Quantization dimensions of compactly supported probability measures via Rényi dimensions

Abstract

We provide a complete picture of the upper quantization dimension in terms of the Rényi dimension by proving that the upper quantization dimension of order <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="r greater-than 0"> <mml:semantics> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">r&gt;0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for an arbitrary compactly supported Borel probability measure <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="nu"> <mml:semantics> <mml:mi> ν </mml:mi> <mml:annotation encoding="application/x-tex">\nu</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is given by its Rényi dimension at the point <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q Subscript r"> <mml:semantics> <mml:msub> <mml:mi>q</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>r</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">q_{r}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> where the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript q"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>q</mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">L^{q}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -spectrum of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="nu"> <mml:semantics> <mml:mi> ν </mml:mi> <mml:annotation encoding="application/x-tex">\nu</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and the line through the origin with slope <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="r"> <mml:semantics> <mml:mi>r</mml:mi> <mml:annotation encoding="application/x-tex">r</mml:annotation> </mml:semantics> </mml:math> </inline-formula> intersect. In particular, this proves the continuity of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="r right-arrow from bar upper D overbar Subscript r Baseline left-parenthesis reverse-solidus nu right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo stretchy="false"> ↦ </mml:mo> <mml:msub> <mml:mover> <mml:mi>D</mml:mi> <mml:mo accent="false"> ¯ </mml:mo> </mml:mover> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>r</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mtext>\nu )</mml:mtext> </mml:mrow> <mml:annotation encoding="application/x-tex">r\mapsto \overline {D}_{r}(\text {\nu )}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> as conjectured by Lindsay [ <italic>Quantization dimension for probability distributions</italic> , ProQuest LLC, Ann Arbor, MI, 2001. Thesis (Ph.D.)-University of North Texas]. This viewpoint also sheds new light on the connection of the quantization problem with other concepts from fractal geometry in that we obtain a one-to-one correspondence of the upper quantization dimension and the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript q"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>q</mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">L^{q}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -spectrum restricted to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis 0 comma 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\left (0,1\right )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We give sufficient conditions in terms of the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript q"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>q</mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">L^{q}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -spectrum for the existence of the quantization dimension. In this way we show as a byproduct that the quantization dimension exists for every Gibbs measure with respect to a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper C Superscript 1"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">C</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">\mathcal {C}^{1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -self-conformal iterated function system on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R Superscript d"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>d</mml:mi> </mml:

Topics & Concepts