arXiv subjects

Tarakanta Nayak

Publications and source records attributed to Tarakanta Nayak.

23 records · Page 2Linked to original sources

arXiv2021

Rotation domains and Stable Baker omitted value

A Baker omitted value, in short \textit{bov} of a transcendental meromorphic function $f$ is an omitted value such that there is a disk $D$ centered at the bov for which each component of the boundary of $f^{-1}(D)$ is bounded. Assuming all the iterates $f^n$ are analytic in a neighborhood of its bov, this article proves that the number of Herman rings of a particular period is finite and every Julia component intersects the boundaries of at most finitely many Herman rings. Further, if the bov is the only limit point of the critical values then it is shown that $f$ has infinitely many repelling fixed points. If a repelling periodic point of period $p$ is on the boundary of a $p$-periodic rotation domain then the periodic point is shown to be on the boundary of infinitely many Fatou components. Under additional assumptions on the critical points, a sufficient condition is found for a Julia component to be singleton. As a consequence, it is proved that if the boundary of a wandering domain $W$ accumulates at some point of the plane under the iteration of $f$ then each limit of $f^n$ on $W$ is either a parabolic periodic point or in the $ω$-limit set of recurrent critical points. Using the same ideas, the boundary of rotation domains are shown to be in the $ω$-limit set of recurrent critical points.

math.DS↗

arXiv2020

On periods of Herman rings and relevant poles

Possible periods of Herman rings are studied for general meromorphic functions with at least one omitted value. A pole is called $H$-relevant for a Herman ring $H$ of such a function $f$ if it is surrounded by some Herman ring of the cycle containing $H$. In this article, a lower bound on the period $p$ of a Herman ring $H$ is found in terms of the number of $H$-relevant poles, say $h$. More precisely, it is shown that $p\geq \frac{h(h+1)}{2}$ whenever $f^j(H)$, for some $j$, surrounds a pole as well as the set of all omitted values of $f$. It is proved that $p \geq \frac{h(h+3)}{2}$ in the other situation. Sufficient conditions are found under which equalities hold. It is also proved that if an omitted value is contained in the closure of an invariant or a two periodic Fatou component then the function does not have any Herman ring.

math.DS↗

arXiv2015

Baker Omitted Value

We define Baker omitted value, in short bov, of an entire or meromorphic function f in the complex plane as an omitted value for which there exists r0 > 0 such that for each ball Dr(a) centered at a and with radius r satisfying 0 < r < r0, every component of the boundary of f only asymptotic value. An entire function has bov if and only if the image of every unbounded curve is unbounded. It follows that an entire function has bov whenever it has a Baker wandering domain. Functions with bov has at most one completely invariant Fatou component.

math.DS↗

arXiv2015

Herman rings of meromorphic maps with an omitted value

We investigate the existence and distribution of Herman rings of transcendental meromorphic functions which have at least one omitted value. If all the poles of such a function are multiple then it has no Herman ring. Herman rings of period one or two do not exist. Functions with a single pole or with at least two poles one of which is an omitted value have no Herman ring. Every doubly connected periodic Fatou component is a Herman ring.

math.DS↗

arXiv2012

Omitted values and dynamics of meromorphic functions

Let $M$ be the class of all transcendental meromorphic functions $f: \mathbb{C} \to \mathbb{C} \bigcup\{\infty\}$ with at least two poles or one pole that is not an omitted value, and $M_o =\{f \in M:f{has at least one omitted value}\}$. Some dynamical issues of the functions in $M_o$ are addressed in this article. A complete classification in terms of forward orbits of all the multiply connected Fatou components is made. As a corollary, it follows that the Julia set is not totally disconnected unless all the omitted values are contained in a single Fatou component. Non-existence of both Baker wandering domains and invariant Herman rings are proved. Eventual connectivity of each wandering domain is proved to exist. For functions with exactly one pole, we show that Herman rings of period two also do not exist. A necessary and sufficient condition for the existence of a dense subset of singleton buried components in the Julia set is established for functions with two omitted values. The conjecture that a meromorphic function has at most two completely invariant Fatou components is confirmed for all $f \in M_o$ except in the case when $f$ has a single omitted value, no critical value and is of infinite order. Some relevant examples are discussed.

math.CV↗