Journal of Indian Acad. Math.

ISSN: 0970-5120

Vol. 48, No. 1 (2026) pp. 102–111.

DYNAMICS AND VALUE DISTRIBUTION OF

COMPLEX VALUED MEROMORPHIC

FUNCTIONS OF HIGHER DIMENSIONS

Sanjib Kumar Datta

Abstract. We generalize fundamental results from complex meromorphic dynamics

and value-distribution theory to bicomplex meromorphic functions. After introducing bi-

complex numbers and bicomplex-holomorphic (meromorphic) functions, we extend the

notions of Fatou and Julia sets to the bicomplex plane. Our main results are: (1) a bi-

complex analogue of quasi-nested wandering domains and Baker omitted values, including

existence and boundedness theorems; (2) the extension of Herman-ring conﬁgurations and

their classiﬁcation to the bicomplex case and (3) bicomplex versions of Picard-Montel

normality theorems for families omitting values. All proofs proceed by reducing to the

idempotent components and applying classical one-variable arguments.

Keywords: Bicomplex analysis, multicomplex functions, bicomplex meromorphic dynam-

ics, normal families, value distribution theory, Nevanlinna theory, Picard theorem, Montel

theorem, Julia set, Fatou set, Baker domains, wandering domains, omitted values, higher-

dimensional complex dynamics, holomorphic iteration, dynamics of bicomplex-valued func-

tions, quasi-normal families.

2010 AMS Subject Classiﬁcation: 30G35, 30D45, 30D30, 37F10.

1. Introduction

Classical one-dimensional complex dynamics and Nevanlinna theory have produced

deep insights into the iteration and value-distribution of meromorphic functions. In par-

ticular, Chakraborty and Datta studied quasi-nested wandering domains and Baker omit-

ted values, conﬁgurations of Herman rings and dynamical properties of families omitting

values {cf. [8]}. These works were formulated for complex meromorphic functions on

ꢀ

C {cf. [20]}. In this paper, we extend these theories to the bicomplex setting, where

functions take values in the algebra of bicomplex numbers BC {cf. [1],[4]}.

While bicomplex analysis extends the classical theory of complex functions by introducing

two commuting imaginary units, the concept can be generalized even further through the

framework of multicomplex analysis. A multicomplex algebra is generated by more than

two commuting imaginary units, typically denoted as i₁, i₂, . . . , i_n, each satisfying i²= −1.

k

The resulting space, known as the multicomplex space C_n, is a ﬁnite-dimensional com-

mutative algebra over R, encompassing the complex, bicomplex and tricomplex systems

as special cases. Multicomplex numbers allow for the representation of multidimensional

analytic structures and naturally decompose into direct sums of complex subspaces using

idempotent elements.

102

DYNAMICS AND VALUE DISTRIBUTION OF COMPLEX VALUED MEROMORPHIC FUNCTIONS

103

This generalization signiﬁcantly enriches the analytical landscape: fundamental con-

cepts such as diﬀerentiability, holomorphicity and singularity theory can all be extended

to multicomplex functions, often revealing behaviours not present in the classical or bi-

complex settings. Moreover, the higher-dimensional structure of multicomplex spaces

enables a uniﬁed framework for modelling complex systems involving multiple indepen-

dent processes. As such, multicomplex analysis is increasingly being explored in areas

ranging from advanced function theory to dynamical systems, diﬀerential equations and

applications in physics and engineering.

We adopt the following plan. In Section 2 we recall the deﬁnitions of bicomplex num-

bers {cf. [1],[4]}, the extended bicomplex plane and bicomplex-holomorphic (meromor-

phic) functions via idempotent decomposition cf. [1],[4]}. We then deﬁne bicomplex

Fatou and Julia sets {cf. [7],[12]} analogously to the complex case by using bispherical

normality of iterates {cf. [7],[13]}. In Section 3 we introduce bicomplex quasi-nested

wandering domains and Baker omitted values and we prove the bicomplex analogues of

Chakraborty-Datta’s theorems on their existence and properties. In Section 4 we gener-

alize the classiﬁcation of Herman-ring cycles to bicomplex maps. Finally, in Section 5 we

establish bicomplex Montel/Picard-type theorems for families of meromorphic functions

with omitted values. We conclude with a discussion of implications and future directions.

