Ergodicity of Maps on the Two-Dimensional Torus

George Smart Nduka; Henry Etaroghene Egbogho

doi:doi:10.11648/j.ijssam.20251001.11

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Ergodicity of Maps on the Two-Dimensional Torus

George Smart Nduka^*

, Henry Etaroghene Egbogho

Published in International Journal of Systems Science and Applied Mathematics (Volume 10, Issue 1)

Received: 31 December 2024 Accepted: 21 January 2025 Published: 10 February 2025

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Abstract

This paper considers the ergodicity of maps on the two-dimensional torus, focusing on transformations where invariant real-valued functions are constant. The study considers both additive and multiplicative transformations, providing a detailed analysis of the conditions required for a map to be classified as ergodic. The investigation is grounded in the theory of dynamical systems and leverages mathematical tools such as orthonormal double sequences in Hilbert spaces and Fourier series to establish necessary and sufficient conditions for ergodicity. By connecting these conditions to the Lebesgue measure, the research outlines the fundamental properties of ergodic transformations on the torus. A key aspect of this study is its treatment of invariant functions and their role in defining ergodic behavior. Invariant functions, which remain unchanged under the dynamics of a given transformation, are examined in depth to understand how their constancy relates to the overall system. The analysis also highlights the interplay between additive and multiplicative transformations and their impact on the ergodic properties of the system. The results of this work not only provide a robust framework for understanding the dynamics of transformations on the two-dimensional torus but also have implications for higher-dimensional systems. This contribution is particularly relevant for studying complex systems in ergodic theory, where the behavior of transformations under the Lebesgue measure often serves as a foundation for further exploration. By addressing these fundamental aspects, the paper lays the groundwork for extending these concepts to more intricate and multidimensional settings in mathematical and applied research.

Published in	International Journal of Systems Science and Applied Mathematics (Volume 10, Issue 1)
DOI	10.11648/j.ijssam.20251001.11
Page(s)	1-6
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2025. Published by Science Publishing Group

Keywords

Ergodic Theory, Two-Dimensional Torus, Invariant Functions, Fourier Series, Lebesgue Measure

1. Introduction

Ergodic theory (ET) is a dynamic branch of mathematics that investigates the long-term behavior of systems preserving a measure. Its roots trace back to Ludwig Boltzmann in the 19th century, who introduced the term "ergodic" in the context of statistical mechanics to describe the motion of gas particles. The term itself is derived from the Greek words ergon (work) and odos (path)

[1]

. Over time, ET has evolved into a versatile framework applied in probability theory, statistical physics, functional analysis, and dynamical systems

[2]

At the foundation of ET lies the concept of transformations that preserve a measure. A transformation is deemed ergodic if, when iterated, the system cannot be decomposed into simpler invariant subsystems

[3]

. A foundational result in this area is the Poincaré Recurrence Theorem, which asserts that almost every point in a measurable subset of the phase space will eventually return to that subset

[4]

. This theorem provided early insights into the deep connections between geometry, dynamics, and measure theory, setting the stage for further developments

[5]

A central aspect of ET is its focus on invariant functions—functions that remain constant under the action of a transformation. For a transformation to be ergodic, any invariant real-valued function must be constant almost everywhere. This property is pivotal for understanding the statistical behavior of systems over time and underpins applications in areas such as statistical physics, number theory, and chaos theory

[6, 7]

The study of ergodic transformations on the torus has attracted considerable attention due to the torus's intrinsic simplicity and structural richness. Often visualized as a doughnut-shaped surface, the torus serves as a model for periodic and quasi-periodic systems

[8]

. While transformations on the one-dimensional torus are well-explored, with established conditions for ergodicity

[9]

, extending these analyses to the two-dimensional torus introduces additional challenges. These include proving ergodicity and analyzing invariant functions in higher-dimensional settings.

Motivated by these complexities, this study builds on foundational concepts in ET to explore the ergodic properties of maps on the two-dimensional torus. In particular, it examines cases where every invariant real-valued function is constant, using FS as a central analytical tool. Fourier analysis facilitates the decomposition of functions into orthonormal sequences in Hilbert spaces, enabling systematic comparison of Fourier coefficients to demonstrate ergodicity

[10]

This research also underscores the interplay between ergodic transformations, Lebesgue measures (LM), and invariant functions. By leveraging these connections, the study establishes robust conditions under which maps on the two-dimensional torus exhibit ergodic behavior

[11]

. The findings not only extend classical results in ET but also contribute to understanding higher-dimensional systems, where ergodicity plays a critical role in analyzing dynamical systems

[12]

In summary, this work provides a systematic approach to proving ergodicity using FS and invariant measures. The implications extend beyond the two-dimensional torus, offering a foundation for analyzing higher-dimensional transformations and advancing the broader study of ergodic properties in dynamical systems

[6]

2. Remakes on Special Torus

Originally, skew-product transformations were considered in the context of the torus. This concept can be traced back to

[13]

, who demonstrated that a skew-product transformation

on the torus

, where

is irrational, has the same spectral type as the cross-product of the shift transformation on an infinite torus. However, while the induced operators in

are unitarily equivalent, the transformations are not isomorphic.

