Dynamical properties of renormalization group flow in projective space representation
We consider four-component fermionic (Grassmann-valued) field on the hierarchical lattice. The Gaussian part of the Hamiltonian in the model is invariant under the block-spin renormalization group transformation with given degree of normalization factor ( renormalization group parameter). The non-Gaussian part of the Hamiltonian is given by the self-interaction forms of the 2-nd and 4-th order. Fermionic hierarchical model has natural continuous version and can be described in terms of the $p$-adic quantum field theory approach and Wilson's renormalization group. We investigate dynamics of the renormalization group in the space of the coefficients determining the Grassmann-valued free measure density. This space is treated as two-dimensional projective space. The action of the renormalization group (RG) transformation in this projective space is reduced to the quadratic homogeneous mapping. If the RG parameter is more then 1, then the unique attractive fixed point of RG transformation is given by the density of the Grassmann delta-function. We describe four different RG-invariant subsets in the small neighborhood of delta-function fixed point . Almost all points of the projective plane tend to the $\delta$-function under the iterations of RG mapping through only one of these subsets. Using the results of this classification we give explicit description of the global RG-flow in the whole projective plane. The exact solution of the fermionic hierarchical model generates non-trivial conjectures for the different fermionic and bosonic hierarchical and Euclidean field models.