Homoclinic Points Cr-Created under Hypotheses by Probability Measures

Let M be a closed manifold and f: M → M be a Cr diffeomor-phism. Let p ∈ M be a hyperbolic fixed point of f. Then the stable and unstable sets of p are denoted respectively by % MathType!MTEF!2!1!+- % feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeqabiqaaa % qaaiaadEfadaahaaWcbeqaaiaadofaaaGcdaqadaqaaiaadchaaiaa % wIcacaGLPaaacqGH9aqpdaGadaqaaiaadIhadaabbaqaamaaxababa % GaciiBaiaacMgacaGGTbaaleaacaWGUbGaeyOKH4QaeyOhIukabeaa % aOGaay5bSdGaamizamaabmaabaGaamOzamaaCaaaleqabaGaamOBaa % aakmaabmaabaGaamiEaaGaayjkaiaawMcaaiaacYcacaWGWbaacaGL % OaGaayzkaaGaeyypa0JaaGimaaGaay5Eaiaaw2haaaqaaiaadEfada % ahaaWcbeqaaiaadwhaaaGcdaqadaqaaiaadchaaiaawIcacaGLPaaa % cqGH9aqpdaGadaqaaiaadIhadaWfqaqaamaaeeaabaGaciiBaiaacM % gacaGGTbaacaGLhWoaaSqaaiaad6gacqGHsgIRcqGHEisPaeqaaOGa % amizamaabmaabaGaamOzamaaCaaaleqabaGaeyOeI0IaamOBaaaakm % aabmaabaGaamiEaaGaayjkaiaawMcaaiaacYcacaWGWbaacaGLOaGa % ayzkaaGaeyypa0JaaGimaaGaay5Eaiaaw2haaaaaaaa!6F1C! $$ \begin{array}{*{20}{c}} {{W^S}\left( p \right) = \left\{ {x\left| {\mathop {\lim }\limits_{n \to \infty } } \right.d\left( {{f^n}\left( x \right),p} \right) = 0} \right\}} \\ {{W^u}\left( p \right) = \left\{ {x\mathop {\left| {\lim } \right.}\limits_{n \to \infty } d\left( {{f^{ - n}}\left( x \right),p} \right) = 0} \right\}} \\ \end{array} $$ which are Cr injectively immersed submanifolds of M. The points of intersection of the closure of Ws(p) with Wu(p) or of the closure of Wu(p) with Ws(p), different from p, is called almost homoclinic points to p.