Ergodicity of Maps on the Two-Dimensional Torus

George Smart Nduka; Henry Etaroghene Egbogho

doi:doi:10.11648/j.ijssam.20251001.11

Research Article |

| Peer-Reviewed

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

Views: Downloads:

Download PDF

Share This Article

Twitter
Linked In
Facebook

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

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.

Cite This Article

Plain Text BibTeX RIS

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

Copy | Download

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

Copy | Download

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  -

Copy | Download

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

Download PDF

Sections

Plain Text BibTeX RIS

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

Copy | Download

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

Copy | Download

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  -

Copy | Download