arXiv subjects

Mayuresh Londhe

Publications and source records attributed to Mayuresh Londhe.

7 recordsLinked to original sources

arXiv2026

Arithmetic probability measures

We consider probability measures on compact subsets of the complex plane arising as limiting distributions of Galois conjugates of certain algebraic integers. These measures are characterized by infinitely many integral inequalities, one for each nonzero integer polynomial. We study a broad family of measures given by particular convex combinations of harmonic and equilibrium measures. We then show that, for measures in this family, it suffices to verify two conditions to guarantee all the required inequalities. Under additional arithmetic constraints on the underlying compact sets, we further show that the infinite collection of inequalities is equivalent to checking the inequalities associated with two explicitly specified integer polynomials.

math.NT↗

arXiv2025

Around Fekete's theorem

A classical result of Fekete gives necessary conditions on a compact set in the complex plane so that it contains infinitely many sets of conjugate algebraic integers. For such sets, we demonstrate the existence of a sequence of algebraic integers such that most of their conjugates eventually lie near the set, while maintaining a bound on heights. Finally, we examine properties satisfied by the limiting distribution of a sequence of algebraic numbers.

math.CV↗

arXiv2025

Equidistribution of the conjugates of algebraic units

We prove an equidistribution result for the zeros of polynomials with integer coefficients and simple zeros. Specifically, we show that the normalized zero measures associated with a sequence of such polynomials, having small height relative to a certain compact set in the complex plane, converge to a canonical measure on the set. In particular, this result gives an equidistribution result for the conjugates of algebraic units, in the spirit of Bilu's work. Our approach involves lifting these polynomials to polynomial mappings in two variables and proving an equidistribution result for the normalized zero measures in this setting.

math.CV↗

arXiv2024

Ergodicity in the dynamics of holomorphic correspondences

This paper studies ergodic properties of certain measures arising in the dynamics of holomorphic correspondences. These measures, in general, are not invariant in the classical sense of ergodic theory. We define a notion of ergodicity, and prove a version of Birkhoff's ergodic theorem in this setting. We also show the existence of ergodic measures when a holomorphic correspondence is defined on a compact complex manifold. Lastly, we give an explicit class of dynamically interesting measures that are ergodic as in our definition.

math.DS↗

arXiv2022

Recurrence in the dynamics of meromorphic correspondences and holomorphic semigroups

This paper studies recurrence phenomena in iterative holomorphic dynamics of certain multi-valued maps. In particular, we prove an analogue of the Poincaré recurrence theorem for meromorphic correspondences with respect to certain dynamically interesting measures associated with them. Meromorphic correspondences present a significant measure-theoretic obstacle: the image of a Borel set under a meromorphic correspondence need not be Borel. We manage this issue using the Measurable Projection Theorem, which is an aspect of descriptive set theory. We also prove a result concerning invariance properties of the supports of the measures mentioned.

math.CV↗

arXiv2021

On a spectral version of Cartan's theorem

For a domain $Ω$ in the complex plane, we consider the domain $S_n(Ω)$ consisting of those $n\times n$ complex matrices whose spectrum is contained in $Ω$. Given a holomorphic self-map $Ψ$ of $S_n(Ω)$ such that $Ψ(A)=A$ and the derivative of $Ψ$ at $A$ is identity for some $A\in S_n(Ω)$, we investigate when the map $Ψ$ would be spectrum-preserving. We prove that if the matrix $A$ is either diagonalizable or non-derogatory then for most domains $Ω$, $Ψ$ is spectrum-preserving on $S_n(Ω)$. Further, when $A$ is arbitrary, we prove that $Ψ$ is spectrum-preserving on a certain analytic subset of $S_n(Ω)$.

math.CV↗

arXiv2021

Dynamics of polynomial semigroups: measures, potentials, and external fields

In this paper, we give a description of a natural invariant measure associated with a finitely generated polynomial semigroup (which we shall call the Dinh--Sibony measure) in terms of potential theory. This requires the theory of logarithmic potentials in the presence of an external field, which, in our case, is explicitly determined by the choice of a set of generators. Along the way, we establish the continuity of the logarithmic potential for the Dinh--Sibony measure, which might be of independent interest. We then use the $F$-functional of Mhaskar and Saff to discuss bounds on the capacity and diameter of the Julia sets of such semigroups.

math.CV↗