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FugledeKadison determinants and entropy for actions of discrete amenable groups
 J. Amer. Math. Soc
"... Consider a discrete group Γ and an element f in the integral group ring ZΓ. Then Γ acts from the left on the discrete additive group ZΓ/ZΓf by automorphisms of groups. Dualizing, we obtain a left Γaction on the compact Pontrjagin dual group Xf = ZΓ/ZΓf by continuous automorphisms of groups. By de ..."
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Cited by 34 (6 self)
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Consider a discrete group Γ and an element f in the integral group ring ZΓ. Then Γ acts from the left on the discrete additive group ZΓ/ZΓf by automorphisms of groups. Dualizing, we obtain a left Γaction on the compact Pontrjagin dual group Xf = ZΓ/ZΓf by continuous automorphisms of groups. By definition, Xf is a
The Entropy Theory of Symbolic Extensions
, 2002
"... Fix a topological system (X; T ), with its space K(X;T ) of T  invariant Borel probabilities. If (Y; S) is a symbolic system (subshift) and ' : (Y; S) ! (X; T ) is a topological extension (factor map), then the function ext on K(X;T ) which assigns to each the maximal entropy of a measur ..."
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Cited by 30 (4 self)
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Fix a topological system (X; T ), with its space K(X;T ) of T  invariant Borel probabilities. If (Y; S) is a symbolic system (subshift) and ' : (Y; S) ! (X; T ) is a topological extension (factor map), then the function ext on K(X;T ) which assigns to each the maximal entropy of a measure on Y mapping to is called the extension entropy function of '. The in mum of such functions over all symbolic extensions is called the symbolic extension entropy function and is denoted by hsex . In this paper we completely characterize these functions in terms of functional analytic properties of an entropy structure on (X; T ). The entropy structure H is a sequence of entropy functions h k de ned with respect to a re ning sequence of partitions of X (or of X Z, for some auxiliary system (Z; R) with simple dynamics) whose boundaries have measure zero for all the invariant Borel probabilities.
Symbolic extensions in smooth dynamical systems
 Inventiones Math. 2005
"... Let f: X → X be a homeomorphism of the compact metric space X. A symbolic extension of (f, X) is a subshift on a finite alphabet (g, Y) which has f as a topological factor. We show that a generic C1 nonhyperbolic (i.e.,nonAnosov) area preserving diffeomorphism of a compact surface has no symbolic ..."
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Cited by 30 (4 self)
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Let f: X → X be a homeomorphism of the compact metric space X. A symbolic extension of (f, X) is a subshift on a finite alphabet (g, Y) which has f as a topological factor. We show that a generic C1 nonhyperbolic (i.e.,nonAnosov) area preserving diffeomorphism of a compact surface has no symbolic extensions. For r> 1, we exhibit a residual subset R of an open set U of Cr diffeomorphisms of a compact surface such that if f ∈ R, then any possible symbolic extension has topological entropy strictly larger than that of f. These results complement the known fact that any C ∞ diffeomorphism has symbolic extensions with the same entropy. We also show that Cr generically on surfaces, homoclinic closures which contain tangencies of stable and unstable manifolds have Hausdorff dimension two. 1
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 19 (13 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.
Measurable chromatic and independence numbers for ergodic graphs and group actions
, 2010
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Moduli space of Brody curves, energy and mean dimension
"... Abstract. We study the mean dimension of the moduli space of Brody curves. We introduce the notion of “mean energy ” and show that this can be used to estimate the mean dimension. 1. Main results 1.1. Moduli space of Brody curves. M. Gromov introduced a remarkable notion of mean dimension in [6] (se ..."
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Cited by 14 (7 self)
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Abstract. We study the mean dimension of the moduli space of Brody curves. We introduce the notion of “mean energy ” and show that this can be used to estimate the mean dimension. 1. Main results 1.1. Moduli space of Brody curves. M. Gromov introduced a remarkable notion of mean dimension in [6] (see also LindenstraussWeiss [8] and Lindenstrauss [7]). In this paper we study the mean dimension of the moduli space of Brody curves. We introduce the notion of mean energy of Brody curves and study the relation between mean energy and mean dimension. Mean energy is, in some sense, an infinite dimensional version of characteristic number, and our approach is an attempt to attack an infinite dimensional index problem. Let CP N be the complex projective space and [z0: z1: · · · : zN] be the homogeneous coordinate in CP N. We define the FubiniStudy metric form ωFS on CP N by (1) ωFS:= −1
Turbulence, representations, and tracepreserving actions
 Proc. Lond. Math. Soc
, 2010
"... Abstract. We establish criteria for turbulence in certain spaces of C ∗algebra representations and apply this to the problem of nonclassifiability by countable structures for group actions on a standard atomless probability space (X, µ) and on the hyperfinite II1 factor R. We also prove that the co ..."
