The new faces of the Feigenbaum point

The new faces of the Feigenbaum point: Dynamical hierarchy, self-similar network, theoretical game and stationary distribution. In this talk we first show that the recently revealed features of the dynamics toward the Feigenbaum attractor form a hierarchical construction with modular organization that leads to a clear-cut emergent property. Then we transcribe the well-known Feigenbaum scenario into families of networks via the horizontal visibility algorithm, derive exact results for their degree distributions, recast them in the context of the renormalization group and find that its fixed points coincide with those of network entropy optimization. Next we study a discrete-time version of the replicator equation for two-strategy theoretical games. Their stationary properties differ from those of continuous time for sufficiently large values of the parameters, where periodic and chaotic behavior replaces the usual fixed-point population solutions. We observe the familiar period-doubling and chaotic-band-splitting attractor cascades of unimodal maps. Finally, we look at the limit distributions of sums of deterministic chaotic variables in unimodal maps and find a remarkable renormalization group structure associated with the operation of increment of summands and rescaling. In this structure—where the only relevant variable is the difference in control parameter from its value at the transition to chaos—the trivial fixed point is the Gaussian distribution and a novel nontrivial fixed point is a multifractal distribution that emulates the Feigenbaum attractor.