Renormalization group dynamics in the plane of the coupling constants of the fermionic hierarchical model

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 with coupling constants $r$ and $g$. The action of the renormalization group transformation in the model is reduced to the rational map in the plane of coupling constants $(r,g)$. The upper half-plane $\{(r,g): g > 0\}$ and the lower half-plane are invariant under the renormalization group transformation We investigate the dynamics of this map in the lower and upper half-planes for different values of renormalization group parameter. To describe the global picture of the renormalization group flow we use also the space of the coefficients of the expansion of free measure density wich is denoted as the c-space. The renormalization group action in c-space is given as a homogeneous quadratic map. This space is treated as a two-dimensional projective space and is visualized as a unit disk. If the renormalization group parameter is greater than the dimension of the lattice, then the only attracting fixed point of the renormalization group transformation is defined by the density of the Grassmann delta-function . We describe two different (left and right) invariant neighborhoods of this fixed point and classify the points on the plane according to the way they tend to this fixed point (from the left or from the right). We describe explicitly the zone structure of the classified domains and show that the global renormalization group flow has a nice description in terms of this zone structure. We discuss also the global behavior of all RG-invariant curves.