The first two parts of this paper concern homeomorphisms of the circle, their associated earthquakes, earthquake laminations and shearing measures. We prove a finite version of Thurston's earthquake theorem Thurston4 and show that it implies the existence of an earthquake realizing any homeomorphism. Our approach gives an effective way to compute the lamination. We then show how to recover the earthquake from the measure, and give examples to show that locally finite measures on given laminations do not necessarily yield homeomorphisms. One of them also presents an example of a lamination $\mathcal {L}$ and a measure $\sigma $ such that the corresponding mapping $h_{\sigma}$ is not a homeomorphism of the circle but $h_{2\sigma}$ is.

The third part of the paper concerns the dependence between the norm $||\sigma ||_{Th}$ of a measure $\sigma$ and the norm $||h||_{cr}$ of its corresponding quasisymmetric circle homeomorphism $h_{\sigma}$. We first show that $||\sigma ||_{Th}$ is bounded by a constant multiple of $||h||_{cr}$. Conversely, we show for any $C_0>0$, there exists a constant $C>0$ depending on $C_0$ such that for any $\sigma $, if $||\sigma ||_{Th}\le C_0$ then $||h||_{cr}\le C||\sigma ||_{Th}$.

The fourth part of the paper concerns the differentiability of the earthquake curve $h_{t\sigma }, t\ge 0,$ on the parameter $t$. We show that for any locally finite measure $\sigma $, $h_{t\sigma }$ satisfies the nonautonomous ordinary differential equation $$\frac{d}{dt} h_{t\sigma}(x)=V_t(h_{t\sigma}(x)), \ t\ge 0,$$ at any point $x$ on the boundary of a stratum of the lamination corresponding to the measure $\sigma.$ We also show that if the norm of $\sigma $ is finite, then the differential equation extends to every point $x$ on the boundary circle, and the solution to the differential equation an initial condition is unique.

The fifth and last part of the paper concerns correspondence of regularity conditions on the measure $\sigma,$ on its corresponding mapping $h_{\sigma},$ and on the tangent vector $$V= V_0 = \frac{d}{dt}\big|_{t=0} h_{t\sigma}.$$ We give equivalent conditions on $\sigma, h_{\sigma}$ and $V$ that correspond to $h_{\sigma }$ being in Diff$^ {\ 1+\alpha}$ classes, where $0\le \alpha < 1$.

In this work we deal with non-discrete subgroups of ${\rm Diff}^{\omega} ({\mathbb S}^1)$, the group of orientation-preserving analytic diffeomorphisms of the circle. If $\Gamma$ is such a group, we consider its natural diagonal action ${\widetilde{\Gamma}}$ on the $n-$dimensional torus ${\mathbb T}^n$. It is then obtained a complete characterization of these groups $\Gamma$ whose corresponding ${\widetilde{\Gamma}}-$action on ${\mathbb T}^n$ is not piecewise ergodic (cf. Introduction) for all $n \in {\mathbb N}$ (cf. Theorem A). Theorem A can also be interpreted as an extension of Lie's classification of Lie algebras on ${\mathbb S}^1$ to general non-discrete subgroups of ${\mathbb S}^1$.

Let $G$ be a real algebraic group defined over $\mathbb{Q}$, let $\Gamma$ be an arithmetic subgroup, and let $T$ be a maximal $\mathbb{R}$-split torus. We classify the closed orbits for the action of $T$ on $G/\Gamma,$ and show that they all admit a simple algebraic description. In particular we show that if $G$ is reductive, an orbit $Tx\Gamma$ is closed if and only if $x^{-1}Tx$ is defined over $\mathbb{Q}$, and is (totally) divergent if and only if $x^{-1}Tx$