By Antonio Pumarino, Angel J. Rodriguez

Even if chaotic behaviour had frequently been saw numerically previous, the 1st mathematical evidence of the lifestyles, with confident chance (persistence) of wierd attractors used to be given via Benedicks and Carleson for the Henon relations, firstly of 1990's. Later, Mora and Viana proven unusual attractor is usually chronic in primary one-parameter households of diffeomorphims on a floor which unfolds homoclinic tangency. This booklet is ready the patience of any variety of unusual attractors in saddle-focus connections. The coexistence and endurance of any variety of unusual attractors in an easy 3-dimensional state of affairs are proved, in addition to the truth that infinitely lots of them exist at the same time.

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**Example text**

Let us consider the maps T~(x) = x + cA and the intervals Am = [e-(r~+l),e-m], A + = Am+l U Am U Am-1. Denote by - A the symmetrical interval of A with respect 1 write I m = T~(Am) and I + = T~(A +) and for m _< - ( A - 1), to 0. For m >_ A - I m = T ~ ( - A - m ) and I + -- T~(-A+m). Finally, define U+ -- Urn-1. 6. Fix 0 < / ~ < < 1. ,p(a, m). As an immediate consequence of this definition we obtain t h a t fP(a'm)+l(U+) > e -z(p(a'm)+l) and f~(U*m) j + <_ 2e-~J for 1 <_ j < p(a, m). 7. The assumption (BAn), as given before, has not been used yet.

Therefore, since m ( ~ , - 1 ) - m ( s m (s \ ~ , ) , we shall obtain m ( ~ , ) > (1 - e-89~") m (f~,-1) - e -~-~~ = m (~,-1 \ s + I~l for each n E N. Then =N j=N provided t h a t N is large enough. 21. _~) Proof. Let w E P~-I and w~c = {a E w : a ~ ~'~}. Whether n is not a return situation or whether n is an inessential return, we have we~c -- 0. Then, suppose t h a t n is an essential return of w. In this case, we~ is an interval contained in U[~l_l and hence, I&(~=o)l < 2~-~+~, Let us define c~ = ~ I ( U ~ ) Mw.

S . ,s. 13. 14. Let N E N be large enough. 1) + (BAn) + (FAn) ~ (EGn). Proof. First, suppose that n >_ it, +p, + i. 3), [D~(a)l (/~q')' (~,,+p,+l(a)) ( E ' + x ) ' (Ira(a)) >_ c~+le-Ae ~~ where F,~(a) = qo + ql --F ... + q~ >_ (1 - c~)n according to (FAn). Once c < co is fixed, take a < < 1 such that c o ( l - a) > c + ~ . C~+l e - a e ~F'(a) > c~+ l e89 Then, if tiN > 2A, we have e ~. (a)l > C~,+1 e~-,<,,,e > /A',-'(~+~o~(lo~,-')) to, ~ _,<~'~'~ ) e > by taking A sufficiently large so that (f7 + log (10A-l)) a -1 logC0 + [ a > 0.