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For real analytic expanding interval maps, a novel method is given for rigorously approximating the diffusion coefficient of real analytic observables. As a theoretical algorithm, our approximation scheme is shown to give quadratic exponential convergence to the diffusion coefficient. The method for converting this rapid convergence into explicit high precision rigorous bounds is illustrated in the setting of Lanford's map $$x\mapsto 2x +\frac{1}{2}x(1-x) \pmod 1 $$ . [ABSTRACT FROM AUTHOR]

The famous Birkhoff ergodic theorem shows that given an ergodic measure the averages of an integrable function along typical orbits converge to the integral of the function. The multifractal spectra describe the sets of points for which the averages converge to another limit. In this note, we will consider the specific setting of conformal repellers and show how to estimate the Hausdorff dimensions of such sets via approximations to their alternative characterizations as zeros of appropriate determinant functions. [ABSTRACT FROM PUBLISHER]