2. Preliminaries on Bicomplex Analysis

Let

BC = { z₁+ z₂i₂: z₁, z₂∈ C(i₁) }

denote the algebra of bicomplex numbers, with independent imaginary units i²₁= i²₂= −1

and i₁i₂= i₂i₁= j so that j²= 1. Every w = z₁+ z₂i₂can be uniquely written using the

orthogonal idempotents as

1 + j

1 − j

e₁=

and e₂=

,

2

namely

w = (z₁− i₁z₂)e₁+ (z₁+ i₁z₂)e₂= P₁(w)e₁+ P₂(w)e₂,

where P₁(w) = z₁− i₁z₂and P₂(w) = z₁+ i₁z₂project w onto two copies of the complex

plane. The set of zero-divisors (the null cone) is

N_C= { w : ꢀwꢀj²= 0},

where |w|j = |P₁(w)|, e₁+|P₂(w)|, e₂is the bicomplex modulus. We also form the extended

bicomplex plane BC ∪ I∞ by adjoining points at inﬁnity. In particular, one can regard

BC = C(i₁) ×_eC(i₁) ∪ I∞,

where

I = { w : ꢀwꢀ_j= ∞ },

∞

analogously to the Riemann sphere.

A function f : Ω ⊂ BC → BC is called bicomplex-holomorphic (BC -holomorphic) if it

satisﬁes the bicomplex Cauchy-Riemann conditions. Equivalently, by Ringleb’s Lemma

{cf. [2],[3]}, f is BC-holomorphic {cf. [2],[4],[5],[6]} if and only if its projected components

f_e1: P₁(Ω) → C(i₁),

f_e2: P₂(Ω) → C(i₁)

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Sanjib Kumar Datta

are holomorphic in the usual sense. We say f is bicomplex meromorphic on Ω if it can be

expressed as the quotient of bicomplex-holomorphic functions, or equivalently if f extends

to BC with poles. In fact one has

Theorem 2.1. Let Ω ⊂ BC be open. A function f : Ω → BC is bicomplex meromorphic

if and only if its components f_e1: P₁(Ω) → C(i₁) and f_e2: P₂(Ω) → C(i₁) are (single-

variable) meromorphic and then

f(w) = f_e1(P₁(w)) e₁+ f_e2(P₂(w)) e₂,

w ∈ Ω.

Accordingly, we deﬁne w₀∈ Ω to be a pole of f if either f_e1or f_e2has a pole at the

corresponding projected point. (Such poles are not isolated in BC.)

We now introduce bicomplex dynamical sets. As usual, the Fatou set F(f) is the set

of points in BC where the iterates fⁿform a normal family with respect to the bicom-

plex chordal metric. Its complement J(f) = BC \ F(f) is the Julia set {cf. [18],[20]}.

Equivalently, for a nondegenerate bicomplex polynomial or rational map,

J₂(f) = {ζ : {fⁿ(ζ)} is not normal}.

By construction J(f) is completely invariant under f.

A family M(BC) of bicomplex meromorphic functions on a domain Ω ⊂ BC is normal

if every sequence in M(BC) has a subsequence converging bispherically uniformly on

compact subsets to a limit in BC. Equivalently, each projection f_e1: f ∈ M(BC) and

f_e2: f ∈ M(BC) must be a normal family in the usual complex sense.

Remark 2.2. The bicomplex modulus ꢀwꢀ_jdoes not deﬁne a norm in the usual sense

because of the presence of zero divisors. Nevertheless, it provides a way to measure “size”

component-wise and to classify points as bounded or unbounded in BC.

3. Bicomplex Quasi-Nested Wandering Domains

Let f : BC → BC be bicomplex meromorphic and let U ⊂ F(f) be a Fatou component.

Recall U is a wandering domain {cf. [19]} if the iterates fⁿ(U) are pairwise disjoint and

never periodic. We adapt the notion of a quasi-nested wandering domain {cf. [8]} to BC.

