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By S. Elaydi, J. Cushing, R. Lasser, V. Papageorgiou, A. Ruffing

This quantity comprises talks given at a joint assembly of 3 groups operating within the fields of distinction equations, distinct features and functions (ISDE, OPSFA, and SIDE). The articles mirror the range of the themes within the assembly yet have distinction equations as universal thread. Articles hide subject matters in distinction equations, discrete dynamical structures, certain features, orthogonal polynomials, symmetries, and integrable distinction equations.

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28. Z. Z. , 250 (1997), 51-60. 29. Z. Z. Zhang and M. X. , 271 (1998), 169-177. 30. Z. Z. Zhang and T. , 283 (1998), 289-299. 17 ON THE RUELLE ZETA FUNCTION OF AN EXPANDING INTERVAL MAP JOAO FERREIRA ALVES AND J. L. FACHADA DEPARTMENT OF MATHEMATICS, INSTITUTO SUPERIOR TECNICO, AV. ROVISCO PAIS 1, 1049-001 LISBOA, PORTUGAL. In the context of expanding piecewise monotone interval maps, we present an alternative proof of a main result of Baladi and Ruelle concerning weigthed zeta functions and wigthed kneading determinants.

26) 34 Proof. ),z)ol we obtain (27) Furthermore, from the explicit expressions of, have R ( l ) ,by using (4),we Then, since u(2,t ) satisfies (5), the inequalities (28), (29) and (27) together imply where a0 is the constant from (6). Further, Ri is estimated on [O,a]and [u, TI separately. We consider first the case u < T/2 and so n = -a,’& In&. In [a,T],which is outside the layer 5 C ( 5 - l e - F 5 1) by (5) and 7-j = T . ,No. ,No. , N0/2. Now consider the case (T = T / 2 and so T / 2 < -a;'& ln&.

5 it is Fix(cp,) = Fix($,) = {u}. In particular, $,(u) = f ( u , w ) = cp,(w) > w = $,(w), in contradiction with the fact that $ , is increasing. Applying a similar reasoning to the case 0 < $,(x) < x < w for all x E (0, w ) , we obtain a new contradiction, and the proof finishes. 1. For all u , cp, and $, are strictly decreasing maps. Proof. 4, either cpz and $= are strictly decreasing for all z , or cpz and $z are strictly increasing for all z. Suppose that both cpz and $z are increasing maps for all z .

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