In the context of

[14]

, the transformation

is given for

irrational, where

is a measurable function. The condition for non-ergodicity in this case is that there exists an integer

and a measurable map g such that

almost everywhere. This condition provides a basis for analyzing non-ergodicity in skew-product systems.

The study of skew-product transformations extends naturally to infinite products. For instance, one can consider the map

, where

represents the skewing function. These transformations are closely linked to representations of groups and serve as valuable tools for understanding the structure of complex dynamical systems

[15, 13]

Clotet in

[16]

examined horocycle flows and showed that they admit measurable reparameterizations that yield irrational flows on the torus. However, such flows are not metrically weakly mixing. Nicol, in

[2]

established that certain skew-product transformations, such as those involving irrational rotations, are uniquely ergodic for almost every initial point on the torus.

Furthermore, translations on the torus and irrational rotations of the circle play a significant role in proving fundamental theorems related to ergodicity. These results underscore the deep connections between skew-product transformations, ergodic behavior, and the broader framework of dynamical systems.

2.1. Properties of Measure-preserving Dynamical Systems and Topological Groups

(i) Let

and let

be the σ-algebra of all Lebesgue measurable subsets of X. Suppose that

mod

where

is irrational. Here

is locally compact (but not compact), however is Hausdorff subspace of the real line

[17]

(ii) Let

be the circle group, that is, the topological group of all complex numbers

with absolute value 1 equipped with the normalized Haar measure

(

the o-algebra of all Borel subset of

) then, the map

, where

is not a root of unity

[18]

2.2. Proposition in Some Cases on Two-torus

[9, 15]

Let

then

holds, and

is almost everywhere, if and only if, their Fourier coefficients are equal i.e.

(1)

Proof:

Proving Ergodicity using FS.

Suppose

, then we associate with

the FS.

(2)

where,

(3)

If we let,

(4)

Then

is measure preserving transformation it follows that

(5)

, it follows that the

is the possible sum of the form

, which gave FS of

In particular if we take the FS for

and evaluate at

, then we obtain the FS

. i.e

(6)

almost everywhere. This prove help us to compare the relationship between Fourier coefficients and show that

is constant

[15]

3. Additive Theorem on Two Torus

Let

be irrational and let

be the mapping of the two dimensional tours

in mod 1 given by the formula.

,(7)

which is uniquely ergodic with invariant function in lebsegue measure.

[5]

Proof:

Consider an irrational rotation

as the base and

(8)

We only need to verify that

is ergodic on

with respect to the lebsegue measure.

Let

be some Borel set with

(9)

Consider the FS decomposition,

(10)

where

(11)

Taking the transformation of

(12)

(13)

(14)

(15)

Comparing the coefficient of (10) and (14), we have the identities.

(16)

In particular,

(17)

Since,

is the square sum on

then

, whenever

For

, we have

(18)

As,

and we conclude that

for all

Thus

is an arbitrary constant on

, which shows that it is an ergodic.

3.1. Lemma in L²(m) Orthonormal Double Sequence

Let

and

be a complete orthonormal double sequence in Hilbert space H,

where ∥x∥ is the norm induced by the inner product, and

are the coefficients corresponding to the orthonormal basis

[18]

Proof:

Let,

and

(19)

(20)

This completes the proof.

3.2. Multiplicative Theorem in L²(m) That Are Ergodic

Let the following be defined:

(21)

, where

are irrational real constants, for all complex

, then

is ergodic

[2]

Proof:

Now, with reference to

[18]

, let

be a complete orthonormal double sequence in Hilbert space H in Hilbert space

. If f is a function in H, i.e

, then

,(22)

holds almost everywhere

for some constants

, called the Fourier coefficient with respect to orthonormal sequence

where,

(23)

Remark:

is a circle group. Hence,

,(24)

where,

Similarly,

Next,

Suppose that

-invariant function in

, then

(25)

,(26)

holds in

Where

Comparing the coefficient of (22) and (26) we get the identities

So,

This implies that

is not an integer, except

and

is not an integer except q=0. Then for

. Thus

holds almost everywhere on

4. Concluding Remarks

This study has explored the ergodic properties of maps on the two-dimensional torus, focusing on cases where every invariant real-valued function is constant. The analysis demonstrated that such ergodic behavior occurs under specific conditions, such as the presence of irrational rotation parameters. These results build upon foundational contributions by

[5, 19]

. The findings offer valuable insights into the interplay between ergodicity, the Lebesgue measure, and transformations on the torus. By leveraging FS and systematically comparing Fourier coefficients, this research presents a robust and effective method for establishing ergodicity in two-dimensional systems.

