Results 1  10
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31
Nonarchimedean amoebas and tropical varieties
, 2004
"... We study the nonarchimedean counterpart to the complex amoeba of an algebraic variety, and show that it coincides with a polyhedral set defined by Bieri and Groves using valuations. For hypersurfaces this set is also the tropical variety of the defining polynomial. Using nonarchimedean analysis a ..."
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Cited by 79 (0 self)
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We study the nonarchimedean counterpart to the complex amoeba of an algebraic variety, and show that it coincides with a polyhedral set defined by Bieri and Groves using valuations. For hypersurfaces this set is also the tropical variety of the defining polynomial. Using nonarchimedean analysis and a recent result of Conrad we prove that the amoeba of an irreducible variety is connected. We introduce the notion of an adelic amoeba for varieties over global fields, and establish a form of the localglobal principle for them. This principle is used to explain the calculation of the nonexpansive set for a related dynamical system.
Zeta functions for elements of entropy rank one actions
, 2006
"... An algebraic Z daction of entropy rank one is one for which each element has finite entropy. Using the structure theory of these actions due to Einsiedler and Lind, this paper investigates dynamical zeta functions for elements of the action. An explicit periodic point formula is obtained leading t ..."
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Cited by 13 (8 self)
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An algebraic Z daction of entropy rank one is one for which each element has finite entropy. Using the structure theory of these actions due to Einsiedler and Lind, this paper investigates dynamical zeta functions for elements of the action. An explicit periodic point formula is obtained leading to a uniform parameterization of the zeta functions that arise in expansive components of an expansive action, together with necessary and sufficient conditions for rationality in a more general setting.
Augmenting dimension group invariants for substitution dynamics
"... We present new invariants for substitutional dynamical systems. Our main contribution is a flow invariant which is strictly finer than, but related and akin to, the dimension groups of Herman, Putnam and Skau. We present this group as a stationary inductive limit of a system associated to an intege ..."
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Cited by 8 (5 self)
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We present new invariants for substitutional dynamical systems. Our main contribution is a flow invariant which is strictly finer than, but related and akin to, the dimension groups of Herman, Putnam and Skau. We present this group as a stationary inductive limit of a system associated to an integer matrix defined from combinatorial data based on the class of special words of the dynamical system.
Entropy geometry and disjointness for zerodimensional algebraic actions
 J. Reine Angew. Math
, 2005
"... Abstract. We show that many algebraic actions of higherrank abelian groups on zerodimensional groups are mutually disjoint. The proofs exploit differences in the entropy geometry arising from subdynamics and a form of Abramov–Rokhlin formula for halfspace entropies. We discuss some mutual disjoint ..."
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Cited by 7 (2 self)
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Abstract. We show that many algebraic actions of higherrank abelian groups on zerodimensional groups are mutually disjoint. The proofs exploit differences in the entropy geometry arising from subdynamics and a form of Abramov–Rokhlin formula for halfspace entropies. We discuss some mutual disjointness properties of algebraic actions of higherrank abelian groups on zerodimensional groups. The tools used are a version of the halfspace entropies introduced by Kitchens and Schmidt [14] and adapted by Einsiedler [7], a basic geometric entropy formula from [7], and the structure of expansive subdynamics for algebraic Z dactions due to Einsiedler, Lind, Miles and Ward [9]. We show that any collection of algebraic Z dactions on zerodimensional groups with entropy rank or corank one that look sufficiently different are mutually disjoint. The main results are the following (here N(·) denotes the set of nonexpansive directions defined in Section 1). Theorem 5.1. Let X1,...,Xn be a collection of irreducible algebraic zerodimensional Z dactions, all with entropy rank one. If N(αj) \ ⋃ k>j N(αk) ̸ = ∅ for j = 1,..., n then the systems are mutually disjoint. The simplest illustration of Theorem 5.1 is the fact that Ledrappier’s Example 2.3 and its mirror image are disjoint. This is shown directly in Section 3 to illustrate how the Abramov–Rokhlin formula for halfspace entropies may be used. Theorem 6.2. Let Y and Z be prime Z dactions with entropy corank one. If N(αY) ̸ = N(αZ), then Y and Z are disjoint.
Arithmetic Dynamical Systems
, 2000
"... The main objects of study in this thesis are Z d actions by automorphisms of compact abelian groups, which arise in a natural arithmetic setting. In particular, to a countable integral domain D and units 1 ; : : : ; d 2 D we associate a Z d action by automorphisms of the compact abelian ..."
