2012/05/08 by Daniel J. Thompson, Daniel Thompson · 151 citations
Biochemistry, Genetics and Molecular Biology · Mathematics · #Advanced Topology and Set Theory #Caveolin-1 and cellular processes #Combinatorics #Compact space #Continuous function (set theory) #Continuous map #Discrete mathematics #Function (biology) #Geometry #Hausdorff dimension #Hausdorff space #Mathematical Dynamics and Fractals #Mathematics #Metric space #Product (mathematics) #Pure mathematics #Topological entropy #Transformation (genetics)
paper · pdf · doi:10.1090/s0002-9947-2012-05540-1
published in Transactions of the American Mathematical Society 364(10), 5395-5414 (American Mathematical Society)
openalex publication_date 2012/05/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/26
Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper X comma d right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:mi>d</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(X,d)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a compact metric space, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f colon upper X right-arrow from bar upper X"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>:</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false"> ↦ </mml:mo> <mml:mi>X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">f:X ↦ X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a continuous map satisfying a property we call almost specification (which is slightly weaker than the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="g"> <mml:semantics> <mml:mi>g</mml:mi> <mml:annotation encoding="application/x-tex">g</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -almost product property of Pfister and Sullivan), and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi colon upper X right-arrow from bar double-struck upper R"> <mml:semantics> <mml:mrow> <mml:mi> φ </mml:mi> <mml:mo>:</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false"> ↦ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">φ : X ↦ \mathbb R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a continuous function. We show that the set of points for which the Birkhoff average of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi"> <mml:semantics> <mml:mi> φ </mml:mi> <mml:annotation encoding="application/x-tex">φ</mml:annotation> </mml:semantics> </mml:math> </inline-formula> does not exist (which we call the irregular set) is either empty or has full topological entropy. Every <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="beta"> <mml:semantics> <mml:mi> β </mml:mi> <mml:annotation encoding="application/x-tex">β</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -shift satisfies almost specification and we show that the irregular set for any <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="beta"> <mml:semantics> <mml:mi> β </mml:mi> <mml:annotation encoding="application/x-tex">β</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -shift or <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="beta"> <mml:semantics> <mml:mi> β </mml:mi> <mml:annotation encoding="application/x-tex">β</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -transformation is either empty or has full topological entropy and Hausdorff dimension.