arXiv research

Every weak Perron number is realized as a stretch factor of a homeomorphism on a surface.

Using an idea of Doug Lind, we give a lower bound for the Perron-Frobenius degree of a Perron number that is not totally-real. As an application, we prove that there are cubic Perron numbers whose Perron-Frobenius degrees are arbitrary large; a result known to Lind, McMullen and Thurston. A similar result is proved for…

Modernizes Thurston's proof of entropy theorem for traintrack maps.

Characterizes bi-Perron numbers with specific Galois conjugates.

The paper explores growth rates and Perron numbers in Coxeter systems with low-dimensional Davis complexes.

In this paper we consider the growth rates of 3-dimensional hyperbolic Coxeter polyhedra some of its dihedral angles are $\fracπ{m}$ for $m\geq{7}$. By combining with the classical result by Parry \cite{Pa} and the main result of \cite{Y}, we prove that the growth rates of 3-dimensional hyperbolic Coxeter groups are Pe…

We define a large class of abstract Coxeter groups, that we call $\infty$--spanned, and for which the word growth rate and the geodesic growth rate appear to be Perron numbers. This class contains a fair amount of Coxeter groups acting on hyperbolic spaces, thus corroborating a conjecture by Kellerhals and Perren. We a…

We prove existence of solutions to boundary value problems and obstacle problems for degenerate-elliptic, linear, second-order partial differential operators with partial Dirichlet boundary conditions using a new version of the Perron method. The elliptic operators considered have a degeneracy along a portion of the do…

In this note, we deduce a partial answer to the question in the title. In particular, we show that asymptotically almost all bi-Perron algebraic unit whose characteristic polynomial has degree at most $2n$ do not correspond to dilatations of pseudo-Anosov maps on a closed orientable surface of genus $n$ for $n\geq 10$.…

We construct infinite series of non-simple ideal hyperbolic Coxeter 4-polytopes whose growth rates are Perron numbers. This infinite series is the first example of such a non-compact infinite polytopal series.

Deep learning framework for kernel methods using RKHM and Perron-Frobenius operators.

Edge subdivision affects the Perron eigenvalue of tree Ricci matrices.

In this paper, we give sufficient conditions for a Perron number, given as the leading eigenvalue of an aperiodic matrix, to be a pseudo-Anosov dilatation of a compact surface. We give an explicit construction of the surface and the map when the sufficient condition is met.

We apply stochastic Perron's method to a singular control problem where an individual targets at a given consumption rate, invests in a risky financial market in which trading is subject to proportional transaction costs, and seeks to minimize her probability of lifetime ruin. Without relying on the dynamic programming…

Solves Monge-Ampère equations on compact Hessian manifolds using the Perron method.

Study complex Monge-Ampère flows on Kähler manifolds using Perron method.

Study of hypersurfaces with specific expansion properties.

We study arithmetic properties of the growth rates of cofinite 3-dimensional hyperbolic Coxeter groups whose dihedral angles are of the form $\fracπ{m}$ for $m=2,3,4,5,6$ and show that the growth rates are always Perron numbers.

Lehmer's question is equivalent to one about generalized growth rates of Lefschetz numbers of iterated pseudo-Anosov surface homeomorphisms. One need consider only homeomorphisms that arise as monodromies of fibered knots in lens spaces L(n,1), n>0. Lehmer's question for Perron polynomials is equivalent to one about ge…

A pseudo-Anosov surface automorphism $φ$ has associated to it an algebraic unit $λ_φ$ called the dilatation of $φ$. It is known that in many cases $λ_φ$ appears as the spectral radius of a Perron-Frobenius matrix preserving a symplectic form $L$. We investigate what algebraic units could potentially appear as dilatatio…

We obtain asymptotic counting results with error terms for complex orthospectrum for Schottky groups and orbit counting function for quadratic polynomials. Moreover, we prove equidistribution of holonomy associated to these dynamical systems. Our results are obtained by considering generalized $L$-functions coming from…

The k-dimensional Dehn (or isoperimetric) function of a group bounds the volume of efficient ball-fillings of k-spheres mapped into k-connected spaces on which the group acts properly and cocompactly; the bound is given as a function of the volume of the sphere. We advance significantly the observed range of behavior f…

Paper proves existence and uniqueness of circle patterns on surfaces with assigned geodesic curvatures.

