Fixed Point Sets of <i>k</i>-Continuous Self-Maps of <i>m</i>-Iterated Digital Wedges

• Main Author: Sang-Eon Han
• Format: Article
• Language: English
• Published: MDPI AG 2020-09-01
• Series: Mathematics
• Subjects: digital wedge; alignment; perfect; k-contractibility; digital k-curve; fixed point set
• Online Access: https://www.mdpi.com/2227-7390/8/9/1617

Let <inline-formula><math display="inline"><semantics><msubsup><mi>C</mi><mi>k</mi><mrow><mi>n</mi><mo>,</mo><mi>l</mi></mrow></msubsup></semantics></math></inline-formula> be a simple closed <i>k</i>-curves with <i>l</i> elements in <inline-formula><math display="inline"><semantics><msup><mrow><mi mathvariant="double-struck">Z</mi></mrow><mi>n</mi></msup></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><mrow><mi>W</mi><mo>:</mo><mo>=</mo><msup><mover accent="true"><mrow><msubsup><mi>C</mi><mi>k</mi><mrow><mi>n</mi><mo>,</mo><mi>l</mi></mrow></msubsup><mo>∨</mo><mo>⋯</mo><mo>∨</mo><msubsup><mi>C</mi><mi>k</mi><mrow><mi>n</mi><mo>,</mo><mi>l</mi></mrow></msubsup></mrow><mo>︷</mo></mover><mrow><mi mathvariant="normal">m</mi><mo>-</mo><mi>times</mi></mrow></msup></mrow></semantics></math></inline-formula> be an <i>m</i>-iterated digital wedges of <inline-formula><math display="inline"><semantics><msubsup><mi>C</mi><mi>k</mi><mrow><mi>n</mi><mo>,</mo><mi>l</mi></mrow></msubsup></semantics></math></inline-formula>, and <inline-formula><math display="inline"><semantics><mrow><mi>F</mi><mo>(</mo><mi>C</mi><mi>o</mi><msub><mi>n</mi><mi>k</mi></msub><mrow><mo>(</mo><mi>W</mi><mo>)</mo></mrow><mo>)</mo></mrow></semantics></math></inline-formula> be an alignment of fixed point sets of <i>W</i>. Then, the aim of the paper is devoted to investigating various properties of <inline-formula><math display="inline"><semantics><mrow><mi>F</mi><mo>(</mo><mi>C</mi><mi>o</mi><msub><mi>n</mi><mi>k</mi></msub><mrow><mo>(</mo><mi>W</mi><mo>)</mo></mrow><mo>)</mo></mrow></semantics></math></inline-formula>. Furthermore, when proceeding with this work, this paper addresses several unsolved problems. To be specific, we firstly formulate an alignment of fixed point sets of <inline-formula><math display="inline"><semantics><msubsup><mi>C</mi><mi>k</mi><mrow><mi>n</mi><mo>,</mo><mi>l</mi></mrow></msubsup></semantics></math></inline-formula>, denoted by <inline-formula><math display="inline"><semantics><mrow><mi>F</mi><mo>(</mo><mi>C</mi><mi>o</mi><msub><mi>n</mi><mi>k</mi></msub><mrow><mo>(</mo><msubsup><mi>C</mi><mi>k</mi><mrow><mi>n</mi><mo>,</mo><mi>l</mi></mrow></msubsup><mo>)</mo></mrow><mo>)</mo></mrow></semantics></math></inline-formula>, where <inline-formula><math display="inline"><semantics><mrow><mi>l</mi><mo>(</mo><mo>≥</mo><mn>7</mn><mo>)</mo></mrow></semantics></math></inline-formula> is an odd natural number and <inline-formula><math display="inline"><semantics><mrow><mi>k</mi><mo>≠</mo><mn>2</mn><mi>n</mi></mrow></semantics></math></inline-formula>. Secondly, given a digital image <inline-formula><math display="inline"><semantics><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mi>k</mi><mo>)</mo></mrow></semantics></math></inline-formula> with <inline-formula><math display="inline"><semantics><mrow><msup><mi>X</mi><mo>♯</mo></msup><mo>=</mo><mi>n</mi></mrow></semantics></math></inline-formula>, we find a certain condition that supports <inline-formula><math display="inline"><semantics><mrow><mi>n</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>n</mi><mo>−</mo><mn>2</mn><mo>∈</mo><mi>F</mi><mo>(</mo><mi>C</mi><mi>o</mi><msub><mi>n</mi><mi>k</mi></msub><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mo>)</mo></mrow></semantics></math></inline-formula>. Thirdly, after finding some features of <inline-formula><math display="inline"><semantics><mrow><mi>F</mi><mo>(</mo><mi>C</mi><mi>o</mi><msub><mi>n</mi><mi>k</mi></msub><mrow><mo>(</mo><mi>W</mi><mo>)</mo></mrow><mo>)</mo></mrow></semantics></math></inline-formula>, we develop a method of making <inline-formula><math display="inline"><semantics><mrow><mi>F</mi><mo>(</mo><mi>C</mi><mi>o</mi><msub><mi>n</mi><mi>k</mi></msub><mrow><mo>(</mo><mi>W</mi><mo>)</mo></mrow><mo>)</mo></mrow></semantics></math></inline-formula> perfect according to the (even or odd) number <i>l</i> of <inline-formula><math display="inline"><semantics><msubsup><mi>C</mi><mi>k</mi><mrow><mi>n</mi><mo>,</mo><mi>l</mi></mrow></msubsup></semantics></math></inline-formula>. Finally, we prove that the perfectness of <inline-formula><math display="inline"><semantics><mrow><mi>F</mi><mo>(</mo><mi>C</mi><mi>o</mi><msub><mi>n</mi><mi>k</mi></msub><mrow><mo>(</mo><mi>W</mi><mo>)</mo></mrow><mo>)</mo></mrow></semantics></math></inline-formula> is equivalent to that of <inline-formula><math display="inline"><semantics><mrow><mi>F</mi><mo>(</mo><mi>C</mi><mi>o</mi><msub><mi>n</mi><mi>k</mi></msub><mrow><mo>(</mo><msubsup><mi>C</mi><mi>k</mi><mrow><mi>n</mi><mo>,</mo><mi>l</mi></mrow></msubsup><mo>)</mo></mrow><mo>)</mo></mrow></semantics></math></inline-formula>. This can play an important role in studying fixed point theory and digital curve theory. This paper only deals with <i>k</i>-connected digital images <inline-formula><math display="inline"><semantics><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mi>k</mi><mo>)</mo></mrow></semantics></math></inline-formula> such that <inline-formula><math display="inline"><semantics><mrow><msup><mi>X</mi><mo>♯</mo></msup><mo>≥</mo><mn>2</mn></mrow></semantics></math></inline-formula>.