Deﬁnition 3.1. A wandering domain U ⊂ F(f) is called quasi-nested if there exists a

subsequence of iterates n_k→ ∞ such that each U_n= fⁿ^k(U) is j-bounded and as k → ∞

k

the domains U_ncluster onto a single point a ∈ BC. Equivalently, for some a ∈ BC one

k

has U_nsurrounding a in each projection and max_w∈U

|w − a|_j→ 0.

n

k

Next we deﬁne a bicomplex version of a Baker omitted value.

Deﬁnition 3.2. An omitted value a ∈ BC of f is a Baker omitted value if for all suﬃ-

ciently small r > 0, the preimage f⁻¹(B(a; r)) is of the form

∞

ꢁ

BC \

D_i,

i=1

DYNAMICS AND VALUE DISTRIBUTION OF COMPLEX VALUED MEROMORPHIC FUNCTIONS

105

where the D_iare pairwise disjoint bounded domains in BC. In other words, f⁻¹(B(a; r))

has bounded complement for each small disk B(a; r).

We can now state the bicomplex analogue of Chakraborty-Datta’s result on quasi-nested

domains {cf. [8]}.

Theorem 3.3. Let a ∈ BC be a bicomplex Baker omitted value of f. Then f has a

bicomplex quasi-nested wandering domain if and only if there exists a sequence of iterates

U_nsuch that each U_nsurrounds a and U_n→ a as k → ∞.

k

Proof. Write f(w) = f_e1(P₁(w))e₁+ f_e2(P₂(w))e₂by Theorem 2.1. If a is a bicomplex

Baker omitted value, then its projections a₁= P₁(a) and a₂= P₂(a) are Baker omitted

values of the complex functions f_e1and f_e2. By f_e1has a quasi-nested wandering domain

if and only if a sequence of iterates surrounds a₁and tends to a₁and similarly for f_e2.

If U ⊂ BC is a bicomplex wandering domain quasi-nested around a, then each projection

P_i(U) is quasi-nested around a_i. Hence there exist iterates P_i(U_n) surrounding a_iand

k

tending to a_i. This implies U_nsurrounds a in BC and U_n→ a. Conversely, if such

k

a sequence of U_nexists, then projecting shows each f_eihas a quasi-nested wandering

k

domain around a_i. Therefore U is bicomplex quasi-nested.

□

We can generalize the Theorem 3.1 in multicomplex space as follows.

Theorem 3.4. Let a ∈ C_nbe a multicomplex Baker omitted value of a meromorphic

function F : C_n→ C_n. Then F has a multicomplex quasi-nested wandering domain if

and only if there exists a sequence of iterates {U_n} such that each U_nsurrounds a and

k

U_n−→ a as k → ∞.

k

Proof. Write the multicomplex algebra in its idempotent decomposition

2 ⁿ⁻¹

ꢂ

C_n=

C e_α,

α=1

with canonical projections P_α: C_n→ C and mutually orthogonal idempotents e_α. By the

multicomplex Ringleb decomposition (the obvious generalisation of Ringleb’s Lemma to

C_n), the meromorphic map

F : C_n−→ C_n

can be written on its domain Ω as

2 ⁿ⁻¹

ꢃ

ꢄ

ꢅ

F(W) =

f_αP_α(W) e_α,

α=1

where each component f_α: P_α(Ω) → C is a (single-variable) meromorphic function {cf.

[17]}.

Assume ﬁrst that a ∈ C_nis a multicomplex Baker omitted value of F. Then for each

idempotent index α the projection a_α:= P_α(a) is a Baker omitted value of the complex

meromorphic function f_α. By the classical one-variable theory (the complex version of the

Chakraborty-Datta result), for each α the function f_αadmits a quasi-nested wandering

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Sanjib Kumar Datta

domain precisely when there exists a sequence of iterates of f_αwhose images are bounded

annular neighbourhoods surrounding a_αand which converge to a_α. Consequently, if F is a

multicomplex map with Baker omitted value a, then the idempotent components f_αadmit

corresponding sequences of iterates surrounding a_α. Combining these componentwise

sequences yields a sequence of bicomplex (indeed multicomplex) iterates {U_n} for F

k

such that each U_nsurrounds a in C_n(i.e. each projection P_α(U_n) surrounds a_α) and

k

U_n−→ a as k → ∞,

k

so F has a multicomplex quasi-nested wandering domain.

