I—
Some Properties of the Jacobian

1. We consider a multiplicative system involving t types of particles, in which a particle of type i has a fixed probability pi(i;jl, .. jt) of producing a total of jl +... +jt particles, j, of type v, upon transformation. The corresponding generating transformation G(x) of the unit cube It:gi(x)- =-Pi (i; il,..*,jt) xⁱ '*6 xtit5

has the property that, upon iteration k times, the resulting transformation Gk(x): gi)() = Pk (i;jl,.. .,jt)x...xt has as its coefficient pk(i;j) the probability that the state (jl,...,jt) shall exist in the k-th generation of progeny from one particle of type i.

2. The transformation G(x) has for its Jacobian at x = 1 the first moment matrix J(G(1)) = [Ogi/axj = [mij][m,i j and, more generally, J(Gk(l)) = [gk)/axj] =[^mij ] where mk) is the expectation of particles of type j in the k-th generation of progeny from one particle of type i. Under our assumptions on G (see I), all these moments are positive. Moreover, we have seen that [mij] = [mij]

3. The importance of matrices with positive elements required study of their properties. We found that for such a matrix M, there is one and only one solution, r, v of the relations vM = rv, r > O, v > 0 (i.e., all vi > 0) In similar fashion one shows the existence and uniqueness of r', v' such that Mv' =r'v', r' > 0, v' > . That r = r' in these equations is evident from the following

Lemma. If vM = rv, r > 0, v > 0, and R is an arbitrary positive characteristic root of M, then r > R. Similarly, Mv' = r'v', r' > 0, v' >0, IM - RI_I=, R > 0 implies r'>R.

Proof. First statement: Let wM = Rw, w 5 0, and define b= min{wi/vi}, B = max{wi/vi}. Since all vi > O, bvi<wi <Bvi and hence, bVm(jk) <Em < BVim(k) and brkvj <RkW<BrkVj, all j. Since r > O, also bvj<(R/r)kwj<Bvj. Suppose R/r > 1. If at least one wj > 0 the right member yields a contradiction for large k. If at least one wj < 0 then the left member does. Hence all wj = 0, contradicting choice of w. Thus R/r< 1.

The second statement of the lemma is proved similarly.

We include for later use the trivial remark: If M is a matrix with positive elements, and Mv' = rv', r > O, v' > O, we have Mnv' = r"v^' for all positive integers n, and thus rnvj = m > min(vl ). Hence . m7(n) < r~v'/ min (v) -= rVi < rr max(vi) Vr", where V is a positive constant.

Similarly, if vM = ri, r > 0, v > 0, we have Ei m () < Wrn, where W is a positive constant.

4. We shall also need the following

Theorem. If M is a matrix of positive elements with Mv' = rv', r > 0, v' > 0, and Tk(v) is the transformation Mkv/s (Mkv), then limTk(v) = v' uniformly for all v $ 0, vi> 0.

Here s(w) indicates the sum E wi of the components of the vector w. The proof is entirely analogous to that used in I to prove the same result for row vectors.

5. We proved in I that lim Gk(O) = x^° exists: lin gk)(0) = limpk(i; 0)=X_i, and defined G as supercritical in case all x°< 1. Under our conditions on G, the alternative case is that all x = 1, hence x°= 1, and this case we called subcritical. Most of the present report will be devoted to systems of the latter type, in which the probability of death in generation k rises to limit one.

6. We have seen in I that a system is subcritical if and only if the maximal root r of the first moment matrix M = J(G(1)) is less than or equal to one. Equivalently G is subcritical if and only if the determinants IAnl of the upper principal minors An = [mij - Oij]₁ of the matrix A = [mij - ijl = M - I satisfy the relations: (-1)n+1lA1l < 0, n = 1,... ,t-1, (-1)t+lIM - I < .

7. We say a system is just-critical in case it is subcritical and the maximum positive characteristic root r of M is equal to one. A subcritical system with r < 1 is said to be below-critical. The justcritical case, while of theoretical interest is refractory, and we have limited ourselves for the most part to systems which are below-critical.

Theorem. A subcritical system is just-critical if and only if IM-II = 0. If r = 1, then of course 0 = \M-rI = IM-II, r being a characteristic root of M. Conversely, if IM - II = O, r > 1 by maximality of r, and r< 1 by assumption of subcriticality; hence r = 1.

8. We include for future reference the trivial remark: If M is a matrix of positive elements with Mv' = rv', 1 > r > O, v' > O, then there exists an e > 0 such that all vectors w in the e-neighborhood of v' are positive and satisfy the inequalities mi Wj < -(1 + r)Wi < Wi .

It suffices to note that the functions Ri(w) =_ mijwj/wi are continuous at w = v' and there have value Ri(v') = r < 1. Hence there is an e > 0 so that whenever llw - v'll< e, w will have positive components and Ri(w)< 1/2(1+ r).