  We study the heating dynamics of a generic one dimensional critical system
when driven quasiperiodically. Specifically, we consider a Fibonacci drive
sequence comprising the Hamiltonian of uniform conformal field theory (CFT)
describing such critical systems and its sine-square deformed counterpart. The
asymptotic dynamics is dictated by the Lyapunov exponent which has a fractal
structure embedding Cantor lines where the exponent is exactly zero. Away from
these Cantor lines, the system typically heats up fast to infinite energy in a
non-ergodic manner where the quasiparticle excitations congregate at a small
number of select spatial locations resulting in a build up of energy at these
points. Periodic dynamics with no heating for physically relevant timescales is
seen in the high frequency regime. As we traverse the fractal region and
approach the Cantor lines, the heating slows enormously and the quasiparticles
completely delocalise at stroboscopic times. Our setup allows us to tune
between fast and ultra-slow heating regimes in integrable systems.
