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Ultraproducts and metastability
 NEW YORK J. MATH. 19 (2013) 713–727.
, 2013
"... Given a convergence theorem in analysis, under very general conditions a modeltheoretic compactness argument implies that there is a uniform bound on the rate of metastability. We illustrate with three examples from ergodic theory. ..."
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Given a convergence theorem in analysis, under very general conditions a modeltheoretic compactness argument implies that there is a uniform bound on the rate of metastability. We illustrate with three examples from ergodic theory.
Effective results on nonlinear ergodic averages in CAT(κ) spaces
, 2013
"... In this paper we apply proof mining techniques to compute, in the setting of CAT(κ) spaces (with κ> 0), effective and highly uniform rates of asymptotic regularity and metastability for a nonlinear generalization of the ergodic averages, known as the Halpern iteration. In this way, we obtain a un ..."
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In this paper we apply proof mining techniques to compute, in the setting of CAT(κ) spaces (with κ> 0), effective and highly uniform rates of asymptotic regularity and metastability for a nonlinear generalization of the ergodic averages, known as the Halpern iteration. In this way, we obtain a uniform quantitative version of a nonlinear extension of the classical von Neumann mean ergodic theorem.
VARIATIONNORM AND FLUCTUATION ESTIMATES FOR ERGODIC BILINEAR AVERAGES
, 2015
"... For any dynamical system, we show that higher variationnorms for the sequence of ergodic bilinear averages of two functions satisfy a large range of bilinear Lp estimates. It follows that, with probability one, the number of fluctuations along this sequence may grow at most polynomially with respe ..."
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For any dynamical system, we show that higher variationnorms for the sequence of ergodic bilinear averages of two functions satisfy a large range of bilinear Lp estimates. It follows that, with probability one, the number of fluctuations along this sequence may grow at most polynomially with respect to (the growth of) the underlying scale. These results strengthen previous works of Lacey and Bourgain where almost surely convergence of the sequence was proved (which is equivalent to the qualitative statement that the number of fluctuations is finite at each scale). Via transference, the proof reduces to establishing new bilinear Lp bounds for variationnorms of truncated bilinear operators on R, and the main new ingredient of the proof of these bounds is a variationnorm extension of maximal Bessel inequalities of Lacey and Demeter–Tao–Thiele.