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Kuntal Banerjee

Publications and source records attributed to Kuntal Banerjee.

9 recordsLinked to original sources
arXiv2026

Reversibility and symmetry of affine toral automorphisms

We study reversibility and strong reversibility of affine automorphisms of the two-torus, written as $f_{A,\bar{a}}(\bar{x})=A\bar{x}+\bar{a} \ (\mathrm{mod}\ \mathbb{Z}^2)$. We derive explicit criteria for the reversibility of such maps in terms of the matrix $A$ and the translation $\bar{a}$. If $1$ is not an eigenvalue of $A$, reversibility of the affine map coincides with reversibility of $A$. When $1$ is an eigenvalue, additional arithmetic obstructions appear. We also provide a simple geometric condition, based on Pick's Theorem, that guarantees the existence of fixed points, along with a description of the dynamics of affine toral automorphisms. We also compute the entropy and characterize when conjugacy classes in the affine group are finite or uncountable.

math.DS↗
arXiv2025

On the Piecewise Linear Perturbations of the Doubling Map

Inspired by the 2007 work by M.~Misiurewicz and A.~Rodrigues [Double Standard Maps, M. Misiurewicz, A. Rodrigues, Communications in Mathematical Physics], we consider a family of circle maps that are perturbations of the doubling map on the circle by a piecewise linear map. We call this the \textit{piecewise linear perturbation of the doubling map} (PLPDM) and it is given by the formula, $f_{a,b}(x)= \displaystyle \bigl(2x+a+\dfrac{b}{2} S(x) \bigr) // 1 \quad {\text {for }} x, a, b \in [0,1] $, where $y // 1$ means $y \mod 1$ (or simply, the fractional part of $y$) and $S(x)$ is the piecewise linear approximation of $\sin 2\pi(x-1/4)$. The map $S(x)$ is called the straight sine map. Define the hyperbolic set, $\mathcal{H}= \{ (a,b) \in \mathbb{R}/\mathbb{Z} \times [0,1] : f_{a,b} \text{ has an attracting cycle} \}$. Tongues are defined as the components of $\mathcal{H}$ that touch the ceiling $\{b=1\}$ in a non degenerate interval. Any other component is referred to as an Eye. We show the uniqueness of the attracting cycle of $f_{a,b}$ for $(a,b) \in \mathcal{H}$. We then define \textit{type} and prove the existence of the tongues of all types. We also show how combinatorics of the attracting orbit determines if the component is a tongue or an eye. We show that $f_{a,b}$ is conjugate to the doubling map if $(a,b) \notin \overline{\mathcal{H}}$. Some experimental proof of the existence of eyes in the parameter space corresponding to different combinatorics will be shown.

math.DS↗
arXiv2025

Uniformization of tongues in Double Standard Map family and variation of maximal chaotic sets

We study hyperbolic components, also known as tongues, in the Double Standard Map family comprising circle maps of the form: \begin{align*} f_{a,b}(x)=\left(2x+a+\dfrac{b}{\pi} \sin(2\pi x)\right) \mod 1,\ a \in \mathbb{R}/\mathbb{Z},\ 0 \leq b \leq 1. \end{align*} We prove simple connectedness of tongues by providing a dynamically natural real-analytic uniformization for each tongue. For maps in a tongue, we characterize the unique maximal subset of the circle on which $f_{a,b}$ is Devaney chaotic. We also show that the Hausdorff dimension of this maximal chaotic set varies real-analytically inside a tongue.

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arXiv2024

Very stable and wobbly loci for elliptic curves

We explore very stable and wobbly bundles, twisted in a particular sense by a line bundle, over complex algebraic curves of genus $1$. We verify that twisted stable bundles on an elliptic curve are not very stable for any positive twist. We utilize semistability of trivially twisted very stable bundles to prove that the wobbly locus is always a divisor in the moduli space of semistable bundles on a genus $1$ curve. We prove, by extension, a conjecture regarding the closedness and dimension of the wobbly locus in this setting. This conjecture was originally formulated by Drinfeld in higher genus.

