Grouplike minimal sets in ACFA AND inT_A

Abstract

This paper began as a generalization of a part of the author's PhD thesis about ACFA and ended up with a characterization of groups definable in T A . The thesis concerns minimal formulae of the form x ∈ A ∧ σ(x) = f (x) for an algebraic curve A and a dominant rational function f : A → σ(A).These are shown to be uniform in the Zilber trichotomy, and the pairs (A, f ) that fall into each of the three cases are characterized. These characterizations are definable in families. This paper covers approximately half of the thesis, namely those parts of it which can be made purely model-theoretic by moving from ACFA, the model companion of the class of algebraically closed fields with an endomorphism, to T A , the model companion of the class of models of an arbitrary totally-transcendental theory T with an injective endomorphism, if this model-companion exists. A T A analog of the characterization of groups definable in ACFA is obtained in the process. The full characterization of the cases of the Zilber trichotomy in the thesis is obtained from these intermediate results with heavy use of algebraic geometry.

The dynamical Manin–Mumford conjecture and the dynamical Bogomolov conjecture for endomorphisms of

“…The conclusion from Theorem 1.2 covers the main result of Medvedev’s PhD thesis [Med07] (whose main findings were published in [MS14, Proposition 2.21]) who showed that any invariant subvariety under the coordinatewise action of non-exceptional rational functions must have the form (1.3.1). Our result is stronger than the results from [Med07, MS14] since we only assume that a subvariety contains a Zariski dense set of preperiodic points under the action of and then we derive that must have the form (1.3.1) (see also our Theorem 1.4). Medvedev and Scanlon assume that is invariant by (or more generally, preperiodic under the action of ) and then using the model theory of difference fields, they conclude that must have the form (1.3.1).…”

“…Applying the rigidity of the symmetries of the Julia set on the one-dimensional slices, we are able to derive the rigidity of the symmetries of the entire -current, from which we derive the desired conclusion regarding and the dynamical system (see the proof of Theorem 6.1). It is precisely the study of the rigidity of this -current (for ) which provides the new proof of Medvedev’s result [Med07], which otherwise could not have been obtained from the arguments from our previous paper [GNY17].…”

“…We do not use model theory; instead, we use algebraic geometry (including the powerful arithmetic Hodge index theorem of Yuan and Zhang [YZ17]) coupled with a careful analysis for the local symmetries of the Julia set of a rational function. We state below our formal result which covers the main result of [Med07] thus providing the form of any preperiodic subvariety in under the split action of non-exceptional rational functions.…”