Related papers: A renormalization fixed point for Lorenz maps
We develop a Thurston-like theory to characterize geometrically finite rational maps, then apply it to study pinching and plumbing deformations of rational maps. We show that in certain conditions the pinching path converges uniformly and…
Recurrent neural networks (RNNs) are a class of nonlinear dynamical systems often used to model sequence-to-sequence maps. RNNs have excellent expressive power but lack the stability or robustness guarantees that are necessary for many…
Can regularization terms in the training of invertible neural networks lead to known Bayesian point estimators in reconstruction? Invertible networks are attractive for inverse problems due to their inherent stability and interpretability.…
We present a renormalization lemma for certain maps defined on the unit disc of C and taking values in some metric space. We show that the classical renormalization lemmas of Zalcman and Miniowitz can be deduced from our lemma. We also use…
According to Sullivan, a space ${\cal E}$ of unimodal maps with the same combinatorics (modulo smooth conjugacy) should be treated as an infinitely-dimensional Teichm\"{u}ller space. This is a basic idea in Sullivan's approach to the…
Large-$N$ renormalization group equations for one- and two-matrix models are derived. The exact renormalization group equation involving infinitely many induced interactions can be rewritten in a form that has a finite number of coupling…
Self-consistent new renormalization group flow equations for an O(N)-symmetric scalar theory are approximated in next-to-leading order of the derivative expansion. The Wilson-Fisher fixed point in three dimensions is analyzed in detail and…
We construct an approximate renormalization for Hamiltonian systems with two degrees of freedom in order to study the break-up of invariant tori with arbitrary frequency. We derive the equation of the critical surface of the renormalization…
We study the relationship between Lorentz harmonic maps into the hyperbolic plane and spacelike surfaces in anti-de Sitter 3-space. Using loop group techniques, we develop a DPW-type representation for Lorentz harmonic maps and provide an…
We present a scheme for the analytic computation of renormalization functions on the lattice, using a symbolic manipulation computer language. Our first nontrivial application is a new three-loop result for the topological susceptibility.
We consider the Lorenz equations, a system of three dimensional ordinary differential equations modeling atmospheric convection. These equations are chaotic and hard to study even numerically, and so a simpler "geometric model" has been…
We construct an explicit inversion formula for Guillarmou's normal operator on closed surfaces of constant negative curvature. This normal operator can be defined as a weak limit for an "attenuated normal operator", and we prove this…
In this work we study the renormalization operator acting on piecewise smooth homeomorphisms on the circle, that turns out to be essentially the study of Rauzy-Veech renormalizations of generalized interval exchanges maps with genus one. In…
We show that the Feigenbaum-Cvitanovic equation can be interpreted as a linearizing equation, and the domain of analyticity of the Feigenbaum fixed point of renormalization as a basin of attraction. There is a natural decomposition of this…
Entanglement renormalization can be viewed as an encoding circuit for a family of approximate quantum error correcting codes. The logical information becomes progressively more well-protected against erasure errors at larger length scales.…
The Lipschitz constant of the map between the input and output space represented by a neural network is a natural metric for assessing the robustness of the model. We present a new method to constrain the Lipschitz constant of dense deep…
We develop a framework for quantitative convergence analysis of Picard iterations of expansive set-valued fixed point mappings. There are two key components of the analysis. The first is a natural generalization of single-valued averaged…
The multiscale entanglement renormalization ansatz is applied to the study of boundary critical phenomena. We compute averages of local operators as a function of the distance from the boundary and the surface contribution to the ground…
Recently, several authors have studied maps where a function, describing the local diffusion matrix of a diffusion process with a linear drift towards an attraction point, is mapped into the average of that function with respect to the…
We study independent and identically distributed random iterations of continuous maps defined on a connected closed subset $S$ of the Euclidean space $\mathbb{R}^{k}$. We assume the maps are monotone (with respect to a suitable partial…