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A characterization of subvarieties of Shimura varieties which contain aZariski dense subset of weakly special subvarieties has been proved by thesecond author, by combining o-minimality results and functional transcendenceresults. In this paper, we obtain a new proof of this statement by dynamicstechniques on homogeneous spaces in the spirit of the earlier work of Clozeland the second author. The proof combines ergodic theory \`a la Ratner, with astatement on the dimension of a Hausdorff limit of a sequence of definablesubsets (in an o-minimal theory) extracted from a definable family. One obtainsin passing general homogeneous dynamics statements valid on arbitraryarithmetic quotients which are of independent interest, that can be applied inthe study of variations of Hodge structures and their associated perioddomains.

Equidistribution de sous-vari\'et\'es sp\'eciales et o-minimalit\'e: Andr\'e-Oort g\'eom\'etrique