  The robustness of the universality class concept of the chaotic transition
was investigated by analytically obtaining its critical exponent for a wide
class of maps. In particular, we extended the existing one-dimensional chaotic
maps, thereby generalising the invariant density function from the Cauchy
distribution by adding one parameter. This generalisation enables the
adjustment of the power exponents of the density function and superdiffusive
behavior. We proved that these generalised one-dimensional chaotic maps are
exact (stronger condition than ergodicity) to obtain the critical exponent of
the Lyapunov exponent from the phase average. Furthermore, we proved that the
critical exponent of the Lyapunov exponent is $\frac{1}{2}$ regardless of the
power exponent of the density function and is thus universal. This result can
be considered as rigorous proof of the universality of the critical exponent of
the Lyapunov exponent for a countably infinite number of maps.
