http://hdl.handle.net/11422/57 2026-07-30T18:34:39Z http://hdl.handle.net/11422/29866 Title: Hiperbolicidade não uniforme para fluxos singulares e difeomorfismos Author(s)/Inventor(s): Mesquita, Gabriel Reis Machado de Advisor: Salgado, Luciana Silva Abstract: This work aims the study of some forms of weak hyperbolicity and some of their dynamical and ergodic properties, such as the relation between strong and weak hyperbolicity, dominated splittings and partial hyperbolicity, for differential dynamical systems (here understood as diffeomorphisms and flows from vector fields). In particular, we study the concepts of singular/sectional hyperbolicity and non-uniformly sectional hyperbolicity for singular flows. Publisher: Universidade Federal do Rio de Janeiro Type: Dissertação 2025-03-13T00:00:00Z http://hdl.handle.net/11422/29863 Title: The arithmetic and geometry of fibrations in rational and K3 surfaces Author(s)/Inventor(s): Meira, Felipe Zingali Advisor: Salgado, Cecília Abstract: This thesis consists of 3 chapters. The first chapter deals with introducing the main theory used in the subsequent chapters. Chapter 1 introduces basic concepts in the theory of elliptic surfaces, such as its main definition, the correspondence with elliptic curves over function fields and the classification of distinct fiber types. Furthermore, specific results on rational and K3 elliptic surfaces are presented. In Chapter 2, we study the rank of an elliptic curve E, defined over the function field k(T), which is given by a Weierstrass equation with coefficients of degree at most 2. This is done by studying the induced conic and elliptic fibrations on its Kodaira–Néron model R. In Chapter 3, we study K3 surfaces X with a non-symplectic automorphism σ ∈ Aut(X) of prime order. We classify distinct elliptic fibrations on X with respect to the action of σ on its respective fibers. Each type of elliptic fibrations is related to a linear system on the minimal resolution of the quocient ˜R = X/σ. When the action of σ on the Néron–Severi group of X fixes the fiber class of an elliptic fibration π, this method allows us to determine which Kodaira types are admissible as its reducible fibers. Furthermore, we are able to determine equations for its generic fiber. Publisher: Universidade Federal do Rio de Janeiro Type: Tese 2025-06-17T00:00:00Z http://hdl.handle.net/11422/29681 Title: Weighted composition operators that are chaotic in the sense of Li and Yorke Author(s)/Inventor(s): Vasconcellos, Fernanda Mendonça de Advisor: Bernardes Júnior, Nilson da Costa Abstract: We establish complete characterizations of the notion of Li-Yorke chaos for weighted composition operators on C0(X) spaces and on Lp(μ) spaces (1 ≤ p < ∞). As a consequence, we obtain simple characterizations of the Li-Yorke chaotic weighted shifts on c0 and on ℓp (1 ≤ p < ∞) that complement previously known results. We also investigate the notion of Li-Yorke chaos for weighted shifts on Fr´echet sequence spaces. As applications, we obtain characterizations of the Li-Yorke chaotic weighted shifts on K¨othe sequence spaces depending only on the entries of the K¨othe matrix and the weights of the shift. Publisher: Universidade Federal do Rio de Janeiro Type: Tese 2025-02-25T00:00:00Z http://hdl.handle.net/11422/28361 Title: Local and global well-posedness for some dispersive equations in modulation spaces Author(s)/Inventor(s): Balvin, Fidel Cuba Advisor: Paredes, Xavier Carvajal Abstract: We consider the initial value problem (IVP) associated with a system consisting of modified Korteweg-de Vries (mKdV) type equations ∂tv + ∂3 xv + ∂x(vw2) = 0, v(x, 0) = v0(x), ∂tw + α∂3 xw + ∂x(v2w) = 0, w(x, 0) = w0(x), (3) using the theory of T. Oh and Y. Wang [31] we derive trilinear estimates and use it in the contraction argument, so we prove local well-posedness (LWP) to the IVP (3) for given data in the modulation spaces M2,p s (R) ×M2,p s (R), with s ≥ 1 4 , 2 ≤ p < ∞ and α ̸= 0. Also, at the second part of this work we consider the initial value problem (IVP) associated to the higher order nonlinear Schrödinger (h-NLS) equation with variable coefficients ∂tu + ia(t)∂2 xu + b(t)∂3 xu + ic(t)|u|2u + c1(t)|u|2∂xu = 0, u(x, 0) = u0(x), (4) using the theory of Killip, Visan, Zhang [24], Oh, Wang [30], [31], Carvajal, Gamboa and Santos [8], so we establish a priori estimates and prove global well-posedness (GWP) to the IVP (4) for given data in the modulation spaces M2,p s (R), with s ≥ 1 4 and 2 ≤ p < ∞. Publisher: Universidade Federal do Rio de Janeiro Type: Tese 2024-11-19T00:00:00Z