**Let a finite cyclic group $G = \mathbb Z/n$ act continuously on an open $d$-ball $B^d$. Suppose further that this action extends to the closed ball $\overline{B^d}$. Is there necessarily a fixed point in the interior?**


Note that by Brouwer's fixed point theorem, there has to be some fixed point in the closed ball.


It is also true that for $n$ a prime power, every action of $\mathbb Z/n$ on an open ball has a fixed point, this can be seen from Smith theory. 


On the other hand, for $n$ not a prime power, there are actions 
of $\mathbb Z/n$ on open balls without fixed points, see e.g. Theorem 8.3 in Bredon: Introduction to Compact Transformation Groups. But I am not aware of examples of such actions that extend to the closed ball.



Cyclic groups acting on balls, and interior fixed points
There need not be a fixed point in the interior. As in Bredon's book, p. 61, there exist smooth actions of a cyclic group $C\_{r}$, of order $r$, without fixed points on $\mathbb{R}^{n}$ for large enough $n$ (e.g. $n\ge 8$) whenever $r$ is not a prime power. By one-point compactification one obtains a continuous, one-fixed-point action on the $n$-sphere $S^{n}$. Then by coning from the fixed point one obtains a continuous action on the closed ball $D^{n+1}$, smooth on the interior, but with no interior fixed points. One can push $n$ down at least to $7$ (and down to $6$ if $r$ involves at least $3$ primes), but not down to $4$, in this argument. See 
*Richard Haynes, Slawomir Kwasik, Jerrel Mast, and Reinhard Schultz*, [**Periodic maps on $\mathbb{R}^7$ without fixed points**](http://dx.doi.org/10.1017/S0305004101005345), *Math. Proc. Cambridge Philos. Soc.* **132** (2002), no. 1, 131--136.


