Algebraic Framework for Discrete Dynamical Systems over Laurent Series

Ramamonjy Aandriamifidisoa; Loukman Ben Saindou

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arXiv · 2508.04708

Algebraic Framework for Discrete Dynamical Systems over Laurent Series

Ramamonjy Aandriamifidisoa·Loukman Ben Saindou

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Abstract

We generalize the framework of discrete algebraic dynamical systems \cite{Andriamifidisoa2014} to Laurent polynomials and series over $\Z^r$, enabling the modeling of bidirectional discrete systems. By redefining the spaces $\Dprime$ and $\Aprime$, introducing a bilinear mapping (defined as the scalar product in Section 3), and extending the shift operator, we preserve the duality and adjoint properties of \cite{Andriamifidisoa2014}. These properties are rigorously proved and illustrated through examples and a data processing case study on bidirectional sequence transformations. In contrast to Oberst \cite{Ob90}, our algebraic approach emphasizes the structure of Laurent series, providing a streamlined framework for multidimensional systems. This work addresses an open question from \cite{Andriamifidisoa2014} and has applications in multidimensional data processing, such as image filtering and control theory.

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BibTeX RIS

Ramamonjy Aandriamifidisoa, Loukman Ben Saindou. 2025-07-15. Algebraic Framework for Discrete Dynamical Systems over Laurent Series. https://arxiv.org/abs/2508.04708

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