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W1567102972 abstract "Consider a linear Cantor set <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper K> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding=application/x-tex>K</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, which is the attractor of a linear iterated function system (i.f.s.) <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper S Subscript j Baseline x equals rho Subscript j Baseline x plus b Subscript j> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>S</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>j</mml:mi> </mml:mrow> </mml:msub> <mml:mi>x</mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>ρ</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>j</mml:mi> </mml:mrow> </mml:msub> <mml:mi>x</mml:mi> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>b</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>j</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>S_{j}x = rho _{j}x+b_{j}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=j equals 1 comma ellipsis comma m> <mml:semantics> <mml:mrow> <mml:mi>j</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:mi>m</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>j = 1,ldots ,m</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, on the line satisfying the open set condition (where the open set is an interval). It is known that <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper K> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding=application/x-tex>K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has Hausdorff dimension <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=alpha> <mml:semantics> <mml:mi>α</mml:mi> <mml:annotation encoding=application/x-tex>alpha</mml:annotation> </mml:semantics> </mml:math> </inline-formula> given by the equation <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=sigma-summation Underscript j equals 1 Overscript m Endscripts rho Subscript j Superscript alpha Baseline equals 1> <mml:semantics> <mml:mrow> <mml:munderover> <mml:mo>∑</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>j</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>m</mml:mi> </mml:mrow> </mml:munderover> <mml:msubsup> <mml:mi>ρ</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>j</mml:mi> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>α</mml:mi> </mml:mrow> </mml:msubsup> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>sum ^{m}_{j=1} rho ^{alpha }_{j} = 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and that <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper H Subscript alpha Baseline left-parenthesis upper K right-parenthesis> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>H</mml:mi> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>α</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>K</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>mathcal {H}_{alpha }(K)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is finite and positive, where <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper H Subscript alpha> <mml:semantics> <mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>H</mml:mi> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>α</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding=application/x-tex>mathcal {H}_{alpha }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> denotes Hausdorff measure of dimension <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=alpha> <mml:semantics> <mml:mi>α</mml:mi> <mml:annotation encoding=application/x-tex>alpha</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We give an algorithm for computing <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper H Subscript alpha Baseline left-parenthesis upper K right-parenthesis> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>H</mml:mi> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>α</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>K</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>mathcal {H}_{alpha }(K)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> exactly as the maximum of a finite set of elementary functions of the parameters of the i.f.s. When <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=rho 1 equals rho Subscript m> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>ρ</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>ρ</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>m</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>rho _{1} = rho _{m}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (or more generally, if <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=log rho 1> <mml:semantics> <mml:mrow> <mml:mi>log</mml:mi> <mml:mo>⁡</mml:mo> <mml:msub> <mml:mi>ρ</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>log rho _{1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=log rho Subscript m> <mml:semantics> <mml:mrow> <mml:mi>log</mml:mi> <mml:mo>⁡</mml:mo> <mml:msub> <mml:mi>ρ</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>m</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>log rho _{m}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are commensurable), the algorithm also gives an interval <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper I> <mml:semantics> <mml:mi>I</mml:mi> <mml:annotation encoding=application/x-tex>I</mml:annotation> </mml:semantics> </mml:math> </inline-formula> that maximizes the density <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=d left-parenthesis upper I right-parenthesis equals script upper H Subscript alpha Baseline left-parenthesis upper K intersection upper I right-parenthesis slash StartAbsoluteValue upper I EndAbsoluteValue Superscript alpha> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>I</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>=</mml:mo> <mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>H</mml:mi> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>α</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>K</mml:mi> <mml:mo>∩</mml:mo> <mml:mi>I</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>|</mml:mo> </mml:mrow> <mml:mi>I</mml:mi> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>|</mml:mo> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>α</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>d(I) = mathcal {H}_{alpha }(K cap I)/|I|^{alpha }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The Hausdorff measure <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper H Subscript alpha Baseline left-parenthesis upper K right-parenthesis> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>H</mml:mi> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>α</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>K</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>mathcal {H}_{alpha }(K)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is not a continuous function of the i.f.s. parameters. We also show that given the contraction parameters <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=rho Subscript j> <mml:semantics> <mml:msub> <mml:mi>ρ</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>j</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding=application/x-tex>rho _{j}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, it is possible to choose the translation parameters <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=b Subscript j> <mml:semantics> <mml:msub> <mml:mi>b</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>j</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding=application/x-tex>b_{j}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in such a way that <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper H Subscript alpha Baseline left-parenthesis upper K right-parenthesis equals StartAbsoluteValue upper K EndAbsoluteValue Superscript alpha> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>H</mml:mi> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>α</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>K</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>=</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>|</mml:mo> </mml:mrow> <mml:mi>K</mml:mi> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>|</mml:mo> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>α</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>mathcal {H}_{alpha }(K) = |K|^{alpha }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, so the maximum density is one. Most of the results presented here were discovered through computer experiments, but we give traditional mathematical proofs." @default.
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W1567102972 title "Exact Hausdorff measure and intervals of maximum density for Cantor sets" @default.
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