By Dieudonne J.

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Filtering bases, onal Lie algebras Invent. - (1985), (Russian). Fel'shtyn The Leningrad Technology In the paper we define new dynamical to study the Nielsen of the dynamical Institute zeta functions. We continue zeta function [I, 2] . The universal properties zeta functions are investigated. § I. INTRODUCTION We assume everywhere X ~" X namical X to be a connected compact polyhedron to be a continuous systems the following zeta function number of isolated map. In the theory of discrete zeta functions ~(~=~p( ~ are known: ) fixed points of and the Artin-Mazur , where I~ ; the Lefschetz dy- r,, ~j, is the ~[ zeta function ~=0 H~[X~]is ~01 Z Artin-Mazur Mazur and Lefschetz group rings z m the Lefschetz and Lefschetz The above ~eta functions function of algebraic zeta function zeta functions zeta functions, orZz~of ~ [4] ; reduced [ 5 ] ; twisted Artin- which have coefficients an abelian g r o u p H in the [6] are directly analogous manifolds is rational: number of to the Wail zeta over finite fields [ 7 ] .

Thus, (we take sign + if ~ > 0 and sign expanding map ~ and M ~ [ { ] 7 + 4-t - ing maps and hyperbolic endomorphisms of tatives in their homotopy class. Z M,],tl]-- if ~ < 0 ). We see that expand- T are the minimal represen- 51 § 5. ¢£,)k~/c, gJC THEOREM S[~)~ ~ eN , PROOF. 2. / d,lm. D. of map. 12. From = ~')= e~cp(et~~f a,:[ c°(d') ~-t least period function. ) AI(Z)----~'~'P THEOREM S(d,) via the formula: 11. A~Cz)= gl this formula. H map of the • Then (6) 52 m, Since PROOF. I . = 4,([, , F~(I] = Since F~ ~- ~ ( F ~ 5 ({~)) t cl ) ----~ , where ~ I ~ j111, _ )-- such that I R~({),it number L ( ~ $) fol- of the fixed point manifold of equals to the {{ .

1985), (Russian). Fel'shtyn The Leningrad Technology In the paper we define new dynamical to study the Nielsen of the dynamical Institute zeta functions. We continue zeta function [I, 2] . The universal properties zeta functions are investigated. § I. INTRODUCTION We assume everywhere X ~" X namical X to be a connected compact polyhedron to be a continuous systems the following zeta function number of isolated map. In the theory of discrete zeta functions ~(~=~p( ~ are known: ) fixed points of and the Artin-Mazur , where I~ ; the Lefschetz dy- r,, ~j, is the ~[ zeta function ~=0 H~[X~]is ~01 Z Artin-Mazur Mazur and Lefschetz group rings z m the Lefschetz and Lefschetz The above ~eta functions function of algebraic zeta function zeta functions zeta functions, orZz~of ~ [4] ; reduced [ 5 ] ; twisted Artin- which have coefficients an abelian g r o u p H in the [6] are directly analogous manifolds is rational: number of to the Wail zeta over finite fields [ 7 ] .