Asymptotic behavior of invariants of syzygies of maximal Cohen-Macaulay modules

Let $(A,\mathfrak{m})$ be a complete intersection ring of codimension $c\ge 2$ and dimension $d\ge 1$. Let $M$ be a finitely generated maximal Cohen-Macaulay $A$-module. Set $M_i={\text{Syz}}_i^A(M)$. Let $e_i^{\mathfrak{m}}(M)$ be the $i$-th Hilbert coefficient of $M$ with respect to $\mathfrak{m}$. We prove for all $i\gg 0$, the function $i\mapsto e_j^{\mathfrak{m}}(M_i)$ is a quasi-polynomial type with period $2$ and degree $\text{cx}(M)-1$ for $j=0,1$, where $\text{cx}(M)$ is the complexity of $M\mathrm{.}$ For $\text{cx}(M)=2,$ we prove $$\underset{n\to \mathrm{\infty }}{\lim }{\displaystyle \frac{e_1^{\mathfrak{m}}(M_{2n+j})}{n}}\ge \underset{n\to \mathrm{\infty }}{\lim }{\displaystyle \frac{e_0^{\mathfrak{m}}(M_{2n+j})}{n}}-\underset{n\to \mathrm{\infty }}{\lim }{\displaystyle \frac{\mu (M_{2n+j})}{n}}$$ for $j=0,1$. When equality holds, we prove that the Castelnuovo-Mumford regularity of the associated graded ring of $M_i$ with respect to the maximal ideal $\mathfrak{m}$ is bounded for all $i\ge 0$.