  Haros graphs have been recently introduced as a set of graphs bijectively
related to real numbers in the unit interval. Here we consider the iterated
dynamics of a graph operator $\cal R$ over the set of Haros graphs. This
operator was previously defined in the realm of graph-theoretical
characterisation of low-dimensional nonlinear dynamics, and has a
renormalization group (RG) structure. We find that the dynamics of $\cal R$
over Haros graphs is complex and includes unstable periodic orbits or arbitrary
period and non-mixing aperiodic orbits, overall portraiting a chaotic RG flow.
We identify a single RG stable fixed point whose basin of attraction is the set
of rational numbers, associate periodic RG orbits with (pure) quadratic
irrationals and aperiodic RG orbits with (non-mixing) families of non-quadratic
algebraic irrationals and trascendental numbers. Finally, we show that the
entropy gradients inside periodic RG orbits are constant. We discuss the
possible physical interpretation of such chaotic RG flow and speculate on the
entropy-constant periodic orbits as a possible confirmation of a (quantum
field-theoretic) $c$-theorem applied inside the invariant set of a RG flow.