Conversely, suppose there exists a sequence of Fatou components (iterates) {U_n} for

k

F with the property that each U_nsurrounds a and U_n→ a as k → ∞. Projecting

k

onto the idempotent summands shows that for each α the sequence {P_α(U_n)} consists

k

of Fatou components (iterates) of f_αwhich surround a_αand tend to a_α. Again by the

classical one-variable characterisation, each f_αtherefore possesses a quasi-nested wander-

ing domain around a_α. Reassembling these componentwise quasi-nested domains using

the idempotent decomposition produces a multicomplex wandering domain for F which

is quasi-nested about a. Finally, the equivalence of the componentwise and multicomplex

descriptions of surrounding and convergence (via the projections P_α) yields the claimed

if and only if statement.

This completes the proof.

□

As in the complex case, the existence of a quasi-nested domain forces all Fatou compo-

nents to be bounded.

Proposition 3.5. If f has a quasi-nested wandering domain, then every bicomplex Fatou

component is bounded and in particular f has no Baker domain in BC.

Proof. If U is a bicomplex quasi-nested domain around a, then each projection f_eihas a

complex quasi-nested domain around a_i. By each f_eihas only bounded Fatou components.

Since a bicomplex Fatou component is a Cartesian product of components of f_e1and f_e2,

it follows that all bicomplex Fatou components are bounded. In particular, there is no

bicomplex Baker domain (which would correspond to an unbounded Fatou component).

□

Remark 3.6. Quasi-nested wandering domains in BC may appear as “tubes” in one

component and “discs” in the other. The idempotent decomposition allows one to visualize

these domains as Cartesian products of classical wandering domains in C.

4. Herman Rings in Bicomplex Dynamics

A Herman ring {cf. [9],[10],[11]} for f : BC → BC is a doubly-connected periodic Fatou

component {cf. [14],[15]} on which f acts as an irrational rotation. Concretely, a p-cycle

U₀, . . . , U_p−1of bicomplex Herman rings means each U_kis T-Cartesian (i.e. of the form

U₁e₁+ U₂e₂with each U_ian annulus) and f^pis analytically conjugate to an irrational

rotation on U.

DYNAMICS AND VALUE DISTRIBUTION OF COMPLEX VALUED MEROMORPHIC FUNCTIONS

107

The possible arrangements of a cycle of Herman rings can be described by maximal

nests and nesting, as in the complex theory. We recall: an outermost ring in the cycle is

one not contained in the closure of any other. A maximal nest is a sub-collection of rings

consisting of an outermost ring and all rings nested inside it. The cycle is nested if it has

exactly one maximal nest, strictly nested if each ring except the outermost is properly

inside another and strictly non-nested if no ring lies inside another.

The following lemma relates bicomplex Herman rings to their projections.

Lemma 4.1. If U = U₁e₁+ U₂e₂⊂ BC is a p-periodic bicomplex Herman ring of f, then

U₁and U₂are (period-p) Herman rings for the complex functions f_e1and f_e2. Conversely,

if U₁, U₂are Herman rings of the same period p for f_e1, f_e2, then U = U₁e₁+ U₂e₂is a

bicomplex Herman ring of f.

Proof. If f^p: U → U is conjugate to w → e^2πiθw, then projecting onto each idempotent

e2

shows f_e^p₁is conjugate to rotation by θ on U₁and similarly f^pon U₂. Thus each U_iis

an invariant annulus with irrational rotation, i.e. a Herman ring. Conversely, suppose U₁

and U₂are Herman rings for f_e1, f_e2with angles θ₁, θ₂. One can construct a bicomplex

conjugacy

Φ(w) = φ₁(P₁(w)) e₁+ φ₂(P₂(w)) e₂,

using the conjugating maps φ_ifor each component. Then Φ conjugates f^pon U₁e₁+ U₂e₂

2πiθ₂

to the bicomplex rotation e^2πiθe₁+ e

e₂, yielding a bicomplex Herman ring of period

1

p.

□

Using Lemma 4.1 and known complex results, we derive restrictions on bicomplex

Herman rings.