In conclusion, the results presented in this work have broader implications for extending ET to higher-dimensional tori. The methodologies and principles outlined here provide a systematic framework for analyzing measure-preserving transformations in increasingly complex dynamical systems. This contribution deepens our understanding of invariant functions and their role in ergodicity, setting the stage for further investigations into multidimensional dynamical systems and their ergodic properties.

Abbreviations

ET	Ergodic Theory
FS	Fourier Series
LM	Lebesgue Measure

Author Contributions

George Smart Nduka: Conceptualization, Data curation, Formal Analysis, Methodology, Resources, Supervision, Writing – original draft

Henry Etaroghene Egbogho: Methodology, Project administration, Validation, Writing – review & editing

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Hawkins, J. (2021). Ergodic dynamics. Springer International Publishing.
[2]	Nicol, M., & Petersen, K. (2023). Ergodic theory: Basic examples and constructions. In Ergodic theory (pp. 3–34). New York, NY: Springer US.
[3]	Barreira, L., & Pesin, Y. (2023). Introduction to smooth ergodic theory (Vol. 231). American Mathematical Society.
[4]	Frantzikinakis, N., & McCutcheon, R. (2023). Ergodic theory: Recurrence. In Ergodic theory (pp. 61–78). New York, NY: Springer US.
[5]	del Junco, A. (2023). Ergodic theorems. In Ergodic theory (pp. 79–107). New York, NY: Springer US.
[6]	Buzzi, J. (2023). Chaos and ergodic theory. In Ergodic theory (pp. 633–664). New York, NY: Springer US.
[7]	Aravinda, C. S., & Bhat, V. S. (2022). Basic ergodic theory. In Elements of dynamical systems: Lecture notes from NCM School (pp. 73–107). Singapore: Springer Nature Singapore.
[8]	Adams, T., & Quas, A. (2023). Ergodicity and mixing properties. In Ergodic theory (pp. 35–60). New York, NY: Springer US. https://doi.org/10.1007/978-1-0716-2388-6_175
[9]	Damanik, D., & Fillman, J. (2022). One-dimensional ergodic Schrödinger operators: I. General theory (Vol. 221). American Mathematical Society.
[10]	Cohen, G., & Lin, M. (2023). Uniform ergodicity and the one-sided ergodic Hilbert transform. arXiv preprint arXiv: 2310.15561.
[11]	Okubo, K. I., & Umeno, K. (2021). Infinite ergodicity that preserves the Lebesgue measure. Chaos: An Interdisciplinary Journal of Nonlinear Science, 31(3).
[12]	Sambarino, A. (2024). A report on an ergodic dichotomy. Ergodic Theory and Dynamical Systems, 44(1), 236–289.
[13]	Silva, A. F. D. (2023). Contributions to phase transition of intermittent skew-product and piecewise monotone dynamics on the circle.
[14]	Liu, N., & Liu, X. (2024). On the variational principle for a class of skew product transformations. arXiv preprint arXiv: 2406.16883.
[15]	Danilenko, A. I., & Silva, C. E. (2023). Ergodic theory: Nonsingular transformations. In Ergodic theory (pp. 233–292). New York, NY: Springer US.
[16]	Clotet, S. B. (2022). Sorbonne Université (Doctoral dissertation, Université de Lorraine).
[17]	Yu, H. (2021). An improvement on Furstenberg’s intersection problem. Transactions of the American Mathematical Society, 374(9), 6583–6610.
[18]	Feldman, G. (2023). Characterization of probability distributions on locally compact Abelian groups (Vol. 273). American Mathematical Society.
[19]	Bruin, H. (2022). Topological and ergodic theory of symbolic dynamics (Vol. 228). American Mathematical Society.

