Let $(X,d)$ be an infinite compact [metric space](/page/Metric%20Space), and let $f:X\to X$ be a continuous map. Suppose that the periodic points of $f$, namely the points $p\in X$ for which there exists $m\in\mathbb{N}$ with $f^m(p)=p$, are dense in $X$. Suppose also that $f$ is residue-refined topologically transitive: for every $m\in\mathbb{N}$, every residue $j\in\{0,\dots,m-1\}$, and every pair of non-empty open sets $U,V\subset X$, there exists $n\in\mathbb{N}$ with $n\equiv j\pmod m$ such that $f^n(U)\cap V\neq\varnothing$. Then $f$ has sensitive dependence on initial conditions: there exists $\varepsilon>0$ such that for every $x\in X$ and every open neighbourhood $U\subset X$ of $x$, there are $y\in U$ and $n\in\mathbb{N}$ satisfying $d(f^n(x),f^n(y))>\varepsilon$.