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The universality of the route to chaos is analytically proven for countablyinfinite number of maps by proposing the Super Generalized Boole (SGB)transformations. As one of the route to chaos, intermittency route was studiedby Pomeau and Manneville numerically. They conjectured the universality in Type1 intermittency, that the critical exponent of the Lyapunov exponent is $1/2$in Type 1 intermittency. In order to prove their conjecture, we showed that forcertain parameter ranges, the SGB transformations are exact and preserve theCauchy distribution. Using the property of exactness, we proved that thecritical exponent is $1/2$ for countably infinite number of maps where Type 1intermittency occurs.

Universality of the Route to Chaos -- Exact Analysis