Collinear Fractals and Bandt’s Conjecture
For a complex parameter <i>c</i> outside the unit disk and an integer <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>2</mn></mrow...
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| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2024-12-01
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| Series: | Fractal and Fractional |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2504-3110/8/12/725 |
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| Summary: | For a complex parameter <i>c</i> outside the unit disk and an integer <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>2</mn></mrow></semantics></math></inline-formula>, we examine the <i>n-ary collinear fractal</i> <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>E</mi><mo>(</mo><mi>c</mi><mo>,</mo><mi>n</mi><mo>)</mo></mrow></semantics></math></inline-formula>, defined as the attractor of the iterated function system <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mrow><mo>{</mo><mrow><msub><mi>f</mi><mi>k</mi></msub><mo lspace="0pt">:</mo><mi mathvariant="double-struck">C</mi><mo>⟶</mo><mi mathvariant="double-struck">C</mi></mrow><mo>}</mo></mrow><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>f</mi><mi>k</mi></msub><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mo>:</mo><mo>=</mo><mn>1</mn><mo>+</mo><mi>n</mi><mo>−</mo><mn>2</mn><mi>k</mi><mo>+</mo><msup><mi>c</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mi>z</mi></mrow></semantics></math></inline-formula>. We investigate some topological features of the connectedness locus <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="script">M</mi><mi>n</mi></msub></semantics></math></inline-formula> defined as the set of those <i>c</i> for which <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>E</mi><mo>(</mo><mi>c</mi><mo>,</mo><mi>n</mi><mo>)</mo></mrow></semantics></math></inline-formula> is connected. In particular, we provide a detailed answer to an open question posed by Calegari, Koch, and Walker in 2017. We also extend and refine the technique of the “covering property” by Solomyak and Xu to any <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>2</mn></mrow></semantics></math></inline-formula>. We use it to show that a nontrivial portion of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="script">M</mi><mi>n</mi></msub></semantics></math></inline-formula> is regular closed. When <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>21</mn></mrow></semantics></math></inline-formula>, we enhance this result by showing that in fact, the whole <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">M</mi><mi>n</mi></msub><mo>∖</mo><mi mathvariant="double-struck">R</mi></mrow></semantics></math></inline-formula> lies within the closure of its interior, thus proving that the generalized Bandt’s conjecture is true. |
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| ISSN: | 2504-3110 |