Stable/unstable holonomies, density of periodic points, and transitivity for continuum-wise hyperbolic homeomorphisms (Q6576899)

It is commonly believed that sensitivity captures the essence of chaos and it is possible to prove such a claim by assuming transitivity and density of periodic points.\N\NAs hyperbolicity is a central notion in the study of chaos, it is important to understand how several features of the dynamics of hyperbolic systems are present in chaotic systems. In the existing literature, discussions about the density of periodic points in the nonwandering set -- without assuming expansiveness and shadowing -- are not very frequent.\NThis paper deals with such problem. In fact, the authors try to solve it in a more general setting, namely that of \emph{continuum-wise hyperbolicity}, which is beyond the theory of topological hyperbolicity, see [\textit{A. Artigue} et al., J. Differ. Equations 378, 512--538 (2024; Zbl 1536.37035)]. The main result of this paper is the following:\N\NTheorem. If a continuum-wise hyperbolic homeomorphism (see Definition 1.2) has continuous joint stable/unstable holonomies (see Definition 3.5), then it has a dense set of periodic points in its nonwandering set. If local stable/unstable holonomies of a continuum-wise hyperbolic homeomorphism are isometric (see Definition 5.1), then it is transitive.