This paper investigates recent advances in topological dynamics by applying modified open-set topologies to analyze key system behaviors, such as transitivity, minimality, chaos, and mixing. By relaxing standard topological constraints through Λθ-open and Λδ-open sets, these frameworks offer crucial insights into system dynamics where classical transitivity fails. Within this context, we introduce and analyze Λθ-transitivity and Λδ-transitivity, evaluating their fundamental properties alongside existing generalizations. Additionally, we propose Λθ-mixing and Λδ-minimality—variants of θ-mixing and δ-minimality—and establish their interrelationships with other established modes of transitivity, minimality, and chaoticity. The primary contribution of this work lies in establishing a strict hierarchical framework that demonstrates how dynamical properties translate across topologies of varying granularities and establishes sufficient conditions under which generalized forms imply classical properties.
References
[1]
Akin, E. (1993) The General Topology of Dynamical Systems. American Mathematical Society.
Kaki, M.N.M. (2012) Introduction to θ-Type Transitive Maps on Topological Spaces. International Journal of Basic & Applied Sciences, 12, 104-108.
[4]
Kaki, M.N.M. (2012) New Types of δ-Transitive Maps. International Journal of Engineering & Technology, 12, 134-136.
[5]
Kaki, M.N.M. (2014) New Conceptions of Transitivity and Minimal Mappings. Science Research, 2, 1-6. https://doi.org/10.11648/j.sr.20140201.11
[6]
Murad Kaki, M.N. (2014) New Types of Chaotic Maps on Topological Spaces. Inter-national Journal of Electrical and Electronic Science, 1, 1-5.
[7]
Hasselblatt, B. and Katok, A. (2019) A First Course in Dynamics: With a Panorama of Recent Developments. Cambridge University Press.
[8]
Akin, E., Auslander, J. and Berg, K. (1996) When Is a Transitive Map Chaotic? In: Bergelson, V., March, P. and Rosenblatt, J., Eds., Convergence in Ergodic Theory and Probability, Walter de Gruyter, 25-40. https://doi.org/10.1515/9783110889383.25
[9]
Kaki, M.N.M. (2015) New Concepts of Alpha-Chaotic Maps. Journal of Multidisciplinary Engineering Science and Technology, 2, 187-190.
[10]
Fomin, S. (1943) Extensions of Topological Spaces. The Annals of Mathematics, 44, 471-480. https://doi.org/10.2307/1968976
[11]
Velicko, N.V. (1968) H-Closed Topological Spaces. American Mathematical Society Translations, Series 2, 78, 103-118.
[12]
Dickman, R.F. and Porter, J.R. (1975) θ-Closed Subsets of Hausdorff Spaces. Pacific Journal of Mathematics, 59, 407-415. https://doi.org/10.2140/pjm.1975.59.407
[13]
Dickman, R.F. and Porter, J.R. (1977) θ-Perfect and θ-Absolutely Closed Functions. Illinois Journal of Mathematics, 21, 42-60. https://doi.org/10.1215/ijm/1256049499
[14]
Dontchev, J. and Maki, H. (1999) Groups of θ-Generalized Homeomorphisms and the Digital Line. Topology and Its Applications, 95, 113-128.
[15]
Jafari, S. (1998) Some Properties of Quasi θ-Continuous Functions. Far East Journal of Mathematical Sciences, 6, 689-696.
[16]
Auslander, J. (1988) Minimal Flows and Their Extensions. North-Holland.
[17]
Kaki, M.N.M. (2012) New Types of Transitive Functions and Minimal Systems. International Journal of Basic & Applied Sciences, 12, 53-58.
[18]
De Vries, J. (1993) Elements of Topological Dynamics. Kluwer Academic Publishers.
[19]
Taheri, N. and Kaleel, A.K.K. (2023) Symbolic Dynamics and Topological Complexity in Discrete Dynamical Systems. Babylonian Journal of Mathematics, 2023, 66-71. https://doi.org/10.58496/bjm/2023/013
[20]
Leslie, T.M. and Peszek, J. (2026) Topological and Purely Topological Alignment Dynamics. arXiv: 2601.11828v2.
[21]
Rivas, B., Kalies, W., Iwasaki, K., Bloch, A. and Ghaffari, M. (2026) Topological Dynamics via Learned Hybrid Systems. Proceedings of Machine Learning Research, 331, 1663-1674.
