2011/02/28 by Krerley Oliveira, Xueting Tian · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #Cellular Automata and Applications #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #math-ph #math.DS #math.MP
paper · pdf · doi:10.1090/s0002-9947-2013-05819-9
published as Transactions of the American Mathematical Society, 2013, 365(8): 4371-4392 · 21pages, nonuniform specification
arxiv created 2011/06/18 · openalex publication_date 2013/04/02 · arxiv updated 2013/07/18 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
In this paper we deal with an invariant ergodic hyperbolic measure <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu"> <mml:semantics> <mml:mi> μ </mml:mi> <mml:annotation encoding="application/x-tex">μ</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for a diffeomorphism <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f comma"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">f,</mml:annotation> </mml:semantics> </mml:math> </inline-formula> assuming that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f"> <mml:semantics> <mml:mi>f</mml:mi> <mml:annotation encoding="application/x-tex">f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is either <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Superscript 1 plus alpha"> <mml:semantics> <mml:msup> <mml:mi>C</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mi> α </mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">C1+α </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="upper C Superscript 1"> <mml:semantics> <mml:msup> <mml:mi>C</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">C1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and the Oseledec splitting of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu"> <mml:semantics> <mml:mi> μ </mml:mi> <mml:annotation encoding="application/x-tex">μ</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is dominated. We show that this system <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis f comma mu right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>f</mml:mi> <mml:mo>,</mml:mo> <mml:mi> μ </mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(f,μ )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> satisfies a weaker and non-uniform version of specification, related with notions studied in several recent papers. Our main results have several consequences: as corollaries, we are able to improve the results about quantitative Poincaré recurrence, removing the assumption of the non-uniform specification property in the main theorem of “Recurrence and Lyapunov exponents” by Saussol, Troubetzkoy and Vaienti that establishes an inequality between Lyapunov exponents and local recurrence properties. Another consequence is the fact that any such measure is the weak limit of averages of Dirac measures at periodic points, as in a paper by Sigmund. One can show that the topological pressure can be calculated by considering the convenient weighted sums on periodic points whenever the dynamic is positive expansive and every measure with pressure close to the topological pressure is hyperbolic.