Second order ergodic theorems for ergodic transformations of infinite measure spaces

Aaronson, Jon; Denker, Manfred; Fisher, Albert M.

doi:10.1090/s0002-9939-1992-1099339-4

Cited by 18 publications

(9 citation statements)

References 5 publications

(3 reference statements)

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“…The almost sure behaviour of S" (/) for nonnegative integrable functions / has been obtained in [4][5][6][7]. We repeat here the main results for completeness.…”

Section: Jxmentioning

confidence: 69%

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Ergodic theory for Markov fibred systems and parabolic rational maps

Aaronson

Denker²,

Urbański

1993

Trans. Amer. Math. Soc.

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145

149

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A parabolic rational map of the Riemann sphere admits a nonatomic /¡-conformai measure on its Julia set where h = the HausdorfT dimension of the Julia set and satisfies 1/2 < h < 2 . With respect to this measure the rational map is conservative, exact and there is an equivalent cr-finite invariant measure. Finiteness of the measure is characterised. Central limit theorems are proved in the case of a finite invariant measure and return sequences are identified in the case of an infinite one. A theory of Markov fibred systems is developed, and parabolic rational maps are considered within this framework.for every Borel set A c J(T) such that T\A is injective (see [34]).It turns out (Theorem 8.7) that, for h = the HausdorfT dimension of J(T), the unique A-conformal measure for T is nonatomic. This result is obtained

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“…The almost sure behaviour of S" (/) for nonnegative integrable functions / has been obtained in [4][5][6][7]. We repeat here the main results for completeness.…”

Section: Jxmentioning

confidence: 69%

“…The proofs of most of the results can be found in [29,[2][3][4][5][6][7]. The proofs of most of the results can be found in [29,[2][3][4][5][6][7].…”

Section: Ergodic Theory Of Noninvertible Transformationsmentioning

confidence: 99%

Ergodic theory for Markov fibred systems and parabolic rational maps

Aaronson

Denker²,

Urbański

1993

Trans. Amer. Math. Soc.

Self Cite

145

149

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“…At present, it is an open question whether every pointwise dual ergodic transformation has a Darling-Kac set. However, it is desirable to prove pointwise dual ergodicity by identifying Darling-Kac sets, as this facilitates the proof of several strong properties for f (see for instance [1,2,3,5,6,38,40,42] and for the setting of Markov chains [8,10,29]; see also [9] and references therein).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

First and higher order uniform dual ergodic theorems for dynamical systems with infinite measure

Melbourne

Terhesiu

2012

Isr. J. Math.

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We generalize the proof of Karamata's Theorem by the method of approximation by polynomials to the operator case. As a consequence, we offer a simple proof of uniform dual ergodicity for a very large class of dynamical systems with infinite measure, and we obtain bounds on the convergence rate.In many cases of interest, including the Pomeau-Manneville family of intermittency maps, the estimates obtained through real Tauberian remainder theory are very weak. Building on the techniques of complex Tauberian remainder theory, we develop a method that provides second (and higher) order asymptotics. In the process, we derive a higher order Tauberian theorem for scalar power series, which to our knowledge, has not previously been covered.

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“…It turns out that due to the measure µ being infinite, it is impossible to replace functions S n g(x) by constants {a n } [A, Theorem 2.4.2]. However, it was observed earlier [Fi1,ADF,LS] that for some systems, the ratios S n f (x)/a n (for some choice of a n ) still converge to f dµ, though in a weaker sense (second-order averages). The asymptotic behavior of the sequence {a n } is an invariant of the dynamical system.…”

Section: Introductionmentioning

confidence: 99%