Consider a peacewise continuous map $\gamma:X \rightarrow X$ where $X \subset \mathbb{R}^2$. Now suppose we wish to prove that the map converges to a periodic orbit.
Normally, to prove such a statement with such a map, we may construct an interval $I \subset X$ such that if $x \in I$ and $\gamma ^{n}(x) \in I$ for all positive multiples $k$ of $n$ so that $\gamma ^{kn}(x) \in I$, then it may be shown that $\text{d}(\gamma ^n(x), \gamma^n(y) \leq C\text{d}(x, y)$ for $x, y \in I$ and $C$ being a Lipschitz Constant. From here the contraction mapping theorem (Banach Fixed Point Theorem) could be invoked to prove the map converges to periodic orbits of period $n$.
Now consider the case where we know our map $\gamma:X \rightarrow X$ cannot diverge, and it also cannot contract to a fixed point, and we know that distances contract upon iteration of the map. That is, for some point $x \in X$ and a point $y$ in the neighborhood of $x$, we have $d(\gamma (x), \gamma(y)) \leq d(\gamma^2(x), \gamma^2(y))$.
In my case, this map is so general that I am unable to construct intervals $I$ such as that described above, so the route to proving convergence to a periodic orbit is far less clear. In particular, I am wondering if knowledge that the map cannot diverge, does not converge to a fixed point, and contracts distances at every iteration is sufficient in proving the map must converge to a periodic orbit?
I am guessing this is not the case, so what machinery would I need in order to prove convergence to a periodic orbit for such a map? Let me know if any of this is unclear and I will try my best to elucidate.