We study diffeomorphisms that have one-parameter families of continuous symmetries. For general maps, in contrast to the symplectic case, existence of a symmetry no longer implies existence of an invariant. Conversely, a map with an invariant need not have a symmetry. We show that when a symmetry flow has a global Poincar\'{e} section there are coordinates in which the map takes a reduced, skew-product form, and hence allows for reduction of dimensionality. We show that the reduction of a volume-preserving map again is volume preserving. Finally we sharpen the Noether theorem for symplectic maps. A number of illustrative examples are discussed and the method is compared with traditional reduction techniques.
Deep Dive into Symmetry Reduction by Lifting for Maps.
We study diffeomorphisms that have one-parameter families of continuous symmetries. For general maps, in contrast to the symplectic case, existence of a symmetry no longer implies existence of an invariant. Conversely, a map with an invariant need not have a symmetry. We show that when a symmetry flow has a global Poincar'{e} section there are coordinates in which the map takes a reduced, skew-product form, and hence allows for reduction of dimensionality. We show that the reduction of a volume-preserving map again is volume preserving. Finally we sharpen the Noether theorem for symplectic maps. A number of illustrative examples are discussed and the method is compared with traditional reduction techniques.
A symmetry group of a dynamical system may be discrete or continuous. The existence of discrete symmetries implies the existence of related sets of orbits and imposes constraints on bifurcations [CL00,GS02]. Continuous symmetry often results in reduction; for example, the classical results of Sophus Lie are concerned with the reduction of order of an ODE or PDE with symmetry [Olv93]. In the Hamiltonian or Lagrangian context, a continuous symmetry implies, through Noether's theorem, the existence of an invariant, and the reduction of the dynamics by two dimensions [MR99, MMO + 07, HSS09].
In this paper, we are are interested in global reduction theory for maps that have continuous symmetries. A continuous symmetry of a map f : M → M is a vector field whose flow commutes with the map. The set of symmetries forms a Lie algebra. Symmetry reduction for maps seems to have been first studied by Maeda [Mae80,Mae87], who showed that a map with an s-dimensional, Abelian Lie algebra of symmetries can be written locally in a skew-product form
where σ ∈ R n-s and τ ∈ R s , in the neighborhood of any point where the rank of the symmetry group is s.
We will show in §2 that if the symmetry flow has a global Poincaré section Σ that is relatively closed in a manifold M (that is not necessarily compact), then we can find a covering map of the form p : Σ × R → M . If some topological conditions are satisfied, then the map f has a lift F to Σ × R with the skew-product form (1), see Thm. 3. The idea relies on the fact that the topologies of Σ and M may constitute an obstruction for this lift to exist. We call this procedure reduction by lifting; it is a global version of Maeda’s result.
This global reduction procedure is distinct from the general local reduction method due to Palais [Pal61]. He showed that when an s-dimensional symmetry group has a proper action on M , then there is a codimension-s local neighborhood of each point, a Palais slice, such that the group orbit of a slice forms a tube within which the orbit trivializes: there are generalized “flow-box” coordinates. In this paper, we are primarily concerned with one-parameter symmetry groups, though our results can be extended to the case of multidimensional, Abelian groups. Moreover, we will show that the skew-product (1) can be globally valid on M , or at least on an open, dense subset.
In the classical theory of flows with Lie symmetry groups, a flow on a manifold M with symmetry group G that acts properly on M , can be reduced to a flow on the space of group orbits, M/G. When the action of G is free, then the group orbit space is a manifold; alternatively M/G is an “orbifold” and the reduction has singularities.
By contrast, in our reduction procedure, there is no need to assume that the action of the symmetry group is proper or free; it thus circumvents some of the problems with singular reduction. A number of examples are given in §3.
One motivation of our study is to extend some results from the setting of Hamiltonian vector fields or symplectic maps to the setting of divergence-free vector fields or volume-preserving maps. Indeed, whenever a dynamical system falls into a particular structural class (symplectic, volume preserving,…), the question arises whether the reduction by appropriate symmetries can be performed so that the structure is preserved.
We will show in §4 that when reduction by lifting is applied to volume-preserving maps, then the reduced map k of (1) is also volume preserving with respect to a natural volume form on Σ. In particular, in the three-dimensional, volume-preserving case, the reduced map k : Σ → Σ is symplectic.
As we recall in §5, Noether’s theorem implies that whenever a Hamiltonian flow has a Hamiltonian symmetry there is an invariant, and conversely, that every invariant generates a Hamiltonian vector field that is a symmetry. This result also holds for symplectic twist maps with a Lagrangian generating function [Log73,Mae81,WM97,Man06].
We will generalize a result of Bazzani [Baz88] to show that, with the addition of a recurrence condition, Noether’s theorem also applies more generally to symplectic maps. However, it is important to note that symmetries do not generally lead to invariants nor do invariants necessarily give rise to symmetries.
In §6 we compare our procedure with two standard reduction procedures, using as an example a four-dimensional symplectic map with rotational symmetry. We discuss the advantages and shortcomings of reduction by lifting.
In this section we investigate the conditions under which a map or flow that has a symmetry can be written in the skew-product form (1). Whenever this is possible, the process reduces the effective dimension of the dynamics by one. To accomplish this reduction, we suppose that the symmetry vector field generates a complete flow that admits a global Poincaré section. We recall below the relationship between the existence of cross sections, covering spaces and fu
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