Results 1  10
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153
A converse to Dye's theorem
"... Every nonamenable 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 ..."
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Cited by 41 (2 self)
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Every nonamenable 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 nonhyperfinite Borel equivalence relation which is not universal for treeable in the ^B ordering.
Wadge hierarchy and Veblen hierarchy. Part II: Borel sets of infinite rank
, 1998
"... We consider Borel sets of the form A ` ! (with usual topology) where cardinality of is less than some uncountable regular cardinal . We obtain a "normal form" of A, by finding a Borel set\Omega\Gamma ff) such that A and\Omega\Gamma ff) continuously reduce to each other. We do so by defining Bo ..."
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Cited by 25 (11 self)
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We consider Borel sets of the form A ` ! (with usual topology) where cardinality of is less than some uncountable regular cardinal . We obtain a "normal form" of A, by finding a Borel set\Omega\Gamma ff) such that A and\Omega\Gamma ff) continuously reduce to each other. We do so by defining Borel operations which are homomorphic to the first Veblen ordinal functions of base required to compute the Wadge degree of the set A: the ordinal ff.
Thuswaldner, Generalized radix representations and dynamical systems
 II, Acta Arith
"... 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 system ..."
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Cited by 21 (11 self)
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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 wellknown notions of number systems like βexpansions 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.
Invariant percolation and harmonic Dirichlet functions
, 2004
"... The main goal of this paper is to answer question 1.10 and settle conjecture 1.11 of BenjaminiLyonsSchramm [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 per ..."
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Cited by 19 (3 self)
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The main goal of this paper is to answer question 1.10 and settle conjecture 1.11 of BenjaminiLyonsSchramm [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 RandomCluster model. We prove the existence of the nonuniqueness phase for the Bernoulli percolation (and make some progress for RandomCluster model) on unimodular transitive locally finite graphs admitting nonconstant harmonic Dirichlet functions. This is done by using the device of ℓ 2 Betti numbers.
Superrigidity and countable Borel equivalence relations
 Annals Pure Appl. Logic
"... Introduction. These notes are based upon a daylong lecture workshop presented by Simon ..."
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Cited by 14 (6 self)
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Introduction. These notes are based upon a daylong lecture workshop presented by Simon
Located Sets And Reverse Mathematics
 Journal of Symbolic Logic
, 1999
"... 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 loca ..."
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Cited by 13 (5 self)
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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.
A hierarchy of Deterministic Contextfree ωlanguages
"... Twenty years ago, Klaus. W. Wagner came up with a hierarchy of ωregular sets that actually bears his name. It turned out to be exactly the Wadge hierarchy of the sets of ωwords recognized by Deterministic Finite Automata. We describe the Wadge hierarchy of contextfree ωlanguages, which stands as ..."
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Cited by 13 (6 self)
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Twenty years ago, Klaus. W. Wagner came up with a hierarchy of ωregular sets that actually bears his name. It turned out to be exactly the Wadge hierarchy of the sets of ωwords recognized by Deterministic Finite Automata. We describe the Wadge hierarchy of contextfree ωlanguages, which stands as an extension of Wagner's work from Automata to Pushdown Automata.
An Abstract MonteCarlo Method for the Analysis of Probabilistic Programs
, 2001
"... We introduce a new method, combination of random testing and abstract interpretation, for the analysis of programs featuring both probabilistic and nonprobabilistic nondeterminism. After introducing "ordinary" testing, we show how to combine testing and abstract interpretation and give formulas l ..."
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Cited by 13 (3 self)
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We introduce a new method, combination of random testing and abstract interpretation, for the analysis of programs featuring both probabilistic and nonprobabilistic 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.
P ̸= NP∩coNP for infinite time turing machines
 Journal of Logic and Computation
, 2005
"... 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 ∩coNP. Furthermore, NP ∩coNP is exactly the class of hyperarithmetic sets. For the more general classes, we establish that P ..."
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Cited by 10 (3 self)
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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 ∩coNP. Furthermore, NP ∩coNP is exactly the class of hyperarithmetic sets. For the more general classes, we establish that P + = NP + ∩coNP + = NP ∩coNP, though P ++ is properly contained in NP ++ ∩coNP ++. Within any contiguous block of infinite clockable ordinals, we show that Pα ̸ = NPα ∩coNPα, but if β begins a gap in the clockable ordinals, then Pβ = NPβ ∩coNPβ. Finally, we establish that P f ̸ = NP f ∩coNP f for most functions f: R → ord, although we provide examples where P f = NP f ∩coNP f and P f ̸ = NP f.
COMPACT GROUP AUTOMORPHISMS, ADDITION FORMULAS AND FUGLEDEKADISON DETERMINANTS
"... 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 Fu ..."
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Cited by 10 (10 self)
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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 FugledeKadison determinant of f. For the proof, we establish an ℓ pversion of Rufus Bowen’s definition of topological entropy, addition formulas for group extensions of countable amenable group actions, and an approximation formula for the FugledeKadison determinant of f in terms of the determinants of perturbations of the compressions of f. 1.