  We introduce two 1D tight-binding models based on the Tribonacci
substitution, the hopping and on-site Tribonacci chains, which generalize the
Fibonacci chain. For both hopping and on-site models, a perturbative real-space
renormalization procedure is developed. We show that the two models are
equivalent at the fixed point of the renormalization group flow, and that the
renormalization procedure naturally gives the Local Resonator Modes.
Additionally, the Rauzy fractal, inherent to the Tribonacci substitution, is
shown to serve as the analog of conumbering for the Tribonacci chain. The
renormalization procedure is used to repeatedly subdivide the Rauzy fractal
into copies of itself, which can be used to describe the eigenstates in terms
of Local Resonator Modes. Finally, the multifractal dimensions of the energy
spectrum and eigenstates of the hopping Tribonacci chain are computed, from
which it can be concluded that the Tribonacci chains are critical.
