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The subject is the study of polarized self maps f of a projective variety X over a field K. Polarized means that there is an ample line bundle L on X such that f*L is equal to L^q for an integer q > 1. The basic examples are the squarring map on the projective line and the Lattes map obtained from the multiplication by 2 in an elliptic curve. I will review fundamental notions such as preperiodic points and canonical height (Tate , Cole, Silvermann). I will describe theorems of Fakhrudin (K alg.closed) and recent results of Hruchovski and the speaker with their collaborators (K a function field of one variable). We will also explained different conjectures (S.Zhang,T.Tucker) generalizing what is known about abelian varieties (Dynamical Manin-Mumford, Dynamical Mordell-Lang). If I have enough time I will speak of equidistribution and symmetry (K number field).
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