  In quasicrystals, special tiling patterns could give rise to unique physical
phenomena such as critical states distinct from periodic systems. In this
paper, we study how quasi-periodicity in aperiodic systems results in anomalous
phonon modes, especially focusing on thermal transmittance in one-dimensional
quasicrystals. Unlike periodic or compeletly random systems, we classify
certain quasicrystals could host critical phonon modes whose transport
properties are topologically protected based on their pattern equivariant
cohomology group of supertilings. Starting from discussing general rule to find
such critical phonon modes, we discuss classification of topologically distinct
thermal transmittance in quasiperiodic systems. To be more specific, we
exemplify (decorated) metallic-mean tilings and Cantor tiling, and derive
universal features for resonant and decaying phonon modes as a function of
quasi-periodic strength. Our study paves a new way to understand thermal
transmittance of quasi-periodic systems based on the topological classification
and offers quasicrystals as strong candidates to control drastic phonon modes.
