Symmetric Product of ℂ

Möbius Transformations in the Second Symmetric Product of ℂ

Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>F</mi><mn>2</mn></msub><mrow><mo>(</mo><mi mathvariant="double-struck">C&...

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Bibliographic Details
Main Authors:	Gabriela Hinojosa, Ulises Morales-Fuentes, Rogelio Valdez
Format:	Article
Language:	English
Published:	MDPI AG 2025-02-01
Series:	Mathematics
Subjects:	second symmetric product Möbius transformations transitivity conjugacy classes
Online Access:	https://www.mdpi.com/2227-7390/13/5/780
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Summary:	Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>F</mi><mn>2</mn></msub><mrow><mo>(</mo><mi mathvariant="double-struck">C</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> denote the second symmetric product of the complex plane <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">C</mi></semantics></math></inline-formula> endowed with the Hausdorff topology, i.e., <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>F</mi><mn>2</mn></msub><mrow><mo>(</mo><mi mathvariant="double-struck">C</mi><mo>)</mo></mrow><mo>=</mo><mrow><mo>{</mo><mi>A</mi><mo>⊂</mo><mi mathvariant="double-struck">C</mi><mo>:</mo><mo>\|</mo><mi>A</mi><mo>\|</mo><mo>≤</mo><mn>2</mn><mo>,</mo><mi>A</mi><mo>≠</mo><mo>∅</mo><mo>}</mo></mrow></mrow></semantics></math></inline-formula>. In this paper, we extended the concept of Möbius transformations to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>F</mi><mn>2</mn></msub><mrow><mo>(</mo><mi mathvariant="double-struck">C</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. More precisely, given a Möbius transformation <i>T</i> of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">C</mi></semantics></math></inline-formula>, we define the map <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mover accent="true"><mi>T</mi><mo>˜</mo></mover><mrow><mo>(</mo><mrow><mo>{</mo><mi>z</mi><mo>,</mo><mi>w</mi><mo>}</mo></mrow><mo>)</mo></mrow><mo>=</mo><mrow><mo>{</mo><mi>T</mi><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mo>,</mo><mi>T</mi><mrow><mo>(</mo><mi>w</mi><mo>)</mo></mrow><mo>}</mo></mrow></mrow></semantics></math></inline-formula> within <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>F</mi><mn>2</mn></msub><mrow><mo>(</mo><mi mathvariant="double-struck">C</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. We describe some general properties of these maps, including the structure of their generators, characteristics related to transitivity, and the geometry of the conjugacy classes.
ISSN:	2227-7390