We explicitly calculate the induced gravity theory at the boundary of an asymptotically Anti-de Sitter five dimensional Einstein gravity. We also display the action that encodes the dynamics of radial diffeomorphisms. It is found that the induced theory is a four dimensional conformal gravity plus a scalar field. This calculation confirms some previous results found by a different approach.

We study the dynamics of classical particles in different classes of spatially extended self-similar systems consisting of (i) a self-similar Lorentz billiard channel, (ii) a self-similar graph, and (iii) a master equation. In all three systems, the particles typically drift at constant velocity and spread ballistically. These transport properties are analyzed in terms of the spectral properties of the operator evolving the probability densities. For systems (i) and (ii), we explain the drift in terms of the properties of the Pollicott-Ruelle resonance spectrum and the corresponding eigenvectors.

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