.. ?j-=-1,

, N (k1)?1 ), z 1 ) (f j (z 1 ), f j

.. ?j-=-1, m(1) ? 1 k 2 ) (f (r 2,j ), r 2,j+1 ) ?j = 1

, N (k2)?1 ), z 2 ) (f j (z 2 ), f j

.. ?j-=-1,

, Moreover, by construction, for any couple (q, p) given above, the distance d(q, p) < 2. Consequently, applying Lemma 13 in [12], we obtain a homeomorphism ? : M ? M such that d C 0 (? ? f, f ) < 4?? and ?(q) = p for any such a couple (q, p), By perturbing f a little, we can assume that for every (q i , p i ), (q j , p j ) with i = j, it holds that q i = q j and p i = p j

, > 0 of iterations of ? ? f , we have (? ? f ) N (f ?1 (y)) = f (z)

, Let {B ? } be a finite covering of GR(f ) by open convex balls such that, for any ?, B ? ? U and diam(B ? ) < ?

, By shrinking the balls {B ? } if necessary, we can assume that the {U ? } have pairwise disjoint closures. We now introduce the following equivalence relation on pairs of {U ? }. We say that U ?1 is related to U ?2 if either U ?1 = U ?2 or there is a common cycle containing both U ?1 and U ?2, We denote by {U ? } the connected components of ? B ?

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