arXiv subjects

Tarakanta Nayak

Publications and source records attributed to Tarakanta Nayak.

At least 19 recordsLinked to original sources

arXiv2026

Exponential polynomials with Baker omitted value

We consider exponential polynomials of the form $F_l(z)=P_1(z)\exp(Q_1(z))+P_2(z)\exp(Q_2(z))+\cdots+P_{l-1}(z)\exp(Q_{l-1}(z))+P_l(z)$, where $l\geq2$, each $Q_i$ is a non-constant polynomial and each $P_i$, $i=1,2,\ldots,l-1$, is a polynomial (possibly constant), while $P_l$ is a non-constant polynomial. This article studies the topology of the preimages under $F_l$ of neighborhoods of the essential singularity at infinity. We prove that for each neighbourhood $D$ of infinity, the set $F_l ^{-1} (D) $ is connected, and in fact is an infinitely connected domain with all its boundary components bounded. In such a situation, the point at $\infty$ is called the Baker omitted value of the function. Our methods significantly extend those developed by Das and Nayak (in Complex Var. Elliptic Equ. 70:1831-1847, 2025), providing a broader framework for the study of this phenomenon. We further show that these functions do not admit any Baker wandering domain and hence every Fatou component is simply connected. Finally, we conclude by posing some problems arising out of this work.

math.CV↗

arXiv2026

Chebyshev's method applied to polynomials with rotational symmetry

We investigate the dynamics of Chebyshev's method applied to the polynomial family $p_n(z)=z(z^n-1)$ for $n>1$. The resulting map is denoted by $C_n$. It is proved that the immediate basins corresponding to the non-zero roots are unbounded and simply connected. We also show that the Julia set of $C_n$ is connected. It is proved that the immediate basin of the root at the origin exhibits a different behavior: it is unbounded for $n\leq 16$ and bounded for $n\geq 17$. We establish that $C_n$ is convergent whenever $n\leq 16$ or $n$ is odd. Finally, we determine the symmetry group of $C_n$ and prove that it coincides with the symmetry group of the polynomial $p_n$, thereby confirming, for this family, a conjecture proposed by Nayak and Pal.

math.DS↗

arXiv2026

Chebyshev's method for exponential maps

It is proved that the Chebyshev's method applied to an entire function $f$ is a rational map if and only if $f(z) = p(z) e^{q(z)}$, for some polynomials $p$ and $q$. These are referred to as rational Chebyshev maps, and their fixed points are discussed in this article. It is seen that $\infty$ is a parabolic fixed point with multiplicity one bigger than the degree of $q$. Considering $q(z)=p(z)^n+c$, where $p$ is a linear polynomial, $n \in \mathbb{N}$ and $c$ is a non-zero constant, we show that the Chebyshev's method applied to $ pe^q$ is affine conjugate to that applied to $z e^{z^n}$. We denote this by $C_n$. All the finite extraneous fixed points of $C_n$ are shown to be repelling. The Julia set $\mathcal{J}(C_n)$ of $C_n$ is found to be preserved under rotations of order $n$ about the origin. For each $n$, the immediate basin of $0$ is proved to be simply connected. For all $n \leq 16$, we prove that $\mathcal{J}(C_n)$ is connected. For $n$ even, the non-existence of Herman ring and Siegel disk of $C_n$ is proved. Under some additional hypothesis, the same is also proved for odd $n$. The Newton's method applied to $ze^{z^n}$ is found to be conjugate to a polynomial, and its dynamics is also completely determined.

