{"id":326,"date":"2025-07-08T14:38:57","date_gmt":"2025-07-08T13:38:57","guid":{"rendered":"https:\/\/mat.uab.cat\/web\/tfg\/?p=326"},"modified":"2026-05-28T19:05:21","modified_gmt":"2026-05-28T18:05:21","slug":"mesura-i-dimensio-de-hausdorff","status":"publish","type":"post","link":"https:\/\/mat.uab.cat\/web\/tfg\/mesura-i-dimensio-de-hausdorff\/","title":{"rendered":"Mesura i dimensi\u00f3 de Hausdorff"},"content":{"rendered":"\n<p><strong>Prerrequisits<\/strong>: Teoria de la mesura abstracta.<\/p>\n<p>Veurem la definici\u00f3 de dimensions fraccion\u00e0ries a trav\u00e9s de les mesures de Hausdorff, i estudiarem breument les dimensions de conjunts auto-similars, tipus fractals (Sierpi\u0144ski, Cantor,&#8230;). A partir d&#8217;aqu\u00ed tenim possibles l\u00ednies de treball:<\/p>\n<ul>\n<li>Altres tipus de dimensi\u00f3, tipus Minkowski, etc\u00e8tera. [1]<\/li>\n<li>F\u00f3rmules de l&#8217;\u00e0rea i la co\u00e0rea: la f\u00f3rmula de l&#8217;\u00e0rea ens diu que la mesura de Hausdorff de dimensi\u00f3 \\(n\\) de la imatge d&#8217;un conjunt \\(A\\subset\\mathbb R^n\\) per una funci\u00f3 \\(f:\\mathbb R^n\\to\\mathbb R^m\\) comptant multiplicitat, amb \\(m\\geq n\\), coincideix amb la integral del jacobi\u00e0.\u00a0 La f\u00f3rmula de la co\u00e0rea tracta el cas \\(m&lt;n\\). [2]<\/li>\n<li>Densitat i teorema de Marstrand: considerem la densitat \\(\\Theta^s(\\mu,a)=\\lim_{r\\to 0}(2r)^{-s} \\mu(B(a,r)).\\)<br \/>El teorema de Marstrand diu que quan una mesura (de Radon) t\u00e9 densitat positiva i finita en un conjunt de mesura positiva, aleshores \\(s\\) \u00e9s un enter. [1]<\/li>\n<li>Mesures uniformes: si una mesura de Radon satisf\u00e0 que \\(\\mu(B(x,r))=c r^m\\) per a tot \\(x\\in{\\rm supp} \\mu\\) i tot \\(r&gt;0\\), diem que \\(\\mu\\) \u00e9s una mesura uniforme. Veurem que tota mesura uniforme est\u00e0 suportada en el conjunt de zeros d&#8217;un polinomi quadr\u00e0tic. [1]<\/li>\n<\/ul>\n<p>[1] Mattila, P. <i>Geometry of sets and measures in Euclidean spaces: fractals and rectifiability<\/i>. Vol. 44. Cambridge university press, 1999.<\/p>\n<p>[2] Evans, L. Craig, and R. F. Gariepy. <i>Measure theory and fine properties of functions. <\/i>Studies in advanced mathematics. (1992).<\/p>\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Prerrequisits: Teoria de la mesura abstracta. Veurem la definici\u00f3 de dimensions fraccion\u00e0ries a trav\u00e9s de les mesures de Hausdorff, i estudiarem breument les dimensions de conjunts auto-similars, tipus fractals (Sierpi\u0144ski, Cantor,&#8230;). A partir d&#8217;aqu\u00ed tenim possibles l\u00ednies de treball: Altres tipus de dimensi\u00f3, tipus Minkowski, etc\u00e8tera. [1] F\u00f3rmules de l&#8217;\u00e0rea i la co\u00e0rea: la f\u00f3rmula [&hellip;]<\/p>\n","protected":false},"author":53,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[29],"tags":[40],"class_list":["post-326","post","type-post","status-publish","format-standard","hentry","category-analisi-matematica","tag-marti-prats"],"_links":{"self":[{"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/posts\/326","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/users\/53"}],"replies":[{"embeddable":true,"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/comments?post=326"}],"version-history":[{"count":5,"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/posts\/326\/revisions"}],"predecessor-version":[{"id":336,"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/posts\/326\/revisions\/336"}],"wp:attachment":[{"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/media?parent=326"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/categories?post=326"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/tags?post=326"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}