7,597 research outputs found
Symmetries and reversing symmetries of polynomial automorphisms of the plane
The polynomial automorphisms of the affine plane over a field K form a group
which has the structure of an amalgamated free product. This well-known
algebraic structure can be used to determine some key results about the
symmetry and reversing symmetry groups of a given polynomial automorphism.Comment: 27 pages, AMS-Late
The structure of reversing symmetry groups
We present some of the group theoretic properties of reversing symmetry
groups, and classify their structure in simple cases that occur frequently in
several well-known groups of dynamical systems.Comment: 12 page
Linear degree growth in lattice equations
We conjecture recurrence relations satisfied by the degrees of some
linearizable lattice equations. This helps to prove linear growth of these
equations. We then use these recurrences to search for lattice equations that
have linear growth and hence are linearizable
Periodic orbits of linear endomorphisms on the 2-torus and its lattices
Counting periodic orbits of endomorphisms on the 2-torus is considered, with
special focus on the relation between global and local aspects and between the
dynamical zeta function on the torus and its analogue on finite lattices. The
situation on the lattices, up to local conjugacy, is completely determined by
the determinant, the trace and a third invariant of the matrix defining the
toral endomorphism.Comment: 22 page
Reversing and extended symmetries of shift spaces
The reversing symmetry group is considered in the setting of symbolic dynamics. While this group is generally too big to be analysed in detail, there are interesting cases with some form of rigidity where one can determine all symmetries and reversing symmetries explicitly. They include Sturmian shifts as well as classic examples such as the Thue–Morse system with various generalisations or the Rudin–Shapiro system. We also look at generalisations of the reversing symmetry group to higher-dimensional shift spaces, then called the group of extended symmetries. We develop their basic theory for faithful ℤd-actions, and determine the extended symmetry group of the chair tiling shift, which can be described as a model set, and of Ledrappier’s shift, which is an example of algebraic origin
Symmetries and reversing symmetries of toral automorphisms
Toral automorphisms, represented by unimodular integer matrices, are
investigated with respect to their symmetries and reversing symmetries. We
characterize the symmetry groups of GL(n,Z) matrices with simple spectrum
through their connection with unit groups in orders of algebraic number fields.
For the question of reversibility, we derive necessary conditions in terms of
the characteristic polynomial and the polynomial invariants. We also briefly
discuss extensions to (reversing) symmetries within affine transformations, to
PGL(n,Z) matrices, and to the more general setting of integer matrices beyond
the unimodular ones.Comment: 34 page
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