We consider functional equations driven by linear fractional transformations, which are special cases of de Rham's functional equations. We consider Hausdorff dimension of the measure whose distribution function is the solution. We give a necessary and sufficient condition for singularity. We also show that they have a relationship with stationary measures.

We prove that the f -divergences between univariate Cauchy distributions are always symmetric and can be expressed as functions of the chi-squared divergence. We show that this property does not hold anymore for multivariate Cauchy distributions. We then present several metrizations of f -divergences between univariate Cauchy distributions.

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We consider the range of a one-parameter family of self-interacting walks on the integers up to the time of exit from an interval. We derive the weak convergence of an appropriately scaled range. We show that the distribution functions of the limits of the scaled range satisfy a certain class of de Rham's functional equations. We examine the regularity of the limits. * AMS 2000 subject classifications : 60K35. Key words and phrases : self-interacting random walk, range of random walk.

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