Induced maps of hyperbolic Bernoulli systems (Q1864087)

This author considers a standard hyperbolic automorphism \(f\) of the \(n\)-dimensional torus \(T^n\), equipped with the Haar measure \(\mu\). Let \(A \subset T^n\) be an open set with piecewise \(C^1\) boundary \(\partial A\). It is shown that if \(\partial A\) is everywhere transverse to the stable and unstable foliation of \(f\), then the induced map \(f_A:A\to A\) which preserves the restriction of \(\mu\) to \(A\) is Bernoulli. The proof relies on techniques of Liverani-Wojtkowski and Ornstein-Weiss [\textit{D. Ornstein} and \textit{B. Weiss}, Ergodic Theory Dyn. Syst. 18, No. 2, 441-456 (1998; Zbl 0915.58076)]. From a standard construction of local strong stable and strong unstable manifolds for \(f_A\) and the assumption on the singular set \(\partial A\) it follows easily that either \(f_A\) is Bernoulli or Bernoulli times a rotation. The second case is then ruled out with the observation, proved with a Hopf-Chain argument, that for every \(k>0\) the map \(f_A^k\) is ergodic.