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APA Style

Nduka, G. S., Egbogho, H. E. (2025). Ergodicity of Maps on the Two-Dimensional Torus. International Journal of Systems Science and Applied Mathematics, 10(1), 1-6. https://doi.org/10.11648/j.ijssam.20251001.11

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ACS Style

Nduka, G. S.; Egbogho, H. E. Ergodicity of Maps on the Two-Dimensional Torus. Int. J. Syst. Sci. Appl. Math. 2025, 10(1), 1-6. doi: 10.11648/j.ijssam.20251001.11

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AMA Style

Nduka GS, Egbogho HE. Ergodicity of Maps on the Two-Dimensional Torus. Int J Syst Sci Appl Math. 2025;10(1):1-6. doi: 10.11648/j.ijssam.20251001.11

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@article{10.11648/j.ijssam.20251001.11,
  author = {George Smart Nduka and Henry Etaroghene Egbogho},
  title = {Ergodicity of Maps on the Two-Dimensional Torus
},
  journal = {International Journal of Systems Science and Applied Mathematics},
  volume = {10},
  number = {1},
  pages = {1-6},
  doi = {10.11648/j.ijssam.20251001.11},
  url = {https://doi.org/10.11648/j.ijssam.20251001.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijssam.20251001.11},
  abstract = {This paper considers the ergodicity of maps on the two-dimensional torus, focusing on transformations where invariant real-valued functions are constant. The study considers both additive and multiplicative transformations, providing a detailed analysis of the conditions required for a map to be classified as ergodic. The investigation is grounded in the theory of dynamical systems and leverages mathematical tools such as orthonormal double sequences in Hilbert spaces and Fourier series to establish necessary and sufficient conditions for ergodicity. By connecting these conditions to the Lebesgue measure, the research outlines the fundamental properties of ergodic transformations on the torus. A key aspect of this study is its treatment of invariant functions and their role in defining ergodic behavior. Invariant functions, which remain unchanged under the dynamics of a given transformation, are examined in depth to understand how their constancy relates to the overall system. The analysis also highlights the interplay between additive and multiplicative transformations and their impact on the ergodic properties of the system. The results of this work not only provide a robust framework for understanding the dynamics of transformations on the two-dimensional torus but also have implications for higher-dimensional systems. This contribution is particularly relevant for studying complex systems in ergodic theory, where the behavior of transformations under the Lebesgue measure often serves as a foundation for further exploration. By addressing these fundamental aspects, the paper lays the groundwork for extending these concepts to more intricate and multidimensional settings in mathematical and applied research.
},
 year = {2025}
}

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TY  - JOUR
T1  - Ergodicity of Maps on the Two-Dimensional Torus

AU  - George Smart Nduka
AU  - Henry Etaroghene Egbogho
Y1  - 2025/02/10
PY  - 2025
N1  - https://doi.org/10.11648/j.ijssam.20251001.11
DO  - 10.11648/j.ijssam.20251001.11
T2  - International Journal of Systems Science and Applied Mathematics
JF  - International Journal of Systems Science and Applied Mathematics
JO  - International Journal of Systems Science and Applied Mathematics
SP  - 1
EP  - 6
PB  - Science Publishing Group
SN  - 2575-5803
UR  - https://doi.org/10.11648/j.ijssam.20251001.11
AB  - This paper considers the ergodicity of maps on the two-dimensional torus, focusing on transformations where invariant real-valued functions are constant. The study considers both additive and multiplicative transformations, providing a detailed analysis of the conditions required for a map to be classified as ergodic. The investigation is grounded in the theory of dynamical systems and leverages mathematical tools such as orthonormal double sequences in Hilbert spaces and Fourier series to establish necessary and sufficient conditions for ergodicity. By connecting these conditions to the Lebesgue measure, the research outlines the fundamental properties of ergodic transformations on the torus. A key aspect of this study is its treatment of invariant functions and their role in defining ergodic behavior. Invariant functions, which remain unchanged under the dynamics of a given transformation, are examined in depth to understand how their constancy relates to the overall system. The analysis also highlights the interplay between additive and multiplicative transformations and their impact on the ergodic properties of the system. The results of this work not only provide a robust framework for understanding the dynamics of transformations on the two-dimensional torus but also have implications for higher-dimensional systems. This contribution is particularly relevant for studying complex systems in ergodic theory, where the behavior of transformations under the Lebesgue measure often serves as a foundation for further exploration. By addressing these fundamental aspects, the paper lays the groundwork for extending these concepts to more intricate and multidimensional settings in mathematical and applied research.

