Chaos on Peano continua

Abstract

Generalizing the result of Agronsky and Ceder (1991), we prove that every Peano continuum admits a continuous transformation that is exact Devaney chaotic; that is, it has a dense set of periodic points, and every nonempty open set covers the entire space in finitely many iterations. We identify a natural class of Peano continua, containing all one-dimensional continua and all absolute neighborhood retracts, which allows us to create locally small perturbations. Using this method, we prove that within these specific classes of continua, exact Devaney chaotic systems are dense in all chain transitive systems, mixing systems are generic among chain transitive systems and shadowing is generic among all continuous systems.