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Geometric Pressure for Multimodal Maps of the Interval
This paper is an interval dynamics counterpart of three theories founded earlier by the authors, S. Smirnov and others in the setting of the iteration of rational maps on the Riemann sphere: the equivalence of several notions of non-uniform hyperbolicity, Geometric Pressure, and Nice Inducing Schemes methods leading to results in thermodynamical formalism. The authors work in a setting of generalized multimodal maps, that is, smooth maps f of a finite union of compact intervals Iˆ in R into R with non-flat critical points, such that on its maximal forward invariant set K the map f is topologically transitive and has positive topological entropy. They prove that several notions of non-unifor...
Potential Theory and Dynamics on the Berkovich Projective Line
The purpose of this book is to develop the foundations of potential theory and rational dynamics on the Berkovich projective line over an arbitrary complete, algebraically closed non-Archimedean field. In addition to providing a concrete and ``elementary'' introduction to Berkovich analytic spaces and to potential theory and rational iteration on the Berkovich line, the book contains applications to arithmetic geometry and arithmetic dynamics. A number of results in the book are new, and most have not previously appeared in book form. Three appendices--on analysis, $\mathbb{R}$-trees, and Berkovich's general theory of analytic spaces--are included to make the book as self-contained as possib...
A Local Relative Trace Formula for the Ginzburg-Rallis Model: The Geometric Side
Following the method developed by Waldspurger and Beuzart-Plessis in their proofs of the local Gan-Gross-Prasad conjecture, the author is able to prove the geometric side of a local relative trace formula for the Ginzburg-Rallis model. Then by applying such formula, the author proves a multiplicity formula of the Ginzburg-Rallis model for the supercuspidal representations. Using that multiplicity formula, the author proves the multiplicity one theorem for the Ginzburg-Rallis model over Vogan packets in the supercuspidal case.
New Complex Analytic Methods in the Study of Non-Orientable Minimal Surfaces in Rn
All the new tools mentioned above apply to non-orientable minimal surfaces endowed with a fixed choice of a conformal structure. This enables the authors to obtain significant new applications to the global theory of non-orientable minimal surfaces. In particular, they construct proper non-orientable conformal minimal surfaces in Rn with any given conformal structure, complete non-orientable minimal surfaces in Rn with arbitrary conformal type whose generalized Gauss map is nondegenerate and omits n hyperplanes of CPn−1 in general position, complete non-orientable minimal surfaces bounded by Jordan curves, and complete proper non-orientable minimal surfaces normalized by bordered surfaces in p-convex domains of Rn.
On the Stability of Type I Blow Up for the Energy Super Critical Heat Equation
The authors consider the energy super critical semilinear heat equation The authors first revisit the construction of radially symmetric self similar solutions performed through an ode approach and propose a bifurcation type argument which allows for a sharp control of the spectrum of the corresponding linearized operator in suitable weighted spaces. They then show how the sole knowledge of this spectral gap in weighted spaces implies the finite codimensional nonradial stability of these solutions for smooth well localized initial data using energy bounds. The whole scheme draws a route map for the derivation of the existence and stability of self-similar blow up in nonradial energy super critical settings.
Time-Like Graphical Models
The author studies continuous processes indexed by a special family of graphs. Processes indexed by vertices of graphs are known as probabilistic graphical models. In 2011, Burdzy and Pal proposed a continuous version of graphical models indexed by graphs with an embedded time structure— so-called time-like graphs. The author extends the notion of time-like graphs and finds properties of processes indexed by them. In particular, the author solves the conjecture of uniqueness of the distribution for the process indexed by graphs with infinite number of vertices. The author provides a new result showing the stochastic heat equation as a limit of the sequence of natural Brownian motions on time-like graphs. In addition, the author's treatment of time-like graphical models reveals connections to Markov random fields, martingales indexed by directed sets and branching Markov processes.
Quadratic Vector Equations on Complex Upper Half-Plane
The authors consider the nonlinear equation −1m=z+Sm with a parameter z in the complex upper half plane H, where S is a positivity preserving symmetric linear operator acting on bounded functions. The solution with values in H is unique and its z-dependence is conveniently described as the Stieltjes transforms of a family of measures v on R. In a previous paper the authors qualitatively identified the possible singular behaviors of v: under suitable conditions on S we showed that in the density of v only algebraic singularities of degree two or three may occur. In this paper the authors give a comprehensive analysis of these singularities with uniform quantitative controls. They also find a universal shape describing the transition regime between the square root and cubic root singularities. Finally, motivated by random matrix applications in the authors' companion paper they present a complete stability analysis of the equation for any z∈H, including the vicinity of the singularities.
Xi Brazilian Topology Meeting
This volume reflects the work of the Brazilian topology community. It also contains the work of some topologists who either collaborate or interact with a Brazilian topologist. Most of the work has been done in algebraic and geometric topology.
Geometric Optics for Surface Waves in Nonlinear Elasticity
This work is devoted to the analysis of high frequency solutions to the equations of nonlinear elasticity in a half-space. The authors consider surface waves (or more precisely, Rayleigh waves) arising in the general class of isotropic hyperelastic models, which includes in particular the Saint Venant-Kirchhoff system. Work has been done by a number of authors since the 1980s on the formulation and well-posedness of a nonlinear evolution equation whose (exact) solution gives the leading term of an approximate Rayleigh wave solution to the underlying elasticity equations. This evolution equation, which is referred to as “the amplitude equation”, is an integrodifferential equation of nonlo...
