A lower bound for topological entropy of generic non-Anosov symplectic diffeomorphisms (Q2928247)

The authors study the entropy of a diffeomorphism on a compact manifold. Given a \(C^1\) area-preserving surface diffeomorphism \(f\), for a periodic point \(p\) of period \(m\) denote by \(\lambda(p)\) the absolute value of the eigenvalue of \(T_pf^m\) with norm \(>1\) and let \(s(f)=\sup\{\frac1m\log \lambda(p)\}\), where the supremum is taken over all such points \(p\). \textit{S. E. Newhouse} [Astérisque 51, 323--334 (1978; Zbl 0376.58010)] showed that for a generic \(C^1\) non-hyperbolic (non-Anosov) area-preserving diffeomorphism \(f\), its topological entropy \(h(f)\) is bounded from below by \(s(f)\). The authors extend this estimate and prove that for a generic \(C^1\) non-hyperbolic symplectic diffeomorphism \(f\) of a compact connected manifold the topological entropy of \(f\) is bounded from below by \(s(f)=\sup\{\frac1m\log \lambda(p)\}\), where in this more general case \(\lambda(p)\) denotes the absolute value of the smallest eigenvalue of \(T_pf^m\) among those with absolute value \(> 1\) and where the supremum is taken over all hyperbolic periodic points \(p\) of \(f\).NEWLINENEWLINEFurther, they draw consequences for the regularity of the entropy function when varying the diffeomorphism. While it is known that this function \(f \mapsto h(f)\) is upper semi-continuous in the \(C^\infty\)-topology for any compact boundaryless manifold [\textit{S. E. Newhouse}, Ann. Math. (2) 129, No. 2, 215--235 (1989; Zbl 0676.58039)], the authors present an example of a \(C^\infty\) surface diffeomorphism \(f_0\) such that \(f\mapsto h(f)\) is not upper semi-continuous at \(f_0\) in the \(C^1\)-topology. The regularity of the entropy function is closely connected with the existence of symbolic extensions. The authors show that a generic \(C^1\) non-hyperbolic symplectic diffeomorphism \(f\) of a compact connected manifold has no symbolic extension (see [\textit{T. Downarowicz} and \textit{S. Newhouse}, Invent. Math. 160, No. 3, 453--499 (2005; Zbl 1067.37018)] for an earlier result in the case of an area-preserving surface diffeomorphisms).