Verify that stable and unstable manifolds transversal .

Question

I am trying show that $f(\theta) = \theta + \varepsilon sen(2\pi k \theta) \,\,mod \,1$ in $\mathbb{S}^1$ is a Morse-Smale diffeomorphism, that is

(1) the chain recurrent set is a finite set of periodic orbits, each of which is hyperbolic,
and

(2) each pair of stable and unstable manifolds of periodic points is transverse, i.e., if
$p_1, \,p_2$ are periodic points then $W^{u}(p_1)$ is transverse to $W^{s}(p_2)$.

Part (1) already proved that the chain recurrent set have $2n$ periodic points of type $\dfrac{k}{2n}$ with $k = 0, \cdots, 2n-1$ and all this points are hyperbolic. However, part (2) did not understand how I have to do it, and I found no clear example to explain what must be done to show transversality.
My main difficulty is understanding how to calculate the tangente spaces of the definition of transversality, for example, how to calculate $T_{p_1} \mathbb{S}^1$ Can someone help me?

Note: Two submanifolds $V$ and $W$ in $M$ are transverse (in $M$) provided for any point $q \in V \cap W$, we have that $T_qV + T_qW = T_qM$, that is, the tangent space to $M$ is a direct sum of the tangent spaces to $V$ and $W$.

Answer 1

There is no need to compute the tangent spaces:

It follows from the hyperbolicity of the periodic points that you necessarily generate the whole tangent space, whatever it may be (unless the manifolds don't intersect). Note that the manifold has dimension $1$.

Morse-Smale diffeomorphisms are much more complicated in higher-dimensions.

In any event, the way to compute the tangent space is to pass to coordinate charts, although it really depends on your definition of the circle.