math.DS↗

arXiv2026

A Survey of Baker Wandering Domains

Let $f:\mathbb C\to \widehat{\mathbb C}=\mathbb C \cup\{\infty\}$ be a transcendental meromorphic function (possibly without any pole) with a single essential singularity, and that is chosen to be at $\infty$. The set of points $z\in\mathbb{\widehat{C}}$ such that the family of iterates $\{f^n\}_{n\geq 0}$ is defined and forms a normal family in a neighborhood of $z$ is known as the Fatou set of $f$. For a Fatou component $W$, let $W_j$ denote the Fatou component containing $f^j(W)$. A Fatou component $W$ is called wandering if $W_m\bigcap W_n=\emptyset$ for all $m \neq n$. A wandering domain $W$ of $f$ is called a Baker wandering domain, if each $W_n$ is bounded, multiply connected, and $W_n$ surrounds $0$ for all large $n$ and, dist$(W_n,0)\to\infty$ as $n\to\infty$. This paper surveys the current state of knowledge on Baker wandering domains. We revisit the first example of the Baker wandering domain followed by other examples. The influence of Baker wandering domain on the singular values and dynamics of the function is presented. We also discuss some classes of functions that do not possess any Baker wandering domain. Several problems are proposed throughout the article at relevant places.

math.DS↗

arXiv2026

Newton's method applied to rational functions: Fixed points and Julia sets

For a rational function $R$, let $N_R(z)=z-\frac{R(z)}{R'(z)}.$ Any such $N_R$ is referred to as a Newton map. We determine all the rational functions $R$ for which $N_R$ has exactly two attracting fixed points, one of which is an exceptional point. Further, if all the repelling fixed points of any such Newton map are with multiplier $2$, or the multiplier of the non-exceptional attracting fixed point is at most $\frac{4}{5}$, then its Julia set is shown to be connected. If a polynomial $p$ has exactly two roots, is unicritical but not a monomial, or $p(z)=z(z^n+a)$ for some $a \in \mathbb{C}$ and $n \geq 1$, then we have proved that the Julia set of $N_{\frac{1}{p}}$ is totally disconnected. For the McMullen map $f_λ(z)=z^m - \fracλ{z^n}$, $λ\in \mathbb{C}\setminus \{0\}$ and $m,n \geq 1$, we have proved that the Julia set of $N_{f_λ}$ is connected and is invariant under rotations about the origin of order $m+n$. All the connected Julia sets mentioned above are found to be locally connected.

math.DS↗

arXiv2025

Herman Rings: Structure, Dynamics, and Open Problems

The existence of the Herman ring of a function adds interest and complexity to the dynamics of the function. We present a detailed and understandable summary of the core discoveries and recent developments on the Herman ring of rational and transcendental meromorphic functions. It is demonstrated that the Herman ring is intriguing on its own and valuable in terms of overall dynamics. Finally, the results and potential future research problems are briefly discussed.

math.CV↗

arXiv2025

Chebyshev's method applied to polynomials with two distinct roots

The Julia set of the Chebyshev's method applied to polynomials with exactly two distinct roots is shown to be connected, and its Fatou set is proved to be the union of attracting basins corresponding to the two roots. Further, if the two roots have the same multiplicity then the common boundary of the two immediate basins is proved to be a connected subset of the Julia set.

math.DS↗

arXiv2025

Connectedness of independence attractors of graphs with independence number three

An independent set in a simple graph $G$ is a set of pairwise non-adjacent vertices in $G$. The independence polynomial of $G$, denoted by $I_G$ is defined as $1 + a_1 z + a_2 z^2+\cdots+a_d z^{d}$, where $a_i$ denotes the number of independent sets with cardinality $i$ and $d$ is the cardinality of a largest independent set in $G$. This $d$ is known as the independence number of $G$. Let $G^m$ denote the $m$-times lexicographic product of $G$ with itself. The independence attractor of $G$, denoted by $\mathcal{A}(G)$ is defined as $\mathcal{A}(G) = \lim\limits_{m\rightarrow \infty} \{z: I_{G^m}(z)=0\}$, where the limit is taken with respect to the Hausdorff metric defined on the space of all compact subsets of the plane. This paper investigates the connectedness of the independence attractors of all graphs with independence number three. Let the independence polynomial of $G$ be $1+a_1 z +a_2 z^2 +a_3 z^3$. For $a_1 =3$, $\mathcal{A}(G)$ turns out to be $ \{-1\} \cup \{z: |z+1|=1\} $. For $a_1 >3$, we prove the following. If $a_2 ^2 \leq 3 a_1 a_3$, or $3 a_1 a_3 < a_2 ^2 < 4a_3 (a_1 -1)$ then $\mathcal{A}(G)$ is totally disconnected. For $a_2 ^2 =4a_3 (a_1 -1) $, $\mathcal{A}(G)$ is connected when $a_1 =5$ and is disconnected but not totally disconnected for all other values of $a_1$. If $a_2 ^2 > 4a_3 (a_1 -1)$ then $\mathcal{A}(G)$ can be connected, totally disconnected or disconnected but not totally disconnected depending on further conditions involving $a_1, a_2$ and $a_3$. Examples of graphs exhibiting all the possibilities are provided.

