We survey the connections between entropy, chaos, and independence in topological dynamics. We present extensions of two classical results placing the following notions in the context of symbolic dynamics:   1. Equivalence of positive entropy and the existence of a large (in terms of asymptotic and Shnirelman densities) set of combinatorial independence for shift spaces.   2. Existence of a mixing shift space with a dense set of periodic points with topological entropy zero and without ergodic measure with full support, nor any distributionally chaotic pair. Our proofs are new and yield conclusions stronger than what was known before.

From a two-agent, two-strategy congestion game where both agents apply the multiplicative weights update algorithm, we obtain a two-parameter family of maps of the unit square to itself. Interesting dynamics arise on the invariant diagonal, on which a two-parameter family of bimodal interval maps exhibits periodic orbits and chaos. While the fixed point b corresponding to a Nash equilibrium of such map f is usually repelling, it is globally Cesàro attracting on the diagonal, that is, $\underset{n\to \mathrm{\infty }}{lim}\frac{1}{n}\sum _{k=0}^{n-1}{f}^k\left(x\right)=b$ for every $x\in (0,1)$. This solves a known open question whether there exists a “natural” nontrivial smooth map other than $x\mapsto ax{e}^{-x}$ with centers of mass of all periodic orbits coinciding. We also study the dependence of the dynamics on the two parameters.

We consider the concept of generalized Kolmogorov–Sinai entropy, where instead of the Shannon entropy function, we consider an arbitrary concave function defined on the unit interval, vanishing in the origin. Under mild assumptions on this function, we show that this isomorphism invariant is linearly dependent on the Kolmogorov–Sinai entropy.