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Cited by 7 (5 self)
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The main objects of study in this thesis are Z d actions by automorphisms of compact abelian groups, which arise in a natural arithmetic setting. In particular, to a countable integral domain D and units 1 ; : : : ; d 2 D we associate a Z d action by automorphisms of the compact abelian group b D. This generalises the `Sinteger dynamical systems' introduced by Chothi, Everest and Ward, where d = 1 and D is a ring of Sintegers in an A field. Familiar dynamical properties such as expansiveness and entropy are investigated in this setting, together with the emerging theory of expansive subdynamics introduced by Boyle and Lind. Homoclinic points are also examined. The main results are as follows. 1. Using results of Lind, Schmidt and Ward, an explicit entropy formula is given which applies whenever D is an integrally closed domain (Theorems 3.3.4 and 3.3.8). 2. The wellknown expansiveness criteria for toral automorphisms, involving the eigenvalues of associated integer...
Symbolic and Algebraic Dynamical Systems
, 2000
"... This article gives a brief survey of a class of dynamical systems which contains shifts of finite type (both one and multidimensional), as well as actions of one or more commuting automorphisms of a compact abelian group. Here we use the term `dimension' to refer to the rank of the parameter group ..."
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Cited by 7 (0 self)
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This article gives a brief survey of a class of dynamical systems which contains shifts of finite type (both one and multidimensional), as well as actions of one or more commuting automorphisms of a compact abelian group. Here we use the term `dimension' to refer to the rank of the parameter group for the action, not to the space itself. The connection between classical (onedimensional) shifts of finite type and automorphisms of compact abelian groups via Markov partitions is classical and well understood (cf. [2], [5], [61], and [40, Chap. 6]). A perhaps less classical connection between the two was pointed out in [33] and [57], where expansive automorphisms of compact abelian groups (and, more generally, expansive Z
Periodic point data detects subdynamics in entropy rank one
, 2006
"... A framework for understanding the geometry of continuous actions of Z d was developed by Boyle and Lind using the notion of expansive behavior along lowerdimensional subspaces. For algebraic Z dactions of entropy rank one, the expansive subdynamics is readily described in terms of Lyapunov exponen ..."
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Cited by 6 (5 self)
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A framework for understanding the geometry of continuous actions of Z d was developed by Boyle and Lind using the notion of expansive behavior along lowerdimensional subspaces. For algebraic Z dactions of entropy rank one, the expansive subdynamics is readily described in terms of Lyapunov exponents. Here we show that periodic point counts for elements of an entropy rank one action determine the expansive subdynamics. Moreover, the finer structure of the nonexpansive set is visible in the topological and smooth structure of a set of functions associated to the periodic point data.
Expansive algebraic actions of countable abelian groups
 Monatsh. Math
"... This paper gives an algebraic characterization of expansive actions of countable abelian groups on compact abelian groups. This naturally extends the classification of expansive algebraic Z dactions given by Schmidt using complex varieties. Also included is an application to a natural class of exam ..."
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Cited by 5 (4 self)
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This paper gives an algebraic characterization of expansive actions of countable abelian groups on compact abelian groups. This naturally extends the classification of expansive algebraic Z dactions given by Schmidt using complex varieties. Also included is an application to a natural class of examples arising from unit subgroups of integral domains.
Symbolic dynamics, partial dynamical systems, boolean algebras and C ∗ algebras generated by partial isometries, preprint Univ
, 2004
"... Abstract. We associate to each discrete partial dynamical system a universal C ∗algebra generated by partial isometries satisfying relations given by a Boolean algebra connected to the discrete partial dynamical system in question. We show that for symbolic dynamical systems like onesided and two ..."
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Cited by 4 (0 self)
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Abstract. We associate to each discrete partial dynamical system a universal C ∗algebra generated by partial isometries satisfying relations given by a Boolean algebra connected to the discrete partial dynamical system in question. We show that for symbolic dynamical systems like onesided and twosided shift spaces and topological Markov chains with an arbitrary state space the C ∗algebras usually associated to them, can be obtained in this way. As a consequence of this, we will be able to show that for twosided shift spaces having a certain property, the crossed product of the twosided shift space is a quotient of the C ∗algebra associated to the corresponding onesided shift space. 1.