Every non-amenable countable group induces orbit inequivalent ergodic equivalence relations on standard Borel probability spaces. Not every free, ergodic, measure preserving action of F2 on a standard Borel probability space is orbit equivalent to an action of a countable group on an inverse limit of finite spaces. There is a treeable non-hyperfinite Borel equivalence relation which is not universal for treeable in the ^B ordering.

Abstract. For r = (r1,..., rd) ∈ R d the map τr: Z d → Z d given by τr(a1,..., ad) = (a2,..., ad, −⌊r1a1 + · · · + rdad⌋) is called a shift radix system if for each a ∈ Zd there exists an integer k> 0 with τk r (a) = 0. As shown in the first two parts of this series of papers shift radix systems are intimately related to certain well-known notions of number systems like β-expansions and canonical number systems. In the present paper further structural relationships between shift radix systems and canonical number systems are investigated. Among other results we show that canonical number systems related to polynomials d�i piX =0 i ∈ Z[X] of degree d with a large but fixed constant term p0 approximate the set of (d − 1)-dimensional shift radix systems. The proofs make extensive use of the following tools: Firstly, vectors r ∈ Rd which define shift radix systems are strongly connected to monic real polynomials all of whose roots lie inside the unit circle. Secondly, geometric considerations which were established in Part I of this series of papers are exploited. The main results establish two conjectures mentioned in Part II of this series of papers. 1.

Introduction. These notes are based upon a day-long lecture workshop presented by Simon

The main goal of this paper is to answer question 1.10 and settle conjecture 1.11 of Benjamini-Lyons-Schramm [BLS99] relating harmonic Dirichlet functions on a graph to those of the infinite clusters in the uniqueness phase of Bernoulli percolation. We extend the result to more general invariant percolations, including the Random-Cluster model. We prove the existence of the nonuniqueness phase for the Bernoulli percolation (and make some progress for Random-Cluster model) on unimodular transitive locally finite graphs admitting nonconstant harmonic Dirichlet functions. This is done by using the device of ℓ 2 Betti numbers.

Monte-Carlo Method for the Analysis of Probabilistic Programs # David Monniaux Ecole Normale Superieure Laboratoire d'Informatique 45, rue d'Ulm 75230 Paris cedex 5 France David.Monniaux@ens.fr ABSTRACT We introduce a new method, combination of random testing and abstract interpretation, for the analysis of programs featuring both probabilistic and non-probabilistic nondeterminism. After introducing "ordinary" testing, we show how to combine testing and abstract interpretation and give formulas linking the precision of the results to the number of iterations. We then discuss complexity and optimization issues and end with some experimental results. 1 INTRODUCTION We introduce a generic method that lifts an ordinary abstract interpretation scheme to an analyzer yielding upper bounds on the probability of certain outcomes, taking into account both randomness and ordinary nondeterminism. 1.1 Motivations It is sometimes desirable to estimate the probability of certain outcomes...

Analysis, (1969), Springer-Verlag, New York. Hjorth, G. [97]. On the isomorphism problem for measure preserving transformations, (April 1997), preprint. 67 68 References ...

Let X be a compact metric space. A closed set K is located if the distance function d(x, K) exists as a continuous realvalued function on X ; weakly located if the predicate d(x, K) > r is # 1 allowing parameters. The purpose of this paper is to explore the concepts of located and weakly located subsets of a compact separable metric space in the context of subsystems of second order arithmetic such as RCA 0 , WKL 0 and ACA 0 . We also give some applications of these concepts by discussing some versions of the Tietze extension theorem. In particular we prove an RCA 0 version of this result for weakly located closed sets.

We prove a generalization of the ‘Subadditive Limit Theorem’and of the corresponding Berz Theorem in a class of functions that includes both the measurable functions and the ‘Baire functions’(those with the Baire property). The generic subadditive functions are de…ned by a combinatorial property previously introduced for the study of the foundations of regular variation in [BOst1]. By specialization we thus provide the previously unknown Baire variants of the fundamental theorems of subbaditive functions, answering an old question ([BGT], 2.12.4 p. 123).

Abstract. For a countable amenable group Γ and an element f in the integral group ring ZΓ being invertible in the group von Neumann algebra of Γ, we show that the entropy of the shift action of Γ on the Pontryagin dual of the quotient of ZΓ by its left ideal generated by f is the logarithm of the Fuglede-Kadison determinant of f. For the proof, we establish an ℓ p-version of Rufus Bowen’s definition of topological entropy, addition formulas for group extensions of countable amenable group actions, and an approximation formula for the Fuglede-Kadison determinant of f in terms of the determinants of perturbations of the compressions of f. 1.

Abstract. Extending results of Schindler [Sch] and Hamkins and Welch [HW03], we establish in the context of infinite time Turing machines that P is properly contained in NP ∩co-NP. Furthermore, NP ∩co-NP is exactly the class of hyperarithmetic sets. For the more general classes, we establish that P + = NP + ∩co-NP + = NP ∩co-NP, though P ++ is properly contained in NP ++ ∩co-NP ++. Within any contiguous block of infinite clockable ordinals, we show that Pα ̸ = NPα ∩co-NPα, but if β begins a gap in the clockable ordinals, then Pβ = NPβ ∩co-NPβ. Finally, we establish that P f ̸ = NP f ∩co-NP f for most functions f: R → ord, although we provide examples where P f = NP f ∩co-NP f and P f ̸ = NP f.