In these lectures, we will examine a series of conjectures about the geometry of preperiodic points for endomorphisms of $ \mathbb{P}^{N}$. Lecture 1 will focus on the Dynamical ManinMumford Conjecture (DMM), formulated by Shouwu Zhang in the 1990s as an extension of the well-known Manin-Mumford Conjecture (which investigated the geometry of torsion points in abelian varieties and was proved in the early 1980s by Raynaud). The DMM aims to classify the subvarieties of $\mathbb{P}^{N}$ containing a Zariski-dense set of preperiodic points. Lectures 2 and 3 will be devoted to conjectures that treat families of maps on $\mathbb{P}^{N}$. One conjecture in particular was inspired by the recently-proved ”Relative Manin-Mumford” theorem of Gao-Habegger for abelian varieties, but the dynamical version turns out to be closely related to the study of dynamical stability and to contain many previously-existing questions/conjectures/results about moduli spaces of maps on $\mathbb{P}^{N}$. These lectures are based on joint work with Myrto Mavraki.[-]

Our goal is the study of the local dynamics of tangent to the identity biholomorphisms in C2, and more precisely of the existence of invariant manifolds. In the first lecture we will focus on the problem of existence of invariant curves for two-dimensional vector fields and present some classical results: Seidenberg's resolution of singularities, Briot-Bouquet theorem and Camacho-Sad theorem. In the second lecture we will present the first results of existence of 1-dimensional invariant manifolds for tangent to the identity biholomorphisms obtained by Ecalle/Hakim and Abate, connecting them to the corresponding results for vector fields. In the third lecture we will discuss two extensions of the previous results, obtained in collaboration with Jasmin Raissy, Fernando Sanz, Javier Ribon, Rudy Rosas and Liz Vivas.[-]

Our goal is the study of the local dynamics of tangent to the identity biholomorphisms in C2, and more precisely of the existence of invariant manifolds. In the first lecture we will focus on the problem of existence of invariant curves for two-dimensional vector fields and present some classical results: Seidenberg's resolution of singularities, Briot-Bouquet theorem and Camacho-Sad theorem. In the second lecture we will present the first results of existence of 1-dimensional invariant manifolds for tangent to the identity biholomorphisms obtained by Ecalle/Hakim and Abate, connecting them to the corresponding results for vector fields. In the third lecture we will discuss two extensions of the previous results, obtained in collaboration with Jasmin Raissy, Fernando Sanz, Javier Ribon, Rudy Rosas and Liz Vivas.[-]

Our goal is the study of the local dynamics of tangent to the identity biholomorphisms in C2, and more precisely of the existence of invariant manifolds. In the first lecture we will focus on the problem of existence of invariant curves for two-dimensional vector fields and present some classical results: Seidenberg's resolution of singularities, Briot-Bouquet theorem and Camacho-Sad theorem. In the second lecture we will present the first results of existence of 1-dimensional invariant manifolds for tangent to the identity biholomorphisms obtained by Ecalle/Hakim and Abate, connecting them to the corresponding results for vector fields. In the third lecture we will discuss two extensions of the previous results, obtained in collaboration with Jasmin Raissy, Fernando Sanz, Javier Ribon, Rudy Rosas and Liz Vivas.[-]

In these lectures, we will examine a series of conjectures about the geometry of preperiodic points for endomorphisms of $ \mathbb{P}^{N}$. Lecture 1 will focus on the Dynamical ManinMumford Conjecture (DMM), formulated by Shouwu Zhang in the 1990s as an extension of the well-known Manin-Mumford Conjecture (which investigated the geometry of torsion points in abelian varieties and was proved in the early 1980s by Raynaud). The DMM aims to classify the subvarieties of $\mathbb{P}^{N}$ containing a Zariski-dense set of preperiodic points. Lectures 2 and 3 will be devoted to conjectures that treat families of maps on $\mathbb{P}^{N}$. One conjecture in particular was inspired by the recently-proved ”Relative Manin-Mumford” theorem of Gao-Habegger for abelian varieties, but the dynamical version turns out to be closely related to the study of dynamical stability and to contain many previously-existing questions/conjectures/results about moduli spaces of maps on $\mathbb{P}^{N}$. These lectures are based on joint work with Myrto Mavraki.[-]

In these lectures, we will examine a series of conjectures about the geometry of preperiodic points for endomorphisms of $ \mathbb{P}^{N}$. Lecture 1 will focus on the Dynamical ManinMumford Conjecture (DMM), formulated by Shouwu Zhang in the 1990s as an extension of the well-known Manin-Mumford Conjecture (which investigated the geometry of torsion points in abelian varieties and was proved in the early 1980s by Raynaud). The DMM aims to classify the subvarieties of $\mathbb{P}^{N}$ containing a Zariski-dense set of preperiodic points. Lectures 2 and 3 will be devoted to conjectures that treat families of maps on $\mathbb{P}^{N}$. One conjecture in particular was inspired by the recently-proved ”Relative Manin-Mumford” theorem of Gao-Habegger for abelian varieties, but the dynamical version turns out to be closely related to the study of dynamical stability and to contain many previously-existing questions/conjectures/results about moduli spaces of maps on $\mathbb{P}^{N}$. These lectures are based on joint work with Myrto Mavraki.[-]