Results 1  10
of
27
Norm convergence of multiple ergodic averages for commuting transformations
, 2007
"... Let T1,..., Tl: X → X be commuting measurepreserving transformations on a probability space (X, X, µ). We show that the multiple ergodic averages 1 PN−1 N n=0 f1(T n 1 x)... fl(T n l x) are convergent in L2 (X, X, µ) as N → ∞ for all f1,..., fl ∈ L ∞ (X, X, µ); this was previously established fo ..."
Abstract

Cited by 34 (1 self)
 Add to MetaCart
Let T1,..., Tl: X → X be commuting measurepreserving transformations on a probability space (X, X, µ). We show that the multiple ergodic averages 1 PN−1 N n=0 f1(T n 1 x)... fl(T n l x) are convergent in L2 (X, X, µ) as N → ∞ for all f1,..., fl ∈ L ∞ (X, X, µ); this was previously established for l = 2 by Conze and Lesigne [2] and for general l assuming some additional ergodicity hypotheses on the maps Ti and TiT −1 j by Frantzikinakis and Kra [3] (with the l = 3 case of this result established earlier in [29]). Our approach is combinatorial and finitary in nature, inspired by recent developments regarding the hypergraph regularity and removal lemmas, although we will not need the full strength of those lemmas. In particular, the l = 2 case of our arguments are a finitary analogue of those in [2].
A quantitative Mean Ergodic Theorem for uniformly convex Banach spaces
, 2008
"... We provide an explicit uniform bound on the local stability of ergodic averages in uniformly convex Banach spaces. Our result can also be viewed as a finitary version in the sense of T. Tao of the Mean Ergodic Theorem for such spaces and so generalizes similar results obtained for Hilbert spaces by ..."
Abstract

Cited by 8 (7 self)
 Add to MetaCart
We provide an explicit uniform bound on the local stability of ergodic averages in uniformly convex Banach spaces. Our result can also be viewed as a finitary version in the sense of T. Tao of the Mean Ergodic Theorem for such spaces and so generalizes similar results obtained for Hilbert spaces by Avigad, Gerhardy and Towsner [1] and T. Tao [10]. 1
The metamathematics of ergodic theory
 THE ANNALS OF PURE AND APPLIED LOGIC
, 2009
"... The metamathematical tradition, tracing back to Hilbert, employs syntactic modeling to study the methods of contemporary mathematics. A central goal has been, in particular, to explore the extent to which infinitary methods can be understood in computational or otherwise explicit terms. Ergodic theo ..."
Abstract

Cited by 5 (1 self)
 Add to MetaCart
The metamathematical tradition, tracing back to Hilbert, employs syntactic modeling to study the methods of contemporary mathematics. A central goal has been, in particular, to explore the extent to which infinitary methods can be understood in computational or otherwise explicit terms. Ergodic theory provides rich opportunities for such analysis. Although the field has its origins in seventeenth century dynamics and nineteenth century statistical mechanics, it employs infinitary, nonconstructive, and structural methods that are characteristically modern. At the same time, computational concerns and recent applications to combinatorics and number theory force us to reconsider the constructive character of the theory and its methods. This paper surveys some recent contributions to the metamathematical study of ergodic theory, focusing on the mean and pointwise ergodic theorems and the Furstenberg structure theorem for measure preserving systems. In particular, I characterize the extent to which these theorems are nonconstructive, and explain how prooftheoretic methods can be used to locate their “constructive content.”
On the logical analysis of proofs based on nonseparable Hilbert space theory
, 2010
"... Starting in [15] and then continued in [9, 17, 24] and [18], general logical metatheorems were developed that guarantee the extractability of highly uniform effective bounds from proofs of theorems that hold for general classes of structures such as ..."
Abstract

Cited by 4 (4 self)
 Add to MetaCart
Starting in [15] and then continued in [9, 17, 24] and [18], general logical metatheorems were developed that guarantee the extractability of highly uniform effective bounds from proofs of theorems that hold for general classes of structures such as
On the computational content of convergence proofs via Banach limits
"... This paper addresses new developments in the ongoing proof mining program, i.e. the use of tools from proof theory to extract effective quantitative information from prima facie ineffective proofs in analysis. Very recently, the current authors developed a method to extract rates of metastability (i ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
This paper addresses new developments in the ongoing proof mining program, i.e. the use of tools from proof theory to extract effective quantitative information from prima facie ineffective proofs in analysis. Very recently, the current authors developed a method to extract rates of metastability (in the sense of Tao) from convergence proofs in nonlinear analysis that are based on Banach limits and so (for all what is known) rely on the axiom of choice. In this paper we apply this method to a proof due to Shioji and Takahashi on the convergence of Halpern iterations in spaces X with a uniformly Gâteaux differentiable norm. We design a logical metatheorem guaranteeing the extractability of highly uniform rates of metastability under the stronger condition of the uniform smoothness of X. Combined with our method of eliminating Banach limits this yields a full quantitative analysis of the proof by Shioji and Takahashi. We also give a sufficient condition for the computability of the rate of convergence of Halpern iterations.
Computing the speed of convergence of ergodic averages and pseudorandom points in computable dynamical systems
, 2010
"... ..."
DYNAMICS AND ABSTRACT COMPUTABILITY: COMPUTING INVARIANT MEASURES
, 2010
"... (Communicated by Carlangelo Liverani) Abstract. We consider the question of computing invariant measures from an abstract point of view. Here, computing a measure means finding an algorithm which can output descriptions of the measure up to any precision. We work in a general framework (computable m ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
(Communicated by Carlangelo Liverani) Abstract. We consider the question of computing invariant measures from an abstract point of view. Here, computing a measure means finding an algorithm which can output descriptions of the measure up to any precision. We work in a general framework (computable metric spaces) where this problem can be posed precisely. We will find invariant measures as fixed points of the transfer operator. In this case, a general result ensures the computability of isolated fixed points of a computable map. We give general conditions under which the transfer operator is computable on a suitable set. This implies the computability of many “regular enough” invariant measures and among them many physical measures. On the other hand, not all computable dynamical systems have a computable invariant measure. We exhibit two examples of computable dynamics, one having a physical measure which is not computable and one for which no invariant measure is computable, showing some subtlety in this kind of problems. 1. Introduction. An
Algorithmic tests and randomness with respect to a class of measures
, 2011
"... This paper offers some new results on randomness with respect to classes of measures, along with a didactical exposition of their context based on results that appeared elsewhere. We start with the reformulation of the MartinLöf definition of randomness (with respect to computable measures) in term ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
This paper offers some new results on randomness with respect to classes of measures, along with a didactical exposition of their context based on results that appeared elsewhere. We start with the reformulation of the MartinLöf definition of randomness (with respect to computable measures) in terms of randomness deficiency functions. A formula that expresses the randomness deficiency in terms of prefix complexity is given (in two forms). Some approaches that go in another direction (from deficiency to complexity) are considered. The notion of Bernoulli randomness (independent coin tosses for an asymmetric coin with some probability p of head) is defined. It is shown that a sequence is Bernoulli if it is random with respect to some Bernoulli