Proposition 4.2. A bicomplex meromorphic function f cannot have a Herman ring of

period 1 or 2. More generally, if either projection f_e1or f_e2omits an essential singularity

(for example, has only one pole), then f has no bicomplex Herman ring.

Proof. If f had a period-1 Herman ring, then by Lemma 4.1 each f_eiwould have a period-1

Herman ring. But it is known that no meromorphic function can have a 1-cycle Herman

ring. The same argument (and citation) rules out period-2. For the second part, if f_e1

is entire or has a single pole (omitted value) then by it has no Herman rings. Thus f

cannot have a bicomplex Herman ring since one projection would fail to provide a ring.

Similarly if f_e2omits ∞.

□

In summary, the classiﬁcation of Herman-ring cycles for f follows from the classiﬁcations

for f_e1and f_e2. For instance, a bicomplex Herman-cycle is nested only if each projected

cycle is nested, etc.

Remark 4.3. Bicomplex Herman rings can exhibit diﬀerent rotation speeds in each com-

ponent. Therefore, the overall orbit in T might densely ﬁll a torus-like region even if each

component rotates along a simple circle in C.

108

Sanjib Kumar Datta

Example 4.4. For example, Let f(w) = e^iπ/3P₁(w)e₁+ e^iπ/4P₂(w)e₂. Then U₁= {1 <

|z| < 2} and U₂= {1 < |z| < 2} form a bicomplex Herman ring where the rotations in U₁

and U₂are incommensurate, producing a quasi-periodic orbit in BC.

5. Dynamics of Bicomplex Families with Omitted Values

We now consider normal families of bicomplex meromorphic functions with omitted

values. The bicomplex analogue of Montel’s theorem reads as follows.

Theorem 5.1 (Bicomplex Montel Theorem). Let M(BC) be a family of bicomplex mero-

morphic functions on a domain Ω ⊂ BC. Suppose there exist three values α, β, γ ∈ BC

such that α−β, β −γ, γ −α are all invertible in BC and no f attains any of α, β, γ. Then

M(BC) is a normal family on Ω.

Proof. Project onto the idempotent components M(BC)_e1= f_e1: f ∈ M(BC) and M(BC)_e2=

f_e2: f ∈ M(BC). The invertibility conditions imply P_i(α), P_i(β), P_i(γ) are three distinct

complex values. Since no f takes α, β, γ, no f_ei∈ F_eitakes the corresponding three com-

plex values. By the classical Montel theorem, each M(BC)_eiis normal in C(i₁). Hence

M(BC) is normal on Ω by Ringleb’s decomposition (Theorem 2.1).

□

We can generalize the theorem 5.1 in multicomplex space as follows.

Theorem 5.2 (Multicomplex Montel Theorem). Let M(C_n) be a family of multicomplex

meromorphic functions on a domain Ω ⊂ C_n. Suppose there exist three distinct values

α, β, γ ∈ C_nsuch that α − β, β − γ and γ − α are all invertible in C_nand no F ∈ M(C_n)

attains any of the values α, β or γ. Then M(C_n) is a normal family on Ω.

Proof. Write the multicomplex algebra in its canonical idempotent decomposition

2 ⁿ⁻¹

ꢂ

C_n=

C e_α,

α=1

and let P_α: C_n→ C denote the projection onto the α-th complex summand. By the

multicomplex Ringleb decomposition, every

F ∈ M(C_n)

admits the representation

2 ⁿ⁻¹

ꢃ

F(W) =

f_α(P_α(W)) e_α,

W ∈ Ω,

α=1

where for each ﬁxed α the map f_αranges over a family F_α= {f_α: F ∈ M(C_n)} of

single-variable meromorphic functions on the complex domain P_α(Ω).

By hypothesis there are three distinct values α, β, γ ∈ C_nwith α − β, β − γ, γ − α

invertible in C_nand no F ∈ M(C_n) attains any of these three values. Fix an idempotent

index α. Applying the projection P_αto the three multicomplex values yields the three

complex numbers

P_α(α),

P_α(β) and P_α(γ).