math.AG↗
arXiv2023

A generalized spectral correspondence

We explore a strong categorical correspondence between isomorphism classes of sheaves of arbitrary rank on a given algebraic curve and twisted pairs on another algebraic curve, mostly from a linear-algebraic standpoint. In a particular application, we realize a generic elliptic curve as a spectral cover of the complex projective line $\mathbb{P}^1$ and then construct examples of cyclic pairs and co-Higgs bundles over $\mathbb{P}^1$. By appealing to a composite push-pull projection formula, we conjecture an iterated version of spectral correspondence. We prove this conjecture for a particular class of spectral covers of $\mathbb {P}^1$ through Galois-theoretic arguments. The proof relies upon a classification of Galois groups into primitive and imprimitive types. In this context, we revisit a classical theorem of Ritt.

math.AG↗
arXiv2019

Tips of Tongues in the Double Standard Family

We answer a question raised by Misiurewicz and Rodrigues concerning the family of degree 2 circle maps $F_λ:\mathbb{R}/\mathbb{Z}\to \mathbb{R}/\mathbb{Z}$ defined by \[F_λ(x) := 2x + a+ \frac{b}π \sin(2πx){\quad\text{with}\quad} λ:=(a,b)\in \mathbb{R}/\mathbb{Z}\times (0,1).\] We prove that if $F_λ^{\circ n}-{\rm id}$ has a zero of multiplicity $3$ in $\mathbb{R}/\mathbb{Z}$, then there is a system of local coordinates $(α,β):W\to \mathbb{R}^2$ defined in a neighborhood $W$ of $λ$, such that $α(λ) =β(λ)=0$ and $F_μ^{\circ n} - {\rm id}$ has a multiple zero with $μ\in W$ if and only if $β^3(μ) = α^2(μ)$. This shows that the tips of tongues are regular cusps.

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arXiv2014

Boundaries of the Arnol'd tongues and the standard family

For a family $(F_{t,a} : x \mapsto x + t + aϕ(x))$ of increasing homeomorphisms of $\mathbb R$ with $ϕ$ being Lipschitz continuous of period 1, there is a parameter space consisting of the values $(t,a)$ such that the map $F_{t,a}$ is strictly increasing and it induces an orientation preserving circle homeomorphism. For each $θ\in \mathbb R$ there is an \textsf{Arnol'd tongue} $\mathcal T_θ$ of \textsf{translation number} $θ$ in the parameter space. Given a rational $p/q$, it is shown that the boundary $\partial \mathcal T_{p/q}$ is a union of two Lipschitz curves which intersect at $a=0$ and there can be a non zero angle between them. In this direction we compute the first order asymptotic expansion of the boundaries of the rational and irrational tongues in the parameter space around $a=0$. For the standard family $(S_{t,a} : x \mapsto x + t + a \sin(2πx))$, the boundary curves of $\mathcal T_{p/q}$ have the same tangency at $a=0$ for $q\ge 2$ and it is known that $q$ is their \textsf{order of contact}. Using the techniques of \textsf{guided} and \textsf{admissible family}, we give a new proof of this. In particular we relate this to the \textsf{parabolic multiplicity} of the map $s_{p/q} : z \mapsto e^{i2πp/q}ze^{πz}$ at $0$.

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arXiv2009

On the widths of the Arnol'd Tongues

Let $F: \mathbb R \to \mathbb R$ be a real analytic increasing diffeomorphism with $F-{\rm Id}$ being 1 periodic. Consider the translated family of maps $(F_t :\mathbb R \to \mathbb R)_{t\in \mathbbR}$ defined as $F_t(x)=F(x)+t$. Let ${\rm Trans}(F_t)$ be the translation number of $F_t$ defined by: \[{\rm Trans}(F_t) := \lim_{n\to +\infty}\frac{F_t^{\circ n}-{\rm Id}}{n}.\] Assume there is a Herman ring of modulus $2τ$ associated to $F$ and let $p_n/q_n$ be the $n$-th convergent of ${\rm Trans}(F)$. Denoting $\ell_θ$ as the length of the interval $\{t\in \mathbb R | {\rm Trans}(F_t)=θ\}$, we prove that the sequence $(\ell_{p_n/q_n})$ decreases exponentially fast with respect to $q_n$. More precisely \[\limsup_{n \to \infty} \frac{1}{q_n} \log {\ell_{p_n/q_n}} \le -2πτ.\]

math.DS↗