Scaling properties of multifractal functions at an attractor-repeller transition

Multifractal properties of repelling sets generated by hyperbolic maps are studied as a function of a parameter describing a transition to an attracting interval. Critical indices in the scaling behavior of multifractal functions are found when a uniform probability density is assumed. A constant probability is also considered, and the resulting thermodynamiclike functions are investigated close to the critical value of the parameter.