Symmetry of attractors and the Perron-Frobenius operator

We study the stochastic aspects of dynamics for equivariant dynamical systems, in particular those aspects that relate to symmetric chaos. By now, symmetric chaos is a rich field with both numerical evidence and theoretical classification of admissible symmetry types of chaotic attractors and their bifurcations. In this paper, we present a novel framework based upon the Perron-Frobenius (P-F) formalism to study these questions. We define appropriate notions of symmetry for basic objects relevant to the study of stochastic aspects of equivariant dynamics. We use these definitions to show that the resulting P-F operator commutes with the action of the symmetry group. We apply group representation theory to analyse both the algebraic and the stochastic implications of this commutativity. We use the P-F formalism to re-derive the necessary conditions for admissible symmetry types of attractors and for admissible symmetry types for the bifurcation of attractors in the presence of reflections. In particular, we show that these conditions are an easy consequence of the commutativity property of the P-F operator.