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We consider a new class of 5-dimensional dilatonic actions which areinvariant under T-duality transformations along three compact coordinates,provided that an appropriate potential is chosen. We show that the invarianceremains when we add a boundary term corresponding to a moving 3-brane, and westudy the effects of the T-duality symmetry on the brane cosmologicalequations. We find that T-duality transformations in the bulk induce scalefactor duality on the brane, together with a change of sign of the pressure ofthe brane cosmological matter. However, in a remarkable analogy with thePre-Big Bang scenario, the cosmological equations are unchanged. Finally, wepropose a model where the dual phases are connected through a scattering of thebrane induced by an effective potential. We show how this model can realise asmooth, non-singular transition between a pre-Big Bang superinflationaryUniverse and a post-Big Bang accelerating Universe.Comment: 18 pages, minor typos corrected, Sec. 2 expanded with more details on  the self-T-dual background, Sec.4 and 5 revised accordingly. Version to  appear on JCA

Pre-Big Bang scenario on self-T-dual bouncing branes