Are almost all trajectories dense in a chaotic set?

Chaos is the most complex dynamic behavior a system can show. Yet, it is produced by two simple, though nonlinear, deterministic mechanisms, hence the more proper name Deterministic Chaos. The first mechanism, stretching, acts locally and linearly, expanding small sets in the system's state space in (at least) one direction and contracting in the others. The second mechanism, folding, acts globally and is the nonlinear operation of bending the elongated sets to form the forward image of the original sets. These two simple operations are invertible and form a chaotic set for the system's forward and backward map, containing an uncountable infinity of astonishingly complex system's trajectories. The simplest explanation of chaos I ever heard is James A. Yorke "pizza machine" (plenary lecture at the SIAM DS, Snowbird, 2003): kneading the pizza dough is nothing but stretching and folding. The mathematical formalization is Stephen Smale's horseshoe map. Using this nice tool, I revisit the hallmarks of chaos, one of which (in the title) still misses a constructive proof. Copyright (c) 2024 The Authors.