DYNAMICS AND VALUE DISTRIBUTION OF COMPLEX VALUED MEROMORPHIC FUNCTIONS

109

Invertibility of diﬀerences in C_nimplies that for each idempotent index α the diﬀerences

P_α(α) − P_α(β),

P_α(β) − P_α(γ) and P_α(γ) − P_α(α)

are nonzero in C; hence P_α(α), P_α(β), P_α(γ) are three distinct complex values. Moreover,

since no multicomplex map F ∈ M(C_n) attains α, β, γ, it follows that no component

function f_α∈ F_αattains any of the complex values P_α(α), P_α(β) and P_α(γ).

Therefore, for each ﬁxed idempotent index α the family F_αof single-variable mero-

morphic functions omits three distinct complex values in P_α(Ω). By the classical Montel

theorem for meromorphic functions, each family F_αis normal on P_α(Ω).

Normality of every component family F_αimplies normality of the original multicomplex

family M(C_n): indeed, any sequence {F_m} ⊂ M(C_n) has, by diagonal extraction on the

ﬁnitely many (in fact 2ⁿ⁻¹) idempotent indices, a subsequence whose component projec-

tions {f_α,m} converge bispherically (uniformly on compacta) in each complex summand

to meromorphic limits g_α. Recombining these limits via the idempotent decomposition

yields a bispherically convergent subsequence of {F_m} in C_n. Hence M(C_n) is normal on

Ω.

□

An immediate corollary of Theorem 5.1 is a bicomplex Picard theorem.

Theorem 5.3 (Bicomplex Picard Theorem). Let f : BC → BC be a bicomplex meromor-

phic function. If there exist three values α, β, γ with α − β, β − γ, γ − α invertible, such

that f(w) = α, β, γ for all w ∈ BC, then f is constant.

Proof. Consider the constant family M(BC) = f. The Montel theorem implies M(BC) is

normal on BC. Hence each projection f_eiomits three values in C and by Picard’s theorem

must be constant. Therefore f(w) = c₁e₁+ c₂e₂is constant on BC.

□

Similarly, one can obtain bicomplex versions of the Fundamental Normality Test for

two values: e.g. a family omitting two bicomplex values (with invertible diﬀerence) is

normal, by the same reduction.

Remark 5.4. In bicomplex analysis, the notion of “omitting values” must consider in-

vertibility. Two values may coincide in one idempotent component but diﬀer in the other,

so the classical theorems hold only when all three diﬀerences are invertible.

n

1

Example 5.5. For example, Let f_n(w) = _w−1e₁+ _w−2e₂, n ∈ N. Then {f_n} omits 0 in

both components and 1 in the ﬁrst component. The family is normal on BC \ {1, 2} by

Theorem 5.1.

6. Conclusion

We have extended several key aspects of complex meromorphic dynamics into the bi-

complex plane. Using the idempotent decomposition, one essentially obtains that bicom-

plex phenomena mirror two coupled copies of the one-variable case. In particular, bi-

complex analogues of quasi-nested wandering domains, Herman rings and Picard-Montel

110

Sanjib Kumar Datta

theorems follow directly by applying known results to each component. Notably, restric-

tions such as the nonexistence of 1- or 2-cycles of Herman rings, or Montel’s normality

criterion for families missing three values, remain valid in the bicomplex setting.

This framework suggests many directions for further research. One may study other

rotation domains (e.g. bicomplex Siegel disks or Baker domains) {cf. [16]}, bifurcations

of bicomplex rational maps, or dynamics in other commutative hypercomplex algebras.

The bicomplex Picard-Montel theory developed here could be reﬁned or extended and

numerical explorations of bicomplex Julia sets might reveal new structure. We hope this

work lays a foundation for a rich bicomplex dynamics theory.

7. Future Prospects

The present work lays a basic foundation for bicomplex dynamical and value distribu-

tion theory, but the ﬁeld remains largely unexplored. Natural extensions include higher-

order multicomplex systems, where componentwise dynamics and idempotent structure

may produce new behaviours in Fatou-Julia theory; multicomplex Nevanlinna-type re-

sults, Picard phenomena and defect relations, where multiple zero divisors suggest richer

value distribution patterns and interdisciplinary applications in physics, signal analysis

and modelling, where multicomplex representations arise naturally. Computational study

and visualization of multicomplex Julia sets and parameter spaces also remain open.

