Winding Numbers and Fixed Point Theorems

Question

It is known that winding numbers can be used to prove the existence of fixed point theorems in two dimensions. We look at the vector $f(x)-x$ and if the corresponding winding number does not equal $0$ then we can establish the existence of a fixed point. For example, in the following book http://ifts.zju.edu.cn/profiles/xingangwang/Course2010/download/Yorke-chaos.pdf page 208, winding numbers are used to prove the existence of a fixed point in the case of a covering relation, or correctly aligned window, in two dimensions. In the case of covering relations, or in the case of a Brouwer fixed point (where the map is from a ball to itself), I believe that the winding number of the vector $f(x)-x$ is always one, thus we can prove the existence of one fixed point. In the case of a horseshoe map (see the previous book page 214), however, we can have that the winding number is two (one can easily verify this). We also know that the horseshoe map has two fixed points. So my question is the following, does anyone know whether a winding number $i$ establishes $i$ fixed points, or can we have, for example, a case where the winding number is two but only one fixed point exists. Thanks.