(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 48401, 1632]*) (*NotebookOutlinePosition[ 49471, 1667]*) (* CellTagsIndexPosition[ 49427, 1663]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[TextData[ "Measure and Dimension of the Cantor Set"], "Title", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Steven R. Dunbar\nDepartment of Mathematics and Statistics\nUniversity of \ Nebraska-Lincoln"], "Subsubtitle", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "David Fowler\nDepartment of Curriculum and Instruction\nUniversity of \ Nebraska-Lincoln"], "Subsubtitle", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["\[Copyright]", Evaluatable->False, TextAlignment->Right, AspectRatioFixed->True], StyleBox[ " Copyright Steven R. Dunbar, David Fowler, 1992, All rights reserved. T", Evaluatable->False, TextAlignment->Right, AspectRatioFixed->True] }], "SmallText", Evaluatable->False, TextAlignment->Right, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData["Before Starting"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Get[\"CantorSet.ma\"]"], "Input", AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell[TextData["Lebesgue Measure"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData["Measure Zero"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "A subset E of the real line R which can be covered by a countable \ collection of open intervals whose total length is arbitrarily small is said \ to be a set of measure zero. The classical excluded-middle-thirds Cantor set \ C is a set of measure zero. "], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Here are three stages in determining the measure of the classical Cantor \ set. At each stage we use open sets containing the intervals remaining after \ the middle-third removal process."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "lines[l_List] :=\n Map[Line[{{#[[1]],0}, {#[[2]], 0}}] &,l];"], "Input", AspectRatioFixed->True], Cell[TextData[ "superSets[f_,delta_]:=Map[{#[[1]]-delta,#[[2]]+delta} &, f]"], "Input", AspectRatioFixed->True], Cell[TextData[ "rarc[s_]:=Circle[{(s[[1]]+s[[2]])/2,0},\n\t\t\t\t\t (s[[2]]-s[[1]])/2, \ {-Pi/6,Pi/6}];"], "Input", AspectRatioFixed->True], Cell[TextData[ "larc[s_]:=Circle[{(s[[1]]+s[[2]])/2,0},\n\t\t\t\t\t (s[[2]]-s[[1]])/2, \ {5Pi/6,7Pi/6}];"], "Input", AspectRatioFixed->True], Cell[TextData[ "Table[Show\n\t\t[Graphics\n\t\t\t[{Thickness[0.01], \ lines[intervals[k]],{Thickness[.001],\n\t\t\tLine[{{-0.1,0},{1.1,0}}],\n\t\t\t\ Map[larc,superSets[intervals[k], 0.1/2^k]],\n\t\t\t\ Map[rarc,superSets[intervals[k], 0.1/2^k]]\n\t\t\t}}],\n\t\t\t\ PlotRange->{{-0.1,1.1},{-1,1}},\n\t\t\tAspectRatio->.5],{k,3}];"], "Input", AspectRatioFixed->True], Cell[TextData[ "We can show the Cantor has measure zero by using the sets resulting from the \ excluded-middle-thirds process. We directly compute the total length of the \ intervals remaining after each of the stages of the middle-third removal \ process. The reader should verify that this is permissible, since we are \ covering the classical Cantor set with closed intervals instead of open \ intervals as in the definition. "], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Table[ Apply[Plus, Minus[Apply[Subtract, intervals[i], {1}]]],\n {i, 10}]"], "Input", AspectRatioFixed->True], Cell[TextData["N[%]"], "Input", AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell[TextData["Lebesgue Measure"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "In R, the class of Borel sets is the smallest collection of subsets of R \ such that\n\t(1) every open set and every closed set is a Borel set.\n\t(2) \ the union of every finite or countable collection of Borel sets is a Borel \ set, \tand the intersection of any finite or countable collection of Borel \ sets is a Borel set."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["We call ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Mu]", Evaluatable->False, AspectRatioFixed->True], StyleBox[" a measure on R if ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Mu]", Evaluatable->False, AspectRatioFixed->True], StyleBox[" assigns a non-negative number ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Mu]", Evaluatable->False, AspectRatioFixed->True], StyleBox["(A), possibly ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Infinity]", Evaluatable->False, AspectRatioFixed->True], StyleBox[ ", called the measure of A to each set A in some class of subsets of R \ (for instance the Borel sets) such that:\n\t(1)\t m(\[CapitalEGrave]) = 0\n\t\ (2)\tm(A) ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[LessEqual]", Evaluatable->False, AspectRatioFixed->True], StyleBox[" m(B), for A", Evaluatable->False, AspectRatioFixed->True], StyleBox[" \[SubsetEqual] ", Evaluatable->False, AspectRatioFixed->True], StyleBox["B\n\t(3)\tIf A", Evaluatable->False, AspectRatioFixed->True], StyleBox["1", Evaluatable->False, AspectRatioFixed->True, FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[", A", Evaluatable->False, AspectRatioFixed->True], StyleBox["2", Evaluatable->False, AspectRatioFixed->True, FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[", ... is a countable sequence of sets then\n\t\t\t\t", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Mu]", Evaluatable->False, AspectRatioFixed->True], StyleBox["(", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Union]", Evaluatable->False, AspectRatioFixed->True], StyleBox["i=1", Evaluatable->False, AspectRatioFixed->True, FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox["\[Infinity]", Evaluatable->False, AspectRatioFixed->True, FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox[" A", Evaluatable->False, AspectRatioFixed->True], StyleBox["i", Evaluatable->False, AspectRatioFixed->True, FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[") ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[LessEqual]", Evaluatable->False, AspectRatioFixed->True], StyleBox[" ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Sum]", Evaluatable->False, AspectRatioFixed->True], StyleBox["i=1", Evaluatable->False, AspectRatioFixed->True, FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox["\[Infinity]", Evaluatable->False, AspectRatioFixed->True, FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox[" ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Mu]", Evaluatable->False, AspectRatioFixed->True], StyleBox["(A", Evaluatable->False, AspectRatioFixed->True], StyleBox["i", Evaluatable->False, AspectRatioFixed->True, FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[")\n\t\t\t\t\n\t\twith equality if the A", Evaluatable->False, AspectRatioFixed->True], StyleBox["i", Evaluatable->False, AspectRatioFixed->True, FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[" are disjoint Borel sets.