By Antonio Pumarino, Angel J. Rodriguez

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

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**Sample text**

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. It was posed for n > N as long as we always get expansiveness up to time N . However, once (~ is fixed, we may choose N to set the assumption in the following equivalent way: I~j(a) - cA[ > Ae -~j for 1 < j <_ n, (BAn) where A = A(A) = fa(x)(cx) - cx is a constant smaller than one. Henceforth, we shall use this formulation.

THE UNIMODAL FAMILY it suffices to d e m o n s t r a t e t h a t m ({a ~ ~ : Tn(a) > cm}) < e -~-~0~ I~1. In order to prove this last inequality notice t h a t no r e t u r n s i t u a t i o n fii following the escape s i t u a t i o n e~-i d e p e n d s on a C w ~-1. Hence, for each w C P~ we m a y define E~(w) and T ( w ) = El(W 1) + ... + E~+l(wS+l), where e~+l = n and w s+l = w. In short, for each w c P~ we have defined t h e sequences 1 = e0 < el < ... < e8 < e~+l = n w i t h ei = el(w) and s = s ( w , n ) < n, Y = ftl < ft2 < ...

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. It was posed for n > N as long as we always get expansiveness up to time N . However, once (~ is fixed, we may choose N to set the assumption in the following equivalent way: I~j(a) - cA[ > Ae -~j for 1 < j <_ n, (BAn) where A = A(A) = fa(x)(cx) - cx is a constant smaller than one.