338 M.Campanino, S.Isola

and

n - »oo

where h is as above and ^hdv = 1.

Proof . Given Nq large enough, let us consider the sequence of compact spaces Z^, where МеЫ and N>No, whose elements are the sequences a of the form (J = ((To, (Ti,...), 0) e {1,2,..., N}. Obviously, I^ с Г^^^ c ... с Г> for any iVe N. We then define the family of operators J?V,iv • ^(1^) -^ C{I^) by:

i^w , NvK ( T ) ^i , e^^'^^''<'Micr )

i=l

For any t; e CCZ^,) we have

t| ( ^H^ - if^ , ^ ) t ; |L<||t ; |Uell'^ll - r^^O as AT-. oo

where r^y = X,^iv+i ^^^^- As it is well known (see [B2], [R], [P.P]), for any N as above a Ruelle-Perron-Frobenius theorem holds for i^.iv ? ^^^ we shall denote by Я^^, Адг, Vjv the corresponding quantities. We shall prove that these quantities converge uniformly to Я, A, v of the Theorem when N ^ oo.

First observe that

A^A^ ( cr ) = E e''«-^«'<'">A^(ic7) > e-ll"'ll»(l -rj infA^.

Hence , Ajv > e""'^"*(l — Гдг). Moreover, we have that

\^^ , ! , via ) \^ X ••• X eI"=i^(»*>+^(^»-"'^)|i;(4...z»|

1 * 1 = 1 i„=l

i=l

SO that, by the spectral radius formula: Xf^ <, e""""". Hence,

( 2 . 8 ) е-"'^11-(1-г^)^Я^<е11'^11" and thus,

( 2 . 9 ) limAjv = supAjv = A<e"'^l'«

N - ♦00

We now briefly recall the argument used (see e.g. [P.P]) to show the existence of A^^. Consider the family of cones defined by

If veAf^ and <j, a'eE^ with сг,- = a/, 0 <^i <:П, then it is easy to see that: