BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Junyi Xie (Universite de Rennes I) DTSTART:20201110T160000Z DTEND:20201110T165000Z DTSTAMP:20260422T220759Z UID:20w5206/3 DESCRIPTION:Title: On the Zariski dense orbit conjecture\nby Junyi Xie (Universite de Ren nes I) as part of BIRS workshop: Algebraic Dynamics and its Connections to Difference and Differential Equations\n\n\nAbstract\nWe prove the followi ng theorem. Let f be a dominant endomorphism of a projective surface over an algebraically closed field of characteristic 0. If there is no nonconst ant invariant rational function under f\, then there exists a closed point whose orbit under f is Zariski dense. This result gives us a positive ans wer to the Zariski dense orbit conjecture for endomorphisms of projective surfaces.\n\nWe define a new canonical topology on varieties over an algeb raically closed field which has finite transcendence degree over Q. We ca ll it the adelic topology. This topology is stronger than the Zariski topo logy and an irreducible variety is still irreducible in this topology.\nUs ing the adelic topology\, we propose an adelic version of the Zariski dens e orbit conjecture\, which is stronger than the original one and quantifie s how many such orbits there are. We also prove this adelic version for en domorphisms of projective surfaces\, for endomorphisms of abelian varietie s\, and split polynomial maps. This yields new proofs of the original conj ecture in the latter two cases.\n LOCATION:https://researchseminars.org/talk/20w5206/3/ END:VEVENT END:VCALENDAR