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fundamental cocycles and fundamental groups of symbolic Z d actions, Ergodic Theory and Dynamical Systems 18
, 1998
"... Abstract. We prove that certain topologically mixing twodimensional shifts of finite type have a ‘fundamental ’ 1cocycle with the property that every continuous 1cocycle on the shift space with values in a discrete group is continuously cohomologous to a homomorphic image of the fundamental cocyc ..."
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Cited by 17 (4 self)
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Abstract. We prove that certain topologically mixing twodimensional shifts of finite type have a ‘fundamental ’ 1cocycle with the property that every continuous 1cocycle on the shift space with values in a discrete group is continuously cohomologous to a homomorphic image of the fundamental cocycle. These fundamental cocycles are closely connected with representations of the shift space by Wang tilings and the tiling groups of J.H. Conway, J.C. Lagarias and W. Thurston, and they determine the projective fundamental groups of the shift spaces introduced
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.
Adelic amoebas disjoint from open halfpsaces
, 2007
"... Abstract. We show that a conjecture of Einsiedler, Kapranov, and Lind on adelic amoebas of subvarieties of tori and their intersections with open halfspaces of complementary dimension is false for subvarieties of codimension greater than one that have degenerate projections to smaller dimensional to ..."
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Cited by 1 (1 self)
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Abstract. We show that a conjecture of Einsiedler, Kapranov, and Lind on adelic amoebas of subvarieties of tori and their intersections with open halfspaces of complementary dimension is false for subvarieties of codimension greater than one that have degenerate projections to smaller dimensional tori. We prove a suitably modified version of the conjecture using algebraic methods, functoriality of tropicalization, and a theorem of Zhang on torsion points in subvarieties of tori. 1.
ISOMORPHISM RIGIDITY IN ENTROPY RANK TWO
, 2003
"... Abstract. We study the rigidity properties of a class of algebraic Z 3actions with entropy rank two. For this class, conditions are found which force an invariant measure to be the Haar measure on an affine subset. This is applied to show isomorphism rigidity for such actions, and to provide exampl ..."
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Abstract. We study the rigidity properties of a class of algebraic Z 3actions with entropy rank two. For this class, conditions are found which force an invariant measure to be the Haar measure on an affine subset. This is applied to show isomorphism rigidity for such actions, and to provide examples of nonisomorphic Z 3actions with all their Z 2subactions isomorphic. The proofs use lexicographic halfspace entropies and total ergodicity along critical directions. 1.