Universal characteristic factors and Furstenberg averages

2004/03/31 by Tamar Ziegler · 6 citations
Mathematics · #Limits and Structures in Graph Theory #Mathematical Dynamics and Fractals #Random Matrices and Applications #math.DS #msc:37A25 #msc:37A30

paper · pdf · doi:10.1090/s0894-0347-06-00532-7

published as J. Amer. Math. Soc. 20 (2007), 53-97 · 47 pages

arxiv created 2004/10/19 · openalex publication_date 2006/03/17 · arxiv updated 2012/01/04 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X equals left-parenthesis upper X Superscript 0 Baseline comma script upper B comma mu comma upper T right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>X</mml:mi> <mml:mo>=</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>X</mml:mi> <mml:mn>0</mml:mn> </mml:msup> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">B</mml:mi> </mml:mrow> <mml:mo>,</mml:mo> <mml:mi> μ </mml:mi> <mml:mo>,</mml:mo> <mml:mi>T</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">X=(X⁰,\mathcal B,μ ,T)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be an ergodic probability measure-preserving system. For a natural number <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> we consider the averages <disp-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row with Label left-parenthesis asterisk right-parenthesis EndLabel StartFraction 1 Over upper N EndFraction sigma-summation Underscript n equals 1 Overscript upper N Endscripts product Underscript j equals 1 Overscript k Endscripts f Subscript j Baseline left-parenthesis upper T Superscript a Super Subscript j Superscript n Baseline x right-parenthesis EndLayout"> <mml:semantics> <mml:mtable side="left" displaystyle="false"> <mml:mlabeledtr> <mml:mtd> <mml:mtext>(*)</mml:mtext> </mml:mtd> <mml:mtd> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mi>N</mml:mi> </mml:mfrac> <mml:munderover> <mml:mo> ∑ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mi>N</mml:mi> </mml:munderover> <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:mi>k</mml:mi> </mml:munderover> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>j</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>T</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>a</mml:mi> <mml:mi>j</mml:mi> </mml:msub> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mtd> </mml:mlabeledtr> </mml:mtable> <mml:annotation encoding="application/x-tex"> \frac 1N∑ _n=1^N ∏ _j=1^k f_j(T^a_jnx)</mml:annotation> </mml:semantics> </mml:math> </disp-formula> where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f Subscript j Baseline element-of upper L Superscript normal infinity Baseline left-parenthesis mu right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>j</mml:mi> </mml:msub> <mml:mo> ∈ </mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi> μ </mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">f_j ∈ L^∞ (μ )</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="a Subscript j"> <mml:semantics> <mml:msub> <mml:mi>a</mml:mi> <mml:mi>j</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">a_j</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are integers. A factor of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is characteristic for averaging schemes of length <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (or <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -characteristic) if for any nonzero distinct integers <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a 1 comma ellipsis comma a Subscript k Baseline"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>a</mml:mi> <mml:mn>1</mml:mn> </mml:msub>

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