Uniform periodic point growth in entropy rank one

  title={Uniform periodic point growth in entropy rank one},
  author={Richard Miles and Thomas B. Ward},
  journal={arXiv: Dynamical Systems},
  • R. MilesT. Ward
  • Published 18 September 2006
  • Computer Science, Mathematics
  • arXiv: Dynamical Systems
We show that algebraic dynamical systems with entropy rank one have uniformly exponentially many periodic points in all directions. 

Figures from this paper

Directional uniformities, periodic points, and entropy

Dynamical systems generated by $d\ge2$ commuting homeomorphisms (topological $\mathbb{Z}^d$-actions) contain within them structures on many scales, and in particular contain many actions of

Deposited in DRO : 06 October 2015 Version of attached le : Other Peer-review status of attached

Dynamical systems generated by d > 2 commuting homeomorphisms (topological Z-actions) contain within them structures on many scales, and in particular contain many actions of Z for 1 6 k 6 d.

Directional zeta functions: Final report

  • Mathematics
  • 2008
Directional zeta functions detect subdynamics: The main part of this project (Project 1 in the proposal) concerned the relationship between growth properties of periodic points and the directional

Synchronization points and associated dynamical invariants

This paper introduces new invariants for multiparameter dynamical systems. This is done by counting the number of points whose orbits intersect at time n under simultaneous iteration of finitely

A Note on the Growth of Periodic Points for Commuting Toral Automorphisms

In this note we study the growth of the number of periodic points for non-degenerate actions of commuting hyperbolic toral automorophisms.

A directional uniformity of periodic point distribution and mixing

For mixing [\mathbb Z^d] -actions generated by commuting automorphisms of a compact abelian group, we investigate the directional uniformity of the rate of periodic point distribution and mixing.



Periodic point data detects subdynamics in entropy rank one

A framework for understanding the geometry of continuous actions of $\mathbb Z^d$ was developed by Boyle and Lind using the notion of expansive behaviour along lower-dimensional subspaces. For


We show C∞ local rigidity for Z (k ≥ 2) partially hyperbolic actions by toral automorphisms using a generalization of the KAM (KolmogorovArnold-Moser) iterative scheme. We also prove the existence of

Zeta functions for elements of entropy rank-one actions

  • R. Miles
  • Mathematics
    Ergodic Theory and Dynamical Systems
  • 2007
An algebraic $\mathbb{Z}^d$-action 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

Dynamical properties of quasihyperbolic toral automorphisms

  • D. Lind
  • Mathematics
    Ergodic Theory and Dynamical Systems
  • 1982
Abstract We study the dynamical properties of ergodic toral autmorphisms that have some eigenvalues of modulus one. For such automorphisms, all sufficiently fine smooth partitions generate

Dynamical Systems of Algebraic Origin

This chapter discusses group actions by automorphisms fo compact groups, which are actions on compact abelian groups and the consequences of these actions on entropy.

Orbit-counting in non-hyperbolic dynamical systems

Abstract There are well-known analogues of the prime number theorem and Mertens' Theorem for dynamical systems with hyperbolic behaviour. Here we consider the same question for the simplest

Algebraic ℤ^{}-actions of entropy rank one

The measure entropy of a class of skew products is computed, where the fiber maps are elements from an algebraic Z d -action of entropy rank one, and this leads to a formula for the topological entropy of continuous skew products as the maximum of a finite number of topological pressures.

S-integer dynamical systems: periodic points.

We associate via duality a dynamical system to each pair (RS,x), where RS is the ring of S-integers in an A-field k, and x is an element of RS\{0}. These dynamical systems include the circle doubling

An Uncountable Family of Group Automorphisms, and a Typical Member

We describe an uncountable family of compact group automorphisms with entropy log 2. Each member of the family has a distinct dynamical zeta function, and the members are parametrised by a

Periodic points of endomorphisms on solenoids and related groups

This paper investigates the problem of finding the possible sequences of periodic point counts for endomorphisms of solenoids. For an ergodic epimorphism of a solenoid, a closed formula is given that