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Cited by 14 (1 self)
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Abstract. We establish criteria for turbulence in certain spaces of C ∗algebra representations and apply this to the problem of nonclassifiability by countable structures for group actions on a standard atomless probability space (X, µ) and on the hyperfinite II1 factor R. We also prove that the conjugacy action on the space of free actions of a countably infinite amenable group on R is turbulent, and that the conjugacy action on the space of ergodic measurepreserving flows on (X, µ) is generically turbulent. 1.
HOMOCLINIC GROUP, IE GROUP, AND EXPANSIVE ALGEBRAIC ACTIONS
"... Abstract. We give algebraic characterizations for expansiveness of algebraic actions of countable groups. The notion of pexpansiveness is introduced for algebraic actions, and we show that for countable amenable groups, a finitely presented algebraic action is 1expansive exactly when it has finite ..."
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Cited by 11 (7 self)
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Abstract. We give algebraic characterizations for expansiveness of algebraic actions of countable groups. The notion of pexpansiveness is introduced for algebraic actions, and we show that for countable amenable groups, a finitely presented algebraic action is 1expansive exactly when it has finite entropy. We also study the local entropy theory for actions of countable amenable groups on compact groups by automorphisms, and show that the IE group determines the Pinsker factor for such actions. For an expansive algebraic action of a polycyclicbyfinite group on X, it is shown that the entropy of the action is equal to the entropy of the induced action on the Pontryagin dual of the homoclinic group, the homoclinic group is a dense subgroup of the IE group, the homoclinic group is nontrivial exactly when the action has positive entropy, and the homoclinic group is dense in X exactly when the action has completely positive entropy. 1.
SOFICITY, AMENABILITY, AND DYNAMICAL ENTROPY
"... Abstract. In a previous paper the authors developed an operatoralgebraic approach to Lewis Bowen’s sofic measure entropy that yields invariants for actions of countable sofic groups by homeomorphisms on a compact metrizable space and by measurepreserving transformations on a standard probability s ..."
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Cited by 10 (3 self)
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Abstract. In a previous paper the authors developed an operatoralgebraic approach to Lewis Bowen’s sofic measure entropy that yields invariants for actions of countable sofic groups by homeomorphisms on a compact metrizable space and by measurepreserving transformations on a standard probability space. We show here that these measure and topological entropy invariants both coincide with their classical counterparts when the acting group is amenable. 1.
Rokhlin dimension and C∗dynamics
, 2012
"... Abstract. We develop the concept of Rokhlin dimension for integer and for finite group actions on C∗algebras. Our notion generalizes the socalled Rokhlin property, which can be thought of as Rokhlin dimension 0. We show that finite Rokhlin dimension is prevalent and appears in cases in which the ..."
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Cited by 8 (1 self)
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Abstract. We develop the concept of Rokhlin dimension for integer and for finite group actions on C∗algebras. Our notion generalizes the socalled Rokhlin property, which can be thought of as Rokhlin dimension 0. We show that finite Rokhlin dimension is prevalent and appears in cases in which the Rokhlin property cannot be expected: the property of having finite Rokhlin dimension is generic for automorphisms of Zstable C∗algebras, where Z denotes the JiangSu algebra. Moreover, crossed products by automorphisms with finite Rokhlin dimension preserve the property of having finite nuclear dimension, and under a mild additional hypothesis also preserve Zstability. Finally, we prove a topological version of the classical Rokhlin lemma: automorphisms arising from minimal homeomorphisms of finite dimensional compact metrizable spaces always have finite Rokhlin dimension. An action of a countable discrete group G on a (possibly noncommutative) space X has a Rokhlin property if there are systems of subsets of X, indexed by large