Strong persistence index and fluctuations in colon powers of monomial ideals

Let $I$ be an ideal in a commutative Noetherian ring $R$. We say that a positive integer ${\mathrm{\ell }}_0$ is the strong persistence index of $I$ if ${\mathrm{\ell }}_0$ is the smallest integer such that $(I^{\mathrm{\ell }+1}{:}_{R}I)=I^{\mathrm{\ell }}$ for all $\mathrm{\ell }\ge {\mathrm{\ell }}_0$. The first aim of this paper is to study this notion for monomial ideals. We also introduce the notion of fluctuation in colon powers if there exist positive integers $a<b<c$ such that at least one of the following cases occurs: (i) $(I^a:I)=I^{a-1}$, $(I^b:I)\mathrm{\ne }{I}^{b-1}$, but $(I^c:I)=I^{c-1}$. (ii) $(I^a:I)\mathrm{\ne }{I}^{a-1}$, $(I^b:I)=I^{b-1}$, but $(I^c:I)\mathrm{\ne }{I}^{c-1}$. The second purpose of this work is to study this phenomenon for monomial ideals.