{"title":"MODIFIED SHRINKING TARGET PROBLEM FOR MATRIX TRANSFORMATIONS OF TORI","authors":"NA YUAN, SHUAILING WANG","doi":"10.1142/s0218348x24500762","DOIUrl":null,"url":null,"abstract":"<p>In this paper, we calculate the Hausdorff dimension of the fractal set <disp-formula-group><span><math altimg=\"eq-00001.gif\" display=\"block\" overflow=\"scroll\"><mrow><mfenced close=\"}\" open=\"{\" separators=\"\"><mrow><mstyle mathvariant=\"monospace\"><mi>x</mi></mstyle><mo>∈</mo><msup><mrow><mi>𝕋</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>:</mo><munder><mrow><mo>∏</mo></mrow><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>d</mi></mrow></munder><mo>|</mo><msubsup><mrow><mi>T</mi></mrow><mrow><msub><mrow><mi>β</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><mi>n</mi></mrow></msubsup><mo stretchy=\"false\">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy=\"false\">)</mo><mo stretchy=\"false\">−</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>|</mo><mo><</mo><mi>ψ</mi><mo stretchy=\"false\">(</mo><mi>n</mi><mo stretchy=\"false\">)</mo><mtext> </mtext><mstyle><mtext>for infinitely many </mtext></mstyle><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow></mfenced><mspace width=\"-.17em\"></mspace><mo>,</mo></mrow></math></span><span></span></disp-formula-group> where <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>T</mi></mrow><mrow><msub><mrow><mi>β</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msub></math></span><span></span> is the standard <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>β</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span><span></span>-transformation with <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>β</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>></mo><mn>1</mn></math></span><span></span>, <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mi>ψ</mi></math></span><span></span> is a positive function on <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><mi>ℕ</mi></math></span><span></span> and <span><math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"><mo>|</mo><mo stretchy=\"false\">⋅</mo><mo>|</mo></math></span><span></span> is the usual metric on the torus <span><math altimg=\"eq-00008.gif\" display=\"inline\" overflow=\"scroll\"><mi>𝕋</mi></math></span><span></span>. Moreover, we investigate a modified version of the shrinking target problem, which unifies the shrinking target problems and quantitative recurrence properties for matrix transformations of tori. Let <span><math altimg=\"eq-00009.gif\" display=\"inline\" overflow=\"scroll\"><mi>T</mi></math></span><span></span> be a <span><math altimg=\"eq-00010.gif\" display=\"inline\" overflow=\"scroll\"><mi>d</mi><mo stretchy=\"false\">×</mo><mi>d</mi></math></span><span></span> non-singular matrix with real coefficients. Then, <span><math altimg=\"eq-00011.gif\" display=\"inline\" overflow=\"scroll\"><mi>T</mi></math></span><span></span> determines a self-map of the <span><math altimg=\"eq-00012.gif\" display=\"inline\" overflow=\"scroll\"><mi>d</mi></math></span><span></span>-dimensional torus <span><math altimg=\"eq-00013.gif\" display=\"inline\" overflow=\"scroll\"><msup><mrow><mi>𝕋</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>:</mo><mo>=</mo><msup><mrow><mi>ℝ</mi></mrow><mrow><mi>d</mi></mrow></msup><mspace width=\"-.2em\"></mspace><mo stretchy=\"false\">/</mo><msup><mrow><mi>ℤ</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span><span></span>. For any <span><math altimg=\"eq-00014.gif\" display=\"inline\" overflow=\"scroll\"><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>d</mi></math></span><span></span>, let <span><math altimg=\"eq-00015.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>ψ</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span><span></span> be a positive function on <span><math altimg=\"eq-00016.gif\" display=\"inline\" overflow=\"scroll\"><mi>ℕ</mi></math></span><span></span> and <span><math altimg=\"eq-00017.gif\" display=\"inline\" overflow=\"scroll\"><mi mathvariant=\"normal\">Ψ</mi><mo stretchy=\"false\">(</mo><mi>n</mi><mo stretchy=\"false\">)</mo><mo>:</mo><mo>=</mo><mo stretchy=\"false\">(</mo><msub><mrow><mi>ψ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo stretchy=\"false\">(</mo><mi>n</mi><mo stretchy=\"false\">)</mo><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>ψ</mi></mrow><mrow><mi>d</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>n</mi><mo stretchy=\"false\">)</mo><mo stretchy=\"false\">)</mo></math></span><span></span> with <span><math altimg=\"eq-00018.gif\" display=\"inline\" overflow=\"scroll\"><mi>n</mi><mo>∈</mo><mi>ℕ</mi></math></span><span></span>. We obtain the Hausdorff dimension of the fractal set <disp-formula-group><span><math altimg=\"eq-00019.gif\" display=\"block\" overflow=\"scroll\"><mrow><mo stretchy=\"false\">{</mo><mstyle mathvariant=\"monospace\"><mi>x</mi></mstyle><mo>∈</mo><msup><mrow><mi>𝕋</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>:</mo><msup><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msup><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo>∈</mo><mi>L</mi><mo stretchy=\"false\">(</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy=\"false\">(</mo><mstyle mathvariant=\"monospace\"><mi>x</mi></mstyle><mo stretchy=\"false\">)</mo><mo>,</mo><mi mathvariant=\"normal\">Ψ</mi><mo stretchy=\"false\">(</mo><mi>n</mi><mo stretchy=\"false\">)</mo><mo stretchy=\"false\">)</mo><mtext> </mtext><mstyle><mtext>for infinitely many </mtext></mstyle><mi>n</mi><mo>∈</mo><mi>ℕ</mi><mo stretchy=\"false\">}</mo><mo>,</mo></mrow></math></span><span></span></disp-formula-group> where <span><math altimg=\"eq-00020.gif\" display=\"inline\" overflow=\"scroll\"><mi>L</mi><mo stretchy=\"false\">(</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy=\"false\">(</mo><mstyle mathvariant=\"monospace\"><mi>x</mi></mstyle><mo>,</mo><mi mathvariant=\"normal\">Ψ</mi><mo stretchy=\"false\">(</mo><mi>n</mi><mo stretchy=\"false\">)</mo><mo stretchy=\"false\">)</mo><mo stretchy=\"false\">)</mo></math></span><span></span> is a hyperrectangle and <span><math altimg=\"eq-00021.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mo stretchy=\"false\">{</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy=\"false\">}</mo></mrow><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></msub></math></span><span></span> is a sequence of Lipschitz vector-valued functions on <span><math altimg=\"eq-00022.gif\" display=\"inline\" overflow=\"scroll\"><msup><mrow><mi>𝕋</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span><span></span> with a uniform Lipschitz constant.</p>","PeriodicalId":501262,"journal":{"name":"Fractals","volume":"44 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Fractals","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1142/s0218348x24500762","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
In this paper, we calculate the Hausdorff dimension of the fractal set where is the standard -transformation with , is a positive function on and is the usual metric on the torus . Moreover, we investigate a modified version of the shrinking target problem, which unifies the shrinking target problems and quantitative recurrence properties for matrix transformations of tori. Let be a non-singular matrix with real coefficients. Then, determines a self-map of the -dimensional torus . For any , let be a positive function on and with . We obtain the Hausdorff dimension of the fractal set where is a hyperrectangle and is a sequence of Lipschitz vector-valued functions on with a uniform Lipschitz constant.