Functions with a maximal number of finite invariant or internally-1-quasi-invariant sets or supersets
Abstract
A relaxation of the notion of invariant set, known as $k$-quasi-invariant set, has appeared several times in the literature in relation to group dynamics. The results obtained in this context depend on the fact that the dynamic is generated by a group. In our work, we consider the notions of invariant and 1-internally-quasi-invariant sets as applied to an action of a function $f$ on a set $I$. We answer several problems of the following type, where $k \in \{0,1\}$: what are the functions $f$ for which every finite subset of $I$ is internally-$k$-quasi-invariant? More restrictively, if $I = \mathbb{N}$, what are the functions $f$ for which every finite interval of $I$ is internally-$k$-quasi-invariant? Last, what are the functions $f$ for which every finite subset of $I$ admits a finite superset that is internally-$k$-quasi-invariant? This parallels a similar investigation undergone by C. E. Praeger in the context of group actions.
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References
M. Alaeiyan and B. Razzaghmanieshi. Permutation groups with bounded movement having maximum orbits. Proc. Indian Acad. Sci. (Math. Sci.), 122(2):175–179, 2012.
M. Alaeiyan and S. Yoshiara. Permutation groups of minimal movement. Arch. Math., 85(3):211–226, 2005.
L. Brailovsky. Structure of quasi-invariant sets. Arch. Math., 59(4):322–326, 1992.
L. Brailovsky, D. V. Pasechenik, and C. E. Praeger. Subsets close to invariant subsets for group actions. Proc. Am. Math. Soc., 123(8):2283–2295, 1995.
M. Brin and G. Stuck. Introduction to Dynamical Systems. Cambridge University Press, 2002.
N. El Idrissi and S. Kabbaj. Independence, infinite dimension, and operators. Moroccan Journal of Pure and Applied Analysis (MJPAA), 9(1):86–96, 2023.
A. Katok and B. Hasselblatt. Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, 1995.
A. Kharazishvili. On some applications of almost invariant sets. Bull. TICMI, 23(2):115–124, 2019.
P. S. Kim and Y. Kim. Certain groups with bounded movement having the maximal number of orbits. J. Algebra, 252(1):74–83, 2002.
C. E. Praeger. On permutation groups with bounded movement. J. Algebra, 144(2):436–442, 1991.
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