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We investigate ergodic properties of a one-dimensional intermittent map thathas not only an indifferent fixed point but also a singular structure such thata uniform measure is invariant under mapping. The most striking aspect of ourmodel is that stagnant motion around the indifferent fixed point is induced bythe log-Weibull law, which is derived from Nekhoroshev stability in the contextof nearly-integrable Hamiltonian systems. Using renewal analysis, we derive alogarithmic inverse power decay of the correlation function and a$1/\omega$-like power spectral density. We also derive the so-called 1-stablelaw as a component of the time-average distribution of a simple observablefunction. This distributional law enables us to calculate a logarithmic inversepower law of large deviations. Numerical results confirm these analyticalresults. Finally, we discuss the relationship between the parameters of ourmodel and the degrees of freedom in nearly-integrable Hamiltonian systems.

$1/f$ Spectrum and 1-Stable Law in One-Dimensional Intermittent Map with Uniform Invariant Measure and Nekhoroshev Stability