Overall, extending these ideas to the full multicomplex hierarchy promises a substantially

broader analytical framework with signiﬁcant theoretical and applied potential.

Acknowledgement

The author is grateful to the anonymous referee(s) for careful checking in details and also

for helpful comments towards the improvement of the paper. He is thankful to Prof.(Dr.)

M. Dube and Prof.(Dr.) J.K. Maitra, Department of Mathematics and Computer Science,

Rani Durgavati Vishwavidyalaya, Jabalpur, Madhya Pradesh for giving proper suggestions

to enhance the quality of the paper. The author also sincerely acknowledges the ﬁnancial

support under FRG(Faculty Research Grant) during April,2025- March,2026 rendered by

University of Kalyani, P.O.: Kalyani, Dist: Nadia, PIN: 741235, West Bengal, India.

REFERENCES

[1] G. B. Price, An introduction to multicomplex spaces and functions, Marcel Dekker, 1991.

[2] J. D. Riley, Contributions to the theory of functions of a bicomplex variable, Tˆohoku Math. J., 5

(1953), 132–165.

[3] F. Ringleb, Beitr¨age zur Funktionentheorie in hyperkomplexen Systemen I, Rend. Cir. Mat. Palermo,

57 (1933), 311–340.

[4] D. Alpay, M. E. Luna-Elizarrar´as, M. Shapiro and D. C. Struppa, Basics of functional analysis with

bicomplex scalars, Springer, 2014.

[5] M. E. Luna-Elizarrar´as, M. Shapiro, D. C. Struppa and A. Vajiac, Bicomplex holomorphic functions:

The algebra, geometry and analysis, Birkh¨auser, 2015.

[6] K. S. Charak, D. Rochon and N. A. Sharma, Normal families of bicomplex holomorphic functions,

Complex Var. Elliptic Eq., 54(12):1083–1102, 2009.

[7] W. Bergweiler, Iteration of meromorphic functions, Bull. Amer. Math. Soc., 29(2):151–188, 1993.

DYNAMICS AND VALUE DISTRIBUTION OF COMPLEX VALUED MEROMORPHIC FUNCTIONS

111

[8] G. Chakraborty and S. K. Datta, On quasi-nested wandering domains, Filomat, 36(8):2587–2601,

2022.

[9] T. Nayak, Herman rings of meromorphic maps with an omitted value, Proc. Amer. Math. Soc.,

144(2):587–597, 2016.

[10] F. Yang, On formulas of meromorphic functions with periodic Herman rings, arXiv:2011.10935, 2020.

[11] N. Fagella and J. Peter, On the conﬁguration of Herman rings of meromorphic functions, J. Math.

Anal. Appl., 393(1):456–468, 2012.

[12] P. Dom´ınguez, Dynamics of transcendental meromorphic functions, Ann. Acad. Sci. Fenn. Math.,

23:225–250, 1998.

[13] J. H. Zheng, Iteration of functions which are meromorphic outside a small set, Tˆohoku Math. J.,

57:1–18, 2005.

[14] P. Rippon and G. Stallard, On multiply connected wandering domains of meromorphic functions,

preprint, 2007.

[15] J. Huang, C. Wu and J. H. Zheng, Connectivity of Fatou components of meromorphic functions,

arXiv:2211.14258, 2022.

[16] S. Ghora and T. Nayak, Rotation domains and stable Baker omitted value, arXiv:2101.01951, 2021.

[17] W. K. Hayman, Meromorphic functions, Oxford University Press, 1964.

[18] A. E. Eremenko and M. Y. Lyubich, Dynamical properties of some classes of entire functions, Ann.

Inst. Fourier, 42(4):989–1020, 1992.

[19] I. N. Baker, Wandering domains in iteration of entire functions, Ann. Acad. Sci. Fenn. Math.,

12:191–198, 1987.

´

[20] M. Shishikura, On the quasiconformal surgery of rational functions, Ann. Sci. Ec. Norm. Sup´er.,

20:1–29, 1987.

(Received, October 02, 2025)

(Revised, December 02, 2025)

Mathematics Department,

University of Kalyani,

P.O.: Kalyani, Dist: Nadia, PIN: 741235,

West Bengal, India.

Email: sanjibdatta05@gmail.com