Remarks concerning hyperbolic sets (Q1917797)

Let \(A\) is a hyperbolic set of a smooth dynamical system \(\{g^t\}\), defined on some smooth manifold \(M\). For \(x\in A\) let \(W^s(x)\) and \(W^u(x)\) be stable and unstable manifolds, respectively. The stable manifold \(W^s(x)\) is called periodic with period \(\tau\) if \(g^{\tau}W^s(x)\subset W^s(x)\). The periodicity of the unstable manifold \(W^u(x)\) is defined as follows: \(g^{-\tau}W^u(x)\subset W^u(x)\). Theorem 1. Suppose that \(A\) is a hyperbolic set, \(x\in A\) and \(W^s(x)\) (or \(W^u(x)\)) periodic with period \(\tau\). Then \(W^s(x)\) (\(W^u(x)\) respectively) passes through a periodic point \(y\in A\) which has the same period \(\tau\) (of course, in this case, \(W^s(x)=W^s(y)\) or \(W^u(x)=W^u(y)\)). Theorem 2. If the hyperbolic set \(A\) consists of an infinite number of trajectories or if it consists of a finite number of trajectories but not all periodic, then there are trajectories \(L_1,L_2\subset A\) such that \(W^s(L_1)\setminus L_1\cap A\neq \emptyset\), \(W^u(L_2)\setminus L_2\cap A\neq \emptyset\). Theorem 3. If the hyperbolic set \(A\) consists of an infinite number of trajectories, then it consists a nonperiodic trajectory. In this article auxiliary technical material is given and Theorems 1-3 are proved. The article is translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. Vol. 13, Dinamicheskie Sistemy-1, VINITI, Moskva, 1994.