Simion Filip
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View article: Measure rigidity for generalized u-Gibbs states and stationary measures via the factorization method
Measure rigidity for generalized u-Gibbs states and stationary measures via the factorization method Open
We obtain measure rigidity results for stationary measures of random walks generated by diffeomorphisms, and for actions of $\operatorname{SL}(2,\mathbb{R})$ on smooth manifolds. Our main technical result, from which the rest of the theore…
View article: Finiteness of totally geodesic hypersurfaces
Finiteness of totally geodesic hypersurfaces Open
We prove that a closed negatively curved analytic Riemannian manifold that contains infinitely many totally geodesic hypersurfaces is isometric to an arithmetic hyperbolic manifold. Equivalently, any closed analytic Riemannian manifold wit…
View article: Translation surfaces: Dynamics and Hodge theory
Translation surfaces: Dynamics and Hodge theory Open
A translation surface is a multifaceted object that can be studied with the tools of dynamics, analysis, or algebraic geometry. Moduli spaces of translation surfaces exhibit equally rich features. This survey provides an introduction to th…
View article: The volume of a divisor and cusp excursions of geodesics in hyperbolic manifolds
The volume of a divisor and cusp excursions of geodesics in hyperbolic manifolds Open
We give a complete description of the behavior of the volume function at the boundary of the pseudoeffective cone of certain Calabi-Yau complete intersections known as Wehler N-folds. We find that the volume function exhibits a pathologica…
View article: Global properties of some weight 3 variations of Hodge structure
Global properties of some weight 3 variations of Hodge structure Open
We survey results on the global geometry of variations of Hodge structure with Hodge numbers (1, 1, 1, 1). Included are uniformization results of domains in flag manifolds, a strong Torelli theorem, as well as the formula for the sum of Ly…
View article: Gaps in the support of canonical currents on projective K3 surfaces
Gaps in the support of canonical currents on projective K3 surfaces Open
We construct examples of canonical closed positive currents on projective K3 surfaces that are not fully supported on the complex points. The currents are the unique positive representatives in their cohomology classes and have vanishing s…
View article: Canonical currents and heights for K3 surfaces
Canonical currents and heights for K3 surfaces Open
We construct canonical positive currents and heights on the boundary of the ample cone of a K3 surface. These are equivariant for the automorphism group and fit together into a continuous family, defined over an enlarged boundary of the am…
View article: Uniformization of some weight 3 variations of Hodge structure, Anosov representations, and Lyapunov exponents
Uniformization of some weight 3 variations of Hodge structure, Anosov representations, and Lyapunov exponents Open
We develop a class of uniformizations for certain weight 3 variations of Hodge structure (VHS). The analytic properties of the VHS are used to establish a conjecture of Eskin, Kontsevich, Möller, and Zorich on Lyapunov exponents. Additiona…
View article: A cyclotomic family of thin hypergeometric monodromy groups in ${Sp}_4(\mathbb{R})$
A cyclotomic family of thin hypergeometric monodromy groups in ${Sp}_4(\mathbb{R})$ Open
We exhibit an infinite family of discrete subgroups of ${Sp}_4(\mathbb R)$ which have a number of remarkable properties. Our results are established by showing that each group plays ping-pong on an appropriate set of cones. The groups aris…
View article: A cyclotomic family of thin hypergeometric monodromy groups in\n ${Sp}_4(\\mathbb{R})$
A cyclotomic family of thin hypergeometric monodromy groups in\n ${Sp}_4(\\mathbb{R})$ Open
We exhibit an infinite family of discrete subgroups of ${Sp}_4(\\mathbb R)$\nwhich have a number of remarkable properties. Our results are established by\nshowing that each group plays ping-pong on an appropriate set of cones. The\ngroups …
View article: Geometry and dynamics on Riemann and K3 surfaces
Geometry and dynamics on Riemann and K3 surfaces Open
Surfaces are some of the simplest yet geometrically rich manifolds. Geometric structures on surfaces illuminate their topology and are useful for studying dynamical systems on surfaces. We illustrate below how some of these concepts blend …
View article: Canonical currents and heights for K3 surfaces
Canonical currents and heights for K3 surfaces Open
