TITLE: 
Some bifurcations related to homoclinic tangencies for 1-parameter families of 
symplectic diffeomorphisms.

AUTHOR: Joan Carles Tatjer

Departament de Matematica Aplicada i Analisi
Universitat de Barcelona, Gran Via 585, 08007 Barcelona, Spain

ABSTRACT: 
Let $f_a:{\rm R}^{2n}\longrightarrow{\rm R}^{2n},$ $a\in{\rm R},$ be a 
1-parameter family of symplectic diffeomorphisms such that, for $a=a_0$ there
is a hyperbolic fixed point $p(a_0)$ having all its eigenvalues real and 
different. Suppose that the invariant manifolds of $p(a_0)$ have a quadratic 
homoclinic tangency in $p_1\in{\rm R} ,$ which unfolds generically with $a.$ 
When $n=1$ it is possible to prove that, for all $m$ large enough there exist 
$a_m\in{\rm R}$ such that $f_a$ has a $m$-periodic parabolic point near the 
point $p_1,$ for $a=a_m,$ and $a_m\rightarrow\infty$ when 
$m\rightarrow\infty .$ Moreover when $m\rightarrow\infty ,$ the family $f^m_a$ 
tends, in a suitable sense, to the conservative H\'enon map.  In the case 
$n=2$, and with suitable generic conditions, we also see that there exist 
$m$-periodic parabolic points with the same properties as before.