, HyperJumpg] is a sampling of H up to local dimension

, Then for all k 2 N, x k is of local dimension (d 0 + 1 ) , w h e r e HyperJumpg] ( k QQ tt x) = ( t k x k ), t k 2 R x k 2 R d . For all t 2 R, for all x 2 R d, Assume HyperJumpg] is Zeno for some Q 2 P t2 Rx2 R d

, There exists a xed rst order formula F such that for all k 2 N Q2 P t2 Rx2 R d , dHyperJumpg](k + 1 Q t x )e is deenable by formula F from relation R g (HyperJumpg](k QQ tt x)) and from some recursive relations

, There exists clearly a xed rst order formula G such that for all real sequence (g 0 (k 0 Q x )) k 0 2N converging to some g 0 (QQ x) 2 R d , dg 0 (QQ x)e is deenable by f o r m ula G from relation R g 0 (x), where R g 0 denotes the relation associated to g 0 corresponding to x. As a consequence, the last assertion is clear, since deenition 5.8 can be translated directly into a xed rst order formula F that, there exists a xed rst order formula over relation R g (tt x) that tells if g is Zeno for QQ tt x, vol.42

, We prove n o w that HyperJumpg] is a sampling of H up to local dimension

, t k 2 R x k 2 R d , f o r all k 2 N. Let be the trajectory of H starting from x at time t. From the fact that g is a sampling it is easy to show b y induction over k that for all k 2 N (t k ) = x k . N o w, if there is some k 0 with x k0 2 Q NoEvolution(H), since g is a sampling, it is clear than x k = x k0 t k = t k0 for all k k 0 . If for all kkx k 6 2 Q NoEvolution(H), QQ tt x) = ( t k x k )

, QQ tt x. F or all k 2 N, g must be Zeno for QQ t k;1 x k;1 : hence, (t k x k ) is the limit of g(k 0 Q t k;1 x k;1 ) k 0 2 N. Since g is a sampling up to local dimension (d 0 ) + , the local dimension of x k must be >

, By lemma 5.4, the local dimension of x is > (d 0 + 1 ) + . T h i s proves the rst assertion, Denote t = sup k2N t k and x = = ( t )

,

, some x 2 Q d , a r ational polyhedron F not intersecting Q, s u c h that Cycle(x k;1 z 2 H F x ) is true, x k;1 6 2 QQ z 2 6 2 Q, z 2 6 2 F c

, QQ tt x) = Cycle

, r(m) where r(m) i s a n a n H(y + 0 z k;1 + 0 1) d(ttx)e -recursive index of R gk, p.1

, By lemma 5.11, there exists a xed rst order formula F such that for all n 2 N Q2 P t2 Rx2 R d

, By lemma 5.2, there exists y F 2 OOjy F j < ! and a recursive g that maps r(m) t o g(r(m)), where g(r(m)) is an H, nn QQ tt x)) and from some recursive relations

, ))), where r 0 (g(r(m))) is an H(y + 0 z k;1 + 0 1 + o y F ) d(ttx)e -recursively enumerable index of HyperJumpg k

, 0 z k;1 + 0 1 + o y F for all n 2 N. As a consequence, for all n 2 N, R <n HyperJumpgk;1] is semi-recognized by the machine with oracle H d(ttx)e (h(n ; 1)) that on input <n Q P> , compute for i = 1 : : : n ; 1 a n H d(ttx)e (h(i ; 1))-recursively enumerable index m i of HyperJumpg k;1 ] ( ii QQ tt x) from the H d(ttx)e (h(i;2))-recursively enumerable index m i, Denote by h : N ! O the recursive mapping such that r(0) = 1 r (n + 1 ) = r(n) +

, Assume we h a ve H(y) d(ttx)e -recursively enumerable index m of CycleFreeg k;1 ](nn QQ tt x), where m 2 N y2 O. By lemma 5.6, there exists a recursive r that maps m to r(m) where r(m) i s a n H(y + 0 h(n ; 1) + 0 1) d(ttx)e -recursive i n d e x o f R <n HyperJumpgk;1] (CycleFreeg k;1 ](nn QQ tt x)). By lemma 5, vol.12

, As before, by lemma 5.2, and by lemma 5.6, there exists some recursive g and r 0 that maps m to r 0 (g(r(m))) an H(y+ 0 h(n;1)+ 0 1+ o y G ) d(ttx)e -recursively enumerable index of CycleFreeg k, QQ tt x)) and from some recursive relations, vol.1

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