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Abstract: In this seminar, I will introduce the 1-dimensional cellular automata (or CA shortly) on the field $\mathbf{Z}_p$ and the ring $\mathbf{Z}_m$, $m\geq 1$ integer and will give introductory information about some ergodic properties and kind of entropies of the CAs. Firstly, I shall talk about invertibility property of 1DLCA generated linear local rules of radius $r$ over the ring $\mathbf{Z}_{m}$. I will provide a short study of ergodic theory of 1D infinite linear cellular automata over the ring $\mathbb{Z}_m$, focusing on some ergodic properties of such maps with respect to the Bernoulli and Markov measures. Then, I will explain the concept of entropy without going into detail. We study the measure theoretic entropy of 1D infinite LCA with respect to Bernoulli and Markov measures by means of the Kolmogorov-Sinai Theorem [Walters1982]. I will talk about an important numerical quantity called topological entropy of a continuous map defined on a compact metric space. By taking into account the approach given by Damico et al [Damico2003], I will study the quantity of topological entropy of 1D LCA $T_{f[-r,r]}$ over the space $\mathbb{Z}_m^{\mathbb{Z}}.$ We will investigate the directional measure-theoretic entropy of $\mathbb{Z}^{2}$-actions generated by the shift map and 1D-CA with respect to Bernoulli and Markov measures. Finally, we will present the quantity of topological directional entropy of the $\mathbb{Z}^{2}$-actions as similar to the algorithm defined by D’amico at al. [Damico2003] and [Akin2009-DEnt].