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We show that a full solenoid is locally the product of a euclidean component and />-adic components for each rational prime p. An automorphism of a solenoid preserves these components, and its topological entropy is shown to be the sum of the euclidean and p-adic contributions. The p-adic entropy of the corresponding rational matrix is computed using its(More)
We study the non-archimedean counterpart to the complex amoeba of an algebraic variety, and show that it coincides with a polyhe-dral set defined by Bieri and Groves using valuations. For hypersurfaces this set is also the tropical variety of the defining polynomial. Using non-archimedean analysis and a recent result of Conrad we prove that the amoeba of an(More)
Let α be an action of Z d by continuous automorphisms of a compact abelian group X. A point x in X is called homoclinic for α if α n x → 0X as n → ∞. We study the set ∆α(X) of homoclinic points for α, which is a subgroup of X. If α is expansive then ∆α(X) is at most countable. Our main results are that if α is expansive, then (1) ∆α(x) is nontrivial if and(More)
A general framework for investigating topological actions of Z d on compact metric spaces was proposed by Boyle and Lind in terms of expansive behavior along lower-dimensional subspaces of R d. Here we completely describe this expansive behavior for the class of algebraic Z d-actions given by commuting automorphisms of compact abelian groups. The(More)
We give an overview of the field of symbolic dynamics: its history, applications and basic definitions and examples. The field of symbolic dynamics evolved as a tool for analyzing general dynamical systems by discretizing space. Imagine a point following some trajectory in a space. Partition the space into finitely many pieces, each labeled by a different(More)
We introduce a new type of invariant of block isomorphism for Markov shifts, defined by summing the weights of all spanning trees for a presentation of the Markov shift. We give two proofs of invariance. The first uses the Matrix-Tree Theorem to show that this invariant can be computed from a known invariant, the stochastic zeta function of the shift. The(More)
We formulate Lehmer's Problem about the Mahler measure of polynomials for general compact abelian groups, introducing a Lehmer constant for each such group. We show that all nontrivial connected compact groups have the same Lehmer constant, and conjecture the value of the Lehmer constant for finite cyclic groups. We also show that if a group has infinitely(More)