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Thomas Ward

Publications and source records attributed to Thomas Ward.

47 records · Page 3Linked to original sources
arXiv2006

Periodic Point Data Detects Subdynamics in Entropy Rank One

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 lower-dimensional subspaces. For algebraic Z^d-actions 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 non-expansive set is visible in the topological and smooth structure of a set of functions associated to the periodic point data.

math.DS↗
arXiv2003

Isomorphism rigidity in entropy rank two

We study the rigidity properties of a class of algebraic Z^3-actions 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 non-isomorphic Z^3-actions with all their Z^2-sub-actions isomorphic. The proofs use lexicographic half-space entropies and total ergodicity along critical directions.

math.DS↗
arXiv2003

Periodic points for good reduction maps on curves

The periodic points of a morphism of good reduction for a smooth projective curve with good reduction over the p-adics form a discrete set. This is used to give an interpretation of the morphic height in terms of asymptotic properties of periodic points, and a morphic analogue of Jensen's formula.

math.DS↗
arXiv2003

Group automorphisms with few and with many periodic points

For any $C\in[0,\infty]$ a compact group automorphism $T:X\to X$ is constructed with the property that $$ \frac{1}{n}\log|\{x\in X\mid T^n(x)=x\}|\longrightarrow C. $$ This may be interpreted as a combinatorial analogue of the (still open) problem of whether compact group automorphisms exist with any given topological entropy.

math.DS↗
arXiv2002

Integer sequences counting periodic points

An existing dialogue between number theory and dynamical systems is advanced. A combinatorial device gives necessary and sufficient conditions for a sequence of non-negative integers to count the periodic points in a dynamical system. This is applied to study linear recurrence sequences which count periodic points. Instances where the $p$-parts of an integer sequence themselves count periodic points are studied. The Mersenne sequence provides one example, and the denominators of the Bernoulli numbers provide another. The methods give a dynamical interpretation of many classical congruences such as Euler-Fermat for matrices, and suggest the same for the classical Kummer congruences satisfied by the Bernoulli numbers.

math.NT↗
arXiv2002

Morphic heights and periodic points

An approach to the calculation of local canonical morphic heights is described, motivated by the analogy between the classical height in Diophantine geometry and entropy in algebraic dynamics. We consider cases where the local morphic height is expressed as an integral average of the logarithmic distance to the closure of the periodic points of the underlying morphism. The results may be thought of as a kind of morphic Jensen formula.

math.NT↗
arXiv2002

Asymptotic geometry of non-mixing sequences

The exact order of mixing for zero-dimensional algebraic dynamical systems is not entirely understood. Here non-Archimedean norms in function fields of positive characteristic are used to exhibit an asymptotic shape in non-mixing sequences for algebraic $Z^2$-actions. This gives a relationship between the order of mixing and the convex hull of the defining polynomial. Using these methods, we show that an algebraic dynamical system for which any shape of cardinality three is mixing is mixing of order three, and for any $k\ge1$ exhibit examples that are $k$-fold mixing but not $(k+1)$-fold mixing.

math.DS↗
arXiv2001

Expansive subdynamics for algebraic $Z^d$-actions

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 description uses the logarithmic image of an algebraic variety together with a directional version of Noetherian modules over the ring of Laurent polynomials in several commuting variables. We introduce two notions of rank for topological $Z^d$-actions, and for algebraic $Z^d$-actions describe how they are related to each other and to Krull dimension. For a linear subspace of $R^d$ we define the group of points homoclinic to zero along the subspace, and prove that this group is constant within an expansive component.

math.DS↗
arXiv2001

Arithmetic and growth of periodic orbits

We give necessary and sufficient conditions for a sequence to be exactly realizable as the sequence of numbers of periodic points in a dynamical system. Using these conditions, we show that no non-constant polynomial is realizable, and give some conditions on realizable binary recurrence sequences. Realization in rate is always possible for sufficiently rapidly-growing sequences, and is never possible for slowly-growing sequences. Finally, we discuss the relationship between the growth rate of periodic points and the growth rate of points with specified least period.

math.DS↗
arXiv1999

Canonical heights and entropy in arithmetic dynamics

A system of transformations is associated to a rational point on an elliptic curve. The sequence entropy is connected to the canonical height, and in some cases there is a canonically defined quotient system whose entropy is the canonical height and for which the fibre entropy is accounted for by local heights at primes of bad reduction. The proofs use transcendence theory and a strong form of Siegel's theorem. We go on to extend these ideas to the morphic heights of Call and Goldstine.

math.NT↗