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "For open and closed intervals, we take the Lebesgue measure to be ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Mu]", Evaluatable->False, AspectRatioFixed->True], StyleBox["((a,b)) = b-a, and ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Mu]", Evaluatable->False, AspectRatioFixed->True], StyleBox["([a,b]) = b-a. 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Indeed this is the case, since \ at each stage of the construction there are 2", Evaluatable->False, AspectRatioFixed->True], StyleBox["n", Evaluatable->False, AspectRatioFixed->True, FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox[" intervals of length 2 (2/5)", Evaluatable->False, AspectRatioFixed->True], StyleBox["n ", Evaluatable->False, AspectRatioFixed->True, FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox[ "containing the Cantor set under consideration. Thus the measure is zero.", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[TextData["Box-Counting Dimension"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["Consider a subet E of the real line R, and consider the ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Delta]", Evaluatable->False, AspectRatioFixed->True], StyleBox["-mesh on R given by the collection of intervals [n ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Delta]", Evaluatable->False, AspectRatioFixed->True], StyleBox[", (n+1) ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Delta]", Evaluatable->False, AspectRatioFixed->True], StyleBox["] where n is an integer. 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Nevertheless, without proving the \ required theorem validating the double-limit process, we can make \ experimental approximations to the box-counting dimension of the Cantor \ set."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "We will take the list of endpoints generated by the middle-thirds process \ at stage n as an approximation to the Cantor set. Then multiply each point \ of this set by 1/", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Delta]", Evaluatable->False, AspectRatioFixed->True], StyleBox[ " and take the Floor of the result. This gives the box-index of each \ point Then taking the Union gives a list of the distinct box-indices which \ contain an endpoint of the Cantor set. The cardinality is then given by the \ Length of the list. ", Evaluatable->False, AspectRatioFixed->True] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "boxCount[l_List, delta_] :=\n\tLength[ Union[ Floor[(1/delta)*l]]]"], "Input",\ AspectRatioFixed->True], Cell[TextData["boxCount[Flatten[intervals[5]], 1/40]"], "Input", AspectRatioFixed->True], Cell[TextData[ "approxDimension[ l_List, delta_] := \n\tN[ Log[ \ boxCount[l,delta]]/(-Log[delta])]"], "Input", AspectRatioFixed->True], Cell[TextData["approxDimension[ Flatten[intervals[10]], 1/2^10]"], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "Alternative definitions of the box-counting dimension exist, for instance \ one can let N", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Delta]", Evaluatable->False, AspectRatioFixed->True, FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox["(F) be the smallest number of balls of radius ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Delta]", Evaluatable->False, AspectRatioFixed->True], StyleBox[ " that cover F, or the largest number of disjoint balls of radius ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Delta]", Evaluatable->False, AspectRatioFixed->True], StyleBox[ " that cover F. Theorems are needed to establish that these alternate \ definitions are equivalent. Taking the first alternate definition as the \ smallest number of balls of radius ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Delta]", Evaluatable->False, AspectRatioFixed->True], StyleBox[ " needed to cover F, taking F as the classical Cantor set, taking ", Evaluatable->False, AspectRatioFixed->True], StyleBox["\[Delta]", Evaluatable->False, AspectRatioFixed->True], StyleBox[" = 1/3", Evaluatable->False, AspectRatioFixed->True], StyleBox["n, ", Evaluatable->False, AspectRatioFixed->True, FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox["then clearly it requires at most 2", Evaluatable->False, AspectRatioFixed->True], StyleBox["n ", Evaluatable->False, AspectRatioFixed->True, FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox[ "such balls to cover. 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L. Royden, Macmillan, New York, 1988.\n\nOur brief and simplified \ treament is based on the discussion in\n", AspectRatioFixed->True, FontFamily->"Times", FontWeight->"Plain"], StyleBox["Fractal Geometry", AspectRatioFixed->True, FontFamily->"Times", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[", K. J. Falconer, J. Wiley and Sons, New York, 1990.", AspectRatioFixed->True, FontFamily->"Times", FontWeight->"Plain"], StyleBox[" ", AspectRatioFixed->True, FontFamily->"Times", FontWeight->"Plain"] }], "Input", AspectRatioFixed->True], Cell[TextData[{ StyleBox[ "The box-counting dimension is also known as the fractal dimension and the \ capacity dimension and it is a popular measure of a set because it is \ relatively easy to approximate numerically. 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