The symplectic representation of mapping classes is not surjective for certain types of mapping classes.

We study the set G of growth rates of of ideal Coxeter groups in hyperbolic 3-space which consists of real algebraic integers greater than 1. We show that (1) G is unbounded above while it has the minimum, (2) any element of G is a Perron number, and (3) growth rates of of ideal Coxeter groups with $n$ generators are l…

Study of endperiodic maps on infinite graphs, proving homotopy and eigenvalue properties.

We extend the stochastic Perron method to analyze the framework of stochastic target games, in which one player tries to find a strategy such that the state process almost surely reaches a given target no matter which action is chosen by the other player. Within this framework, our method produces a viscosity sub-solut…

Existence and uniqueness of discrete Einstein metrics on trees proven.

In this paper, we adapt stochastic Perron's method to analyze a stochastic target problem with unbounded controls in a jump diffusion set-up. With this method, we construct a viscosity sub-solution and super-solution to the associated Hamiltonian-Jacobi-Bellman (HJB) equations. Under comparison principles, uniqueness o…

Paper finds principal eigenvalue for infinity Laplacian in metric spaces.

In [6], Kellerhals and Perren conjectured that the growth rates of the reflection groups given by hyperbolic Coxeter polyhedra are always Perron numbers. We prove that this conjecture is always true for the case of ideal Coxeter polyhedra in $\mathbb{H}^3$. We also find out the ideal Coxeter polyhedron in $\mathbb{H}^3…

The development of a metric for structural data is a long-term problem in pattern recognition and machine learning. In this paper, we develop a general metric for comparing nonlinear dynamical systems that is defined with Perron-Frobenius operators in reproducing kernel Hilbert spaces. Our metric includes the existing …

Using the Perron method, we prove the existence of hypersurfaces of prescribed special Lagrangian curvature with prescribed boundary inside complete Riemannian manifolds of non-positive curvature.

We shall use the classical Perron envelope method to show a general existence theorem to degenerate complex Monge-Ampère type equations on compact Kähler manifolds.

Let $G$ be a finitely generated group with a finite generating set $S$. For $g\in G$, let $l_S(g)$ be the length of the shortest word over $S$ representing $g$. The growth series of $G$ with respect to $S$ is the series $A(t) = \sum_{n=0}^\infty a_n t^n$, where $a_n$ is the number of elements of $G$ with $l_S(g)=n$. If…

Study of complex Hessian equations using subharmonic functions and geodesics.

Let $f\colon M\to N$ be a continuous map between closed irreducible graph manifolds with infinite fundamental group. Perron and Shalen showed that if $f$ induces a homology equivalence on all finite covers, then $f$ is in fact homotopic to a homeomorphism. Their proof used the statement that every graph manifold is fin…

Investor aims to meet financial goals with deadlines and target amounts, considering stock trading costs.

We describe a novel technique for solving the Plateau problem for constant curvature hypersurfaces based on recent work of Harvey and Lawson. This is illustrated by an existence theorem for hypersurfaces of constant Gaussian curvature in $\Bbb{R}^{n+1}$.

This paper aims to make a new contribution to the study of lifetime ruin problem by considering investment in two hedge funds with high-watermark fees and drift uncertainty. Due to multi-dimensional performance fees that are charged whenever each fund profit exceeds its historical maximum, the value function is expecte…

Study on self-consuming generative models with diverse human curation, focusing on convergence and stability.

IGNN captures long-range graph dependencies using fixed-point equations.

RaNNDy uses randomized neural networks to learn transfer operators efficiently.

New methods for clustering graphs using spectral analysis.

A framework for goal-based investing with penalties for fund transfers.

Paper introduces RKHM for more explicit variable structures analysis.

We study generalization properties of weakly supervised learning. That is, learning where only a few "strong" labels (the actual target of our prediction) are present but many more "weak" labels are available. In particular, we show that having access to weak labels can significantly accelerate the learning rate for th…

The paper defines new homotopy relations on knot projections and classifies certain knot types.