On the growth of topological complexity

Let $mathrm{TC}_r(X)$ denote the $r$-th topological complexity of a space $X$. In many cases, the generating function $sum_{rge 1}mathrm{TC}_{r+1}(X)x^r$ is a rational function $frac{P(x)}{(1-x)^2}$ where $P(x)$ is a polynomial with $P(1)=mathrm{cat}(X)$, that is, the asymptotic growth of $mathrm{TC}_r(X)$ with respect to $r$ is $mathrm{cat}(X)$. In this paper, we introduce a lower bound $mathrm{MTC}_r(X)$ of $mathrm{TC}_r(X)$ for a rational space $X$, and estimate the growth of $mathrm{MTC}_r(X)$.