We construct canonical positive currents and heights on the boundary of the ample cone of a K3 surface. These are equivariant for the automorphism group and fit together into a continuous family, defined over an enlarged boundary of the am…
View article: Kummer rigidity for K3 surface automorphisms via Ricci-flat metrics
Kummer rigidity for K3 surface automorphisms via Ricci-flat metrics Open
We give an alternative proof of a result of Cantat and Dupont, showing that any automorphism of a K3 surface with measure of maximal entropy in the Lebesgue class must be a Kummer example. Our method exploits the existence of Ricci-flat me…
View article: Asymptotic shifting numbers in triangulated categories
Asymptotic shifting numbers in triangulated categories Open
We introduce invariants, called shifting numbers, that measure the asymptotic amount by which an autoequivalence of a triangulated category translates inside the category. The invariants are analogous to Poincare translation numbers that a…
View article: On pseudo-Anosov autoequivalences
On pseudo-Anosov autoequivalences Open
Motivated by results of Thurston, we prove that any autoequivalence of a triangulated category induces a filtration by triangulated subcategories, provided the existence of Bridgeland stability conditions. The filtration is given by the ex…
View article: Tropical Dynamics of Area-Preserving Maps
Tropical Dynamics of Area-Preserving Maps Open
We consider a class of area-preserving, piecewise affine maps on the 2-sphere. These maps encode degenerating families of K3 surface automorphisms and are profitably studied using techniques from tropical and Berkovich geometries.
View article: Tropical dynamics of area-preserving maps
Tropical dynamics of area-preserving maps Open
We consider a class of area-preserving, piecewise affine maps on the 2-sphere. These maps encode degenerating families of K3 surface automorphisms and are profitably studied using techniques from tropical and Berkovich geometries.
View article: The algebraic hull of the Kontsevich–Zorich cocycle
The algebraic hull of the Kontsevich–Zorich cocycle Open
We compute the algebraic hull of the Kontsevich-Zorich cocycle over any GL^+_2(R) invariant subvariety of the Hodge bundle, and derive from this finiteness results on such subvarieties.
View article: Quaternionic covers and monodromy of the Kontsevich–Zorich cocycle in orthogonal groups
Quaternionic covers and monodromy of the Kontsevich–Zorich cocycle in orthogonal groups Open
We give an example of a Teichmüller curve which contains, in a factor of its monodromy, a group which was not observed before. Namely, it has Zariski closure equal to the group SO^*(6) in its standard representation; up to finite index, th…
View article: Notes on the multiplicative ergodic theorem
Notes on the multiplicative ergodic theorem Open
The Oseledets multiplicative ergodic theorem is a basic result with numerous applications throughout dynamical systems. These notes provide an introduction to this theorem, as well as subsequent generalizations. They are based on lectures …
View article: Some analogies between flat and K3 surfaces
Some analogies between flat and K3 surfaces Open
Among all complex two-dimensional manifolds, K3 surfaces are distinguished for having a wealth of extra structures. They admit dynamically interesting automorphisms, have Ricci-flat metrics (by Yau's solution of the Calabi conjecture) and …
View article: Counting special Lagrangian fibrations in twistor families of K3 surfaces
Counting special Lagrangian fibrations in twistor families of K3 surfaces Open
The number of closed billiard trajectories in a rational-angled polygon grows quadratically in the length. This paper gives an analogue on K3 surfaces, by considering special Lagrangian tori. The analogue of the angle of a billiard traject…
View article: Zero Lyapunov exponents and monodromy of the Kontsevich–Zorich cocycle
Zero Lyapunov exponents and monodromy of the Kontsevich–Zorich cocycle Open
We describe all the situations in which the Kontsevich-Zorich cocycle has\nzero Lyapunov exponents. Confirming a conjecture of Forni, Matheus, and Zorich,\nthis only occurs when the cocycle satisfies additional geometric constraints.\nWe a…
View article: Teichmuller Dynamics and Hodge Theory
Teichmuller Dynamics and Hodge Theory Open
This thesis is concerned with applications of Hodge theory in Teichmuller dynamics. Recall that the moduli space pairs (X, ω) of Riemann surfaces with a holomorphic 1-form carries a natural action of the group SL(2,R). The diagonal subgrou…