math.CO↗

arXiv2025

Circles and line segments as independence attractors of graphs

By an independent set in a simple graph $G$, we mean a set of pairwise non-adjacent vertices in $G$. The independence polynomial of $G$ is defined as $I_G(z)=a_0 + a_1 z + a_2 z^2+\cdots+a_αz^α$, where $a_i$ is the number of independent sets in $G$ with cardinality $i$ and $α$ is the cardinality of a largest independent set in $G$, known as the independence number of $G$. Let $G^m$ denote the $m$-times lexicographic product of $G$ with itself. The independence attractor of $G$, denoted by $\mathcal{A}(G)$, is defined as $\mathcal{A}(G) = \lim_{m\rightarrow \infty} \{z: I_{G^m}(z)=0\}$, where the limit is taken with respect to the Hausdorff metric on the space of all compact subsets of the plane. This paper deals with independence attractors that are topologically simple. It is shown that $\mathcal{A}(G)$ can never be a circle. If $\mathcal{A}(G)$ is a line segment then it is proved that the line segment is $[-\frac{4}{k}, 0]$ for some $k \in \{1, 2, 3, 4 \}$. Examples of graphs with independence number four are provided whose independence attractors are line segments.

math.CO↗

arXiv2024

Sum of the exponential and a polynomial: Singular values and Baker wandering domains

This article studies the singular values of entire functions of the form $E^k (z)+P(z)$ where $E^k$ denotes the $k-$times composition of $e^z$ with itself and $P$ is any non-constant polynomial. It is proved that the full preimage of each neighborhood of $\infty$ is an infinitely connected domain without having any unbounded boundary component. Following the literature, the point at $\infty$ is called a Baker omitted value for the function in such a situation. More importantly, there are infinitely many critical values and no finite asymptotic value and in fact, the set of all critical values is found to be unbounded for these functions. We also investigate the iteration of three examples of entire functions with Baker omitted value and prove that these do not have any Baker wandering domain. There is a conjecture stating that the number of completely invariant domains of a transcendental entire function is at most one. How some of these maps are right candidates to work upon in view of this conjecture is demonstrated.

math.CV↗

arXiv2024

Julia sets of rational maps with rotational symmetries

By a symmetry of the Julia set of a polynomial, also referred as polynomial Julia set, we mean an Euclidean isometry preserving the Julia set. Each such symmetry is in fact a rotation about the centroid of the polynomial. In this article, a survey of the symmetries of polynomial Julia sets is made. Then the Euclidean isometries preserving the Julia set of rational maps are considered. A rotation preserving the Julia set of a rational map is called a rotational symmetry of its Julia set. A sufficient condition is provided for a rational map to have rotational symmetries whenever the rational map has an exceptional point. Two classes of rational maps are provided whose Julia sets have rotational symmetries of finite orders. Using this, it is proved that $ z\mapsto μz$ where $μ^{m+n}=1$ is a rotational symmetry of the McMullen map $ z^m+\fracλ{z^n}$ for all $m,n$ with $m\geq 2$ and $λ\in \mathbb{C}\setminus \{0\}$. Assuming that a normalized polynomial has a simple root at the origin, it is shown that the groups of the rotational symmetries of the polynmial coincide with that of its Newton's method and Chebyshev's method.

