Dichotomy Law for a Modified Shrinking Target Problem in Beta Dynamical System
Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>φ</mi><mo>:</mo><mi mathvariant="double-struck">N</mi><mo>→</mo><mo>(</mo><m...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2024-11-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/12/23/3680 |
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| Summary: | Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>φ</mi><mo>:</mo><mi mathvariant="double-struck">N</mi><mo>→</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></semantics></math></inline-formula> be a positive function. We consider the size of the set <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="double-struck">E</mi><mi>f</mi></msub><mrow><mo>(</mo><mi>φ</mi><mo>)</mo></mrow><mo>:</mo><mo>=</mo><mrow><mo>{</mo><mi>β</mi><mo>></mo><mn>1</mn><mo>:</mo><mo>|</mo></mrow><msubsup><mi>T</mi><mi>β</mi><mi>n</mi></msubsup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>−</mo><mi>f</mi><mrow><mo>(</mo><mi>β</mi><mo>)</mo></mrow><mrow><mo>|</mo><mo><</mo><mi>φ</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mspace width="3.33333pt"></mspace><mspace width="3.33333pt"></mspace><mspace width="3.33333pt"></mspace><mi>i</mi><mo>.</mo><mi>o</mi><mo>.</mo><mi>n</mi><mo>}</mo></mrow></mrow></semantics></math></inline-formula>, where “<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>i</mi><mo>.</mo><mi>o</mi><mo>.</mo><mi>n</mi></mrow></semantics></math></inline-formula>” stands for “infinitely often”, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>:</mo><mo>(</mo><mn>1</mn><mo>,</mo><mo>∞</mo><mo>)</mo><mo>→</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></semantics></math></inline-formula> is a Lipschitz function. For any <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>x</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></semantics></math></inline-formula>, it is proved that the Hausdorff measure of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="double-struck">E</mi><mi>f</mi></msub><mrow><mo>(</mo><mi>φ</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> fulfill a dichotomy law according to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><munder><mo movablelimits="false" form="prefix">lim sup</mo><mrow><mi>n</mi><mo>→</mo><mo>∞</mo></mrow></munder><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mo form="prefix">log</mo><mi>φ</mi><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mi>n</mi></mfrac></mstyle><mo>=</mo><mo>−</mo><mo>∞</mo></mrow></semantics></math></inline-formula> or not, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>T</mi><mi>β</mi></msub></semantics></math></inline-formula> is the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula>-transformation. In ergodic theory, the phenomenon of shrinking targets is crucial for understanding the long-term behavior of systems. By studying the shrinking target problem of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula> dynamical system, we can reveal the relationship between randomness and determinism, which is significant for constructing more complex mathematical models. Moreover, there is a close connection between the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula> transformation and number theory. Investigating the contraction target problem helps uncover new properties and patterns in number theory, advancing the development of this field. In this work, we establish a significant relationship between the decay rate of the positive function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>φ</mi><mo>(</mo><mi>n</mi><mo>)</mo></mrow></semantics></math></inline-formula> and the structural properties of the set <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>E</mi><mi>f</mi></msub><mrow><mo>(</mo><mi>φ</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. Specifically, we show that: The Hausdorff dimension of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>E</mi><mi>f</mi></msub><mrow><mo>(</mo><mi>φ</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> either vanishes or is positive based on the behavior of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>φ</mi><mo>(</mo><mi>n</mi><mo>)</mo></mrow></semantics></math></inline-formula> as <i>n</i> approaches infinity. The establishment of this dichotomy can help us more effectively understand the geometric characteristics and dynamical behavior of the system, thereby aiding our acceptance and comprehension of complex theories. Researching this shrinking target problem can help us uncover new properties in number theory, leading to a better understanding of the structure of numbers and promoting the development of related fields in number theory. |
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| ISSN: | 2227-7390 |