VL  - 10
IS  - 1
ER  -

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Author Information

George Smart Nduka

Department of Mathematics, Dennis Osadebay University, Asaba, Nigeria

Contact Email

http://orcid.org/0009-0006-7157-7722
Henry Etaroghene Egbogho

Department of Mathematics, Dennis Osadebay University, Asaba, Nigeria

Contact Email

http://orcid.org/0009-0001-6230-5294

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APA Style

Nduka, G. S., Egbogho, H. E. (2025). Ergodicity of Maps on the Two-Dimensional Torus. International Journal of Systems Science and Applied Mathematics, 10(1), 1-6. https://doi.org/10.11648/j.ijssam.20251001.11

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ACS Style

Nduka, G. S.; Egbogho, H. E. Ergodicity of Maps on the Two-Dimensional Torus. Int. J. Syst. Sci. Appl. Math. 2025, 10(1), 1-6. doi: 10.11648/j.ijssam.20251001.11

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AMA Style

Nduka GS, Egbogho HE. Ergodicity of Maps on the Two-Dimensional Torus. Int J Syst Sci Appl Math. 2025;10(1):1-6. doi: 10.11648/j.ijssam.20251001.11

Copy | Download

@article{10.11648/j.ijssam.20251001.11,
  author = {George Smart Nduka and Henry Etaroghene Egbogho},
  title = {Ergodicity of Maps on the Two-Dimensional Torus
},
  journal = {International Journal of Systems Science and Applied Mathematics},
  volume = {10},
  number = {1},
  pages = {1-6},
  doi = {10.11648/j.ijssam.20251001.11},
  url = {https://doi.org/10.11648/j.ijssam.20251001.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijssam.20251001.11},
  abstract = {This paper considers the ergodicity of maps on the two-dimensional torus, focusing on transformations where invariant real-valued functions are constant. The study considers both additive and multiplicative transformations, providing a detailed analysis of the conditions required for a map to be classified as ergodic. The investigation is grounded in the theory of dynamical systems and leverages mathematical tools such as orthonormal double sequences in Hilbert spaces and Fourier series to establish necessary and sufficient conditions for ergodicity. By connecting these conditions to the Lebesgue measure, the research outlines the fundamental properties of ergodic transformations on the torus. A key aspect of this study is its treatment of invariant functions and their role in defining ergodic behavior. Invariant functions, which remain unchanged under the dynamics of a given transformation, are examined in depth to understand how their constancy relates to the overall system. The analysis also highlights the interplay between additive and multiplicative transformations and their impact on the ergodic properties of the system. The results of this work not only provide a robust framework for understanding the dynamics of transformations on the two-dimensional torus but also have implications for higher-dimensional systems. This contribution is particularly relevant for studying complex systems in ergodic theory, where the behavior of transformations under the Lebesgue measure often serves as a foundation for further exploration. By addressing these fundamental aspects, the paper lays the groundwork for extending these concepts to more intricate and multidimensional settings in mathematical and applied research.
},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - Ergodicity of Maps on the Two-Dimensional Torus

AU  - George Smart Nduka
AU  - Henry Etaroghene Egbogho
Y1  - 2025/02/10
PY  - 2025
N1  - https://doi.org/10.11648/j.ijssam.20251001.11
DO  - 10.11648/j.ijssam.20251001.11
T2  - International Journal of Systems Science and Applied Mathematics
JF  - International Journal of Systems Science and Applied Mathematics
JO  - International Journal of Systems Science and Applied Mathematics
SP  - 1
EP  - 6
PB  - Science Publishing Group
SN  - 2575-5803
UR  - https://doi.org/10.11648/j.ijssam.20251001.11
AB  - This paper considers the ergodicity of maps on the two-dimensional torus, focusing on transformations where invariant real-valued functions are constant. The study considers both additive and multiplicative transformations, providing a detailed analysis of the conditions required for a map to be classified as ergodic. The investigation is grounded in the theory of dynamical systems and leverages mathematical tools such as orthonormal double sequences in Hilbert spaces and Fourier series to establish necessary and sufficient conditions for ergodicity. By connecting these conditions to the Lebesgue measure, the research outlines the fundamental properties of ergodic transformations on the torus. A key aspect of this study is its treatment of invariant functions and their role in defining ergodic behavior. Invariant functions, which remain unchanged under the dynamics of a given transformation, are examined in depth to understand how their constancy relates to the overall system. The analysis also highlights the interplay between additive and multiplicative transformations and their impact on the ergodic properties of the system. The results of this work not only provide a robust framework for understanding the dynamics of transformations on the two-dimensional torus but also have implications for higher-dimensional systems. This contribution is particularly relevant for studying complex systems in ergodic theory, where the behavior of transformations under the Lebesgue measure often serves as a foundation for further exploration. By addressing these fundamental aspects, the paper lays the groundwork for extending these concepts to more intricate and multidimensional settings in mathematical and applied research.

VL  - 10
IS  - 1
ER  -

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