math.DS↗

arXiv2023

On dynamics of the Chebyshev's method for quartic polynomials

Let $p$ be a normalized (monic and centered) quartic polynomial with non-trivial symmetry groups. It is already known that if $p$ is unicritical, with only two distinct roots with the same multiplicity or having a root at the origin then the Julia set of its Chebyshev's method $C_p$ is connected and symmetry groups of $p$ and $C_p$ coincide~[Nayak, T., and Pal, S., Symmetry and dynamics of Chebyshev's method, \cite{Sym-and-dyn}]. Every other quartic polynomial is shown to be of the form $p_a (z)=(z^2 -1)(z^2-a)$ where $a \in \mathbb{C}\setminus \{-1,0,1\}$. Some dynamical aspects of the Chebyshev's method $C_a$ of $p_a$ are investigated in this article for all real $a$. It is proved that all the extraneous fixed points of $C _a$ are repelling which gives that there is no invariant Siegel disk for $C_a$. It is also shown that there is no Herman ring in the Fatou set of $C_a$. For positive $a$, it is proved that at least two immediate basins of $C_a$ corresponding to the roots of $p_a$ are unbounded and simply connected. For negative $a$, it is however proved that all the four immediate basins of $C_a$ corresponding to the roots of $p_a$ are unbounded and those corresponding to $\pm i\sqrt{|a|}$ are simply connected.

math.DS↗

arXiv2023

Symmetry and dynamics of Chebyshev's method

The set of all holomorphic Euclidean isometries preserving the Julia set of a rational map $R$ is denoted by $ΣR$. It is shown in this article that if a root-finding method $F$ satisfies the Scaling theorem, i.e., for a polynomial $p$, $F_p$ is affine conjugate to $F_{λp \circ T}$ for every nonzero complex number $λ$ and every affine map $T$, then for a centered polynomial $p$ of order at least two (which is not a monomial), $Σp\subseteq ΣF_p$. As the Chebyshev's method satisfies the Scaling theorem, we have $Σp \subseteq Σ{C_p}$, where $p$ is a centered polynomial. The rest part of this article is devoted to explore the situations where the equality holds and in the process, the dynamics of $C_p$ is found. We show that the Julia set $\mathcal{J}(C_p)$ of $ C_p$ can never be a line. If a centered polynomial $p$ is (a) unicritical, (b) having exactly two roots with the same multiplicity, (c) cubic and $Σp$ is non-trivial or (d) quartic, $0$ is a root of $p$ and $Σp $ is non-trivial then it is proved that $Σp = ΣC_p$. It is found in all these cases that the Fatou set $\mathcal{F}(C_p)$ is the union of all the attracting basins of $C_p$ corresponding to the roots of $p$ and $\mathcal{J}(C_p)$ is connected. It is observed that $\mathcal{J}(C_p)$ is locally connected in all these cases.

math.DS↗

arXiv2022

Iteration of some topologically hyperbolic maps in the family $ λ+z+\tan z$

Iteration of the function $f_λ(z)=λ+ z+\tan z, z \in \mathbb{C}$ is investigated in this article. It is proved that for every $λ$, the Fatou set of $f_λ$ has a completely invariant Baker domain $B$; we call it the primary Fatou component. The rest of the results deals with $f_λ$ when it is topologically hyperbolic. For all real $λ$ or $λ$ such that $ λ=πk +i λ_2$ for some integer $k$ and $0 < λ_2<1$, the only other Fatou component is shown to be another completely invariant Baker domain. It is proved that if $|2+λ^2|<1$, then the Fatou set is the union of $B$ and infinitely many invariant attracting domains. Every such domain $U$ has exactly one invariant access to infinity and is unbounded in a special way; $\{\Im(z): z\in U\}$ is unbounded whereas $\{\Re(z): z\in U\}$ is bounded. If $\Im(λ)> \sqrt{2}+ \sinh^{-1}1$ then it is found that the primary Fatou component is the only Fatou component and the Julia set is disconnected. For every natural number $k$, the Fatou set of $f_λ$ for $λ=kπ+i\fracπ{2}$ is shown to contain $k$ wandering domains with distinct grand orbits. These wandering domains are found to be escaping. The Fatou set is the union of $B$, these wandering domains and their pre-images.

math.DS↗

arXiv2022

The Julia sets of Chebyshev's method with small degrees

Given a polynomial $p$, the degree of its Chebyshev's method $C_p$ is determined. If $p$ is cubic then the degree of $C_p$ is found to be $4,6$ or $7$ and we investigate the dynamics of $C_p$ in these cases. If a cubic polynomial $p$ is unicritical or non-generic then, it is proved that the Julia set of $C_p$ is connected. The family of all rational maps arising as the Chebyshev's method applied to a cubic polynomial which is non-unicritical and generic is parametrized by the multiplier of one of its extraneous fixed points. Denoting a member of this family with an extraneous fixed point with multiplier $λ$ by $C_λ$, we have shown that the Julia set of $C_λ$ is connected whenever $λ\in [-1,1]$.

math.DS↗

arXiv2021

Quadratic and cubic Newton maps of rational functions

The dynamics of all quadratic Newton maps of rational functions are completely described. The Julia set of such a map is found to be either a Jordan curve or totally disconnected. It is proved that no Newton map with degree at least three of any rational function is conformally conjugate to a unicritical polynomial(i.e., with exactly one finite critical point). However, there are cubic Newton maps which are conformally conjugate to other polynomials. The Julia set of such a Newton map is shown to be a closed curve. It is a Jordan curve whenever the Newton map has two attracting fixed points.

math.CV↗

arXiv2021

On Fatou sets containing Baker omitted value

An omitted value of a transcendental meromorphic function $f$ is called a Baker omitted value, in short \textit{bov} if there is a disk $D$ centered at the bov such that each component of the boundary of $f^{-1}(D)$ is bounded. Assuming that the bov is in the Fatou set of $f$, this article investigates the dynamics of the function. Firstly, the connectivity of all the Fatou components are determined. If $U$ is the Fatou component containing the bov then it is proved that a Fatou component $U'$ is infinitely connected if and only if it lands on $U$, i.e. $f^{k}(U') \subset U$ for some $k \geq 1$. Every other Fatou component is either simply connected or lands on a Herman ring. Further, assuming that the number of critical points in the Fatou set whose forward orbits do not intersect $U$ is finite, we have shown that the connectivity of each Fatou component belongs to a finite set. This set is independent of the Fatou components. It is proved that the Fatou component containing the bov is completely invariant whenever it is forward invariant. Further, if the invariant Fatou component is an attracting domain and compactly contains all the critical values of the function then the Julia set is totally disconnected. Baker domains are shown to be non-existent whenever the bov is in the Fatou set. It is also proved that, if there is a $2$-periodic Baker domain (these are not ruled out when the bov is in the Julia set), or a $2$-periodic attracting or parabolic domain containing the bov then the function has no Herman ring. Some examples exhibiting different possibilities for the Fatou set are discussed. This includes the first example of a meromorphic function with an omitted value which has two infinitely connected Fatou components.

math.DS↗

arXiv2021

The real non-attractive fixed point conjecture and beyond

Is it always true that every polynomial P with the degree at least two has a fixed point z0, the real part of whose multiplier is bigger than or equal to 1, i.e., Real part of (P'(z0))> 1? This question, raised by Coelho and Kalantari in How many real attractive fixed points can a polynomial have? Math. Gaz. 103 (2019), no. 556, 65{76. [3] is answered affirmatively not only for all polynomials but also for all rational functions with a super attracting fixed point. However, this is not true for all rational functions. Some further investigation on distribution of multipliers of fixed points is made. Quadratic and cubic polynomials, all of whose multipliers have real part 1 are characterized. A necessary and sufficient condition is found for cubic and quartic polynomials, all of whose multipliers are equidistant from Is it always true that every polynomial P with the degree at least two has a fixed point z0, the real part of whose multiplier is bigger than or equal to 1, i.e., Real part of P'(z0)> 1? This question, raised by Coelho and Kalantari in How many real attractive fixed points can a polynomial have? Math. Gaz. 103 (2019), no. 556, 65{76. [3] is answered affirmatively not only for all polynomials but also for all rational functions with a super attracting fixed point. However, this is not true for all rational functions. Some further investigation on the distribution of multipliers of fixed points is made. Quadratic and cubic polynomials, all of whose multipliers have real part 1 are characterized. A necessary and sufficient condition is found for cubic and quartic polynomials, all of whose multipliers are equidistant from 1.

math.CV↗