  We study analytically and numerically the role of temperature shifts in the
simplest model where the energy landscape is explicitely hierarchical, namely
the Sinai model. This model has both attractive features (there are valleys
within valleys in a strict self similar sense), but also one important
drawback: there is no phase transition so that the model is, in the large size
limit, effectively at zero temperature. We compute various static chaos
indicators, that are found to be trivial in the large size limit, but exhibit
interesting features for finite sizes. Correspondingly, for finite times, some
interesting rejuvenation effects, related to the self similar nature of the
potential, are observed. Still, the separation of time scales/length scales
with temperatures in this model is much weaker that in experimental
spin-glasses.
