The existence of an invariant stable foliation and the problem of reducing to a one-dimensional map (Q5896416)

In this paper, the result which has been announced in Proc. Jap. Acad., Ser. A 59, 126-129 (1983; Zbl 0522.58039) is proved. Let us consider a map \(H: {\mathbb{R}}^ 2\to {\mathbb{R}}^ 2\) defined by \(H(x,y)=(f(x)+\epsilon_ 1(x,y),\mu y+\epsilon_ 2(x,y)),\) where f is a map of piecewise \(C^ 2\)-class such that f(I)\(\subset I\) for the interval \(I=[0,1]\), \(0<\mu <1\), and each \(\epsilon_ i:{\mathbb{R}}^ 2\to {\mathbb{R}}^ is\) of \(C^ 2\)-class. Then this map has an invariant set \(\Gamma\) near \(I\times\{0\}\) under some conditions on \(\epsilon_ i(x,y)\) and \(\mu\). So, if we could construct an invariant stable foliation on \(\Gamma\), then we could say that the study of the behavior of H near \(\Gamma\) is reduced to the study of the one-dimensional map on \(\Gamma\). In this work, we assume further that H leaves \(I\times\{0\}\) invariant and obtain conditions about H which imply the existence of an invariant stable foliation almost everywhere with respect to Lebesgue measure. Furthermore these conditions are expressed in terms of \(\mu,\epsilon_ i\) and the one-dimensional map f. Recently, Ruelle showed the existence of a stable foliation almost everywhere with respect to invariant measure by using the multiplicative ergodic theorem. In contrast, we only need some assumptions on the ratio of eigenvalues of the tangent maps and the existence of a foliation almost everywhere with respect to Lebesgue measure. Therefore our result is an extension of Ruelle's work. The author has partially extended this result by a different method in Tokyo J. Math. (to appear).