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Kočergin introduced in 1975 a class of smooth flows on the two torus that are mixing. When these flows have one fixed point, they can be viewed as special flows over an irrational rotation of the circle, with a ceiling function having a power-like singularity. Under a Diophantine condition on the rotation’s angle, we prove that the special flows actually have a $t^{-\eta }$-speed of mixing, for some $\eta \>0$.

We present a proof of Herman’s Last Geometric Theorem asserting that if $F$ is a smooth diffeomorphism of the annulus having the intersection property, then any given $F$-invariant smooth curve on which the rotation number of $F$ is Diophantine is accumulated by a positive measure set of smooth invariant curves on which $F$ is smoothly conjugated to rotation maps. This implies in particular that a Diophantine elliptic fixed point of an area preserving diffeomorphism of the plane is stable. The remarkable...

We show that for any irrational number α and a sequence ${m_l}_{l\in \mathbb{N}}$ of integers such that $li{m}_{l\to \infty }\left|\right||m_l\alpha \left|\right||=0$, there exists a continuous measure μ on the circle such that $li{m}_{l\to \infty }{\int }_{}\left|\right||m_l\theta \left|\right||d\mu \left(\theta \right)=0$. This implies that any rigidity sequence of any ergodic transformation is a rigidity sequence for some weakly mixing dynamical system. On the other hand, we show that for any α ∈ ℝ - ℚ, there exists a sequence ${m_l}_{l\in \mathbb{N}}$ of integers such that $\left|\right||m_l\alpha \left|\right||\to 0$ and such that $m_l\theta \left[1\right]$ is dense on the circle if and only if θ ∉ ℚα + ℚ.