# Show that the map $T(x)=x+a \sin^2( \pi x)$ is uniquely ergodic on the unit circle

The task is to prove that the map $$T(x)=x+a \sin^2( \pi x)$$ on the unit circle is uniquely ergodic for $$a < \frac{1}{\pi}$$. I tried to show that Birkhoff sums of the characters with respect to $$T: S^1 \to S^1$$ all converge uniformly to a constant, but I don't think that will work.

• Feb 16, 2022 at 2:34
• Thank you, but this is on the circle and the link is for the line Feb 16, 2022 at 22:21

For posterity, this is (part of) Exr.4.7.3 in Brin & Stuck's Introduction to Dynamical Systems (p.90).

Here is a humble graph: https://www.desmos.com/calculator/tgrhizoot6. It's also useful to have the phase portrait in mind; it looks like this:

(I've taken this image from Hasselblatt & Katok's A First Course in Dynamics, p.130. Note the resemblance to the projective action of shear matrices; indeed horocycle flows are also uniquely ergodic and unique ergodicity is a topological property (see Topological Invariance of Unique Ergodicity for the latter statement).)

For a fixed $$a\in]0,1/\pi[$$, put $$T=T_a: \mathbb{T}\to \mathbb{T}, x\mapsto x+a\sin^2(\pi x)$$. Note that $$T$$ has a unique fixed point $$0\in\mathbb{T}$$, so that $$\delta_0$$ is an invariant Borel probability measure. Thus to show that $$T$$ is uniquely ergodic it suffices to show that for any continuous $$\phi:\mathbb{T}\to \mathbb{R}$$ and for any $$x\in \mathbb{T}$$:

$$\lim_{n\to\infty}\dfrac{1}{n}\sum_{k=0}^{n-1}\phi\circ T^k(x)=\phi(0).$$

Since $$0$$ is a fixed point we may assume $$x\in\mathbb{T}\setminus0$$. Also note that pointwise convergence is sufficient; uniform convergence will be automatic (see e.g. Einsiedler & Ward's Ergodic Theory with a view toward Number Theory, Thm.4.10 on p.105). Let us lift everything to the unit interval $$[0,1]$$ without changing notation.

Observe that $$0< x < T(x) < T^2(x) < \cdots < T^n(x) < \cdots < 1$$ and $$\lim_{n\to\infty} T^n(x)=1$$. Fix $$\epsilon\in\mathbb{R}_{>0}$$. Then since $$\phi$$ is uniformly continuous, for some $$\delta\in\mathbb{R}_{>0}$$ if $$y$$ and $$z$$ are $$\delta$$-close then $$\phi(y)$$ and $$\phi(z)$$ are $$\epsilon$$-close. There is an $$N\in\mathbb{Z}_{>0}$$ such that for any $$n\in\mathbb{Z}_{\geq N}$$ we have $$|T^n(x)-1|<\delta$$; thus

$$\exists N\in\mathbb{Z}_{>0},\forall n\in\mathbb{Z}_{\geq N}: |\phi\circ T^n(x)-\phi(1)|<\epsilon.$$

Then we have, for $$n\in\mathbb{Z}_{\geq N}$$,

\begin{align*} \left|\dfrac{1}{n}\sum_{0\leq k

Taking $$n\to\infty$$ we are done since $$\epsilon$$ was arbitrary. (Note that $$1\sim0$$ when we project back down to the circle.)

• Thank you. I also thought to say that the set of non-wandering points has to have measure 1 for any invariant probability measure. The idea then is to show that the fixed point is the only point that is non-wandering and we have the result. This is basically the proof for that as well. Feb 16, 2022 at 22:27
• Feb 16, 2022 at 22:40
• Barreira's books (with coauthors typically) are also worthwhile overall in my opinion. There are also books about "beyond hyperbolicity", that generalize hyperbolicity in different directions. Overall I think it is more efficient to chase references in papers as needed. Feb 16, 2022 at 22:46
• Finally Anosov's memoir "Geodesic Flows On Closed Riemannian Manifolds With Negative Curvature" is still relevant, readable, and instructive in my opinion. Feb 16, 2022 at 22:53
• (I also have an opinion on whose books should be avoided of course, but strategically I will keep this to myself.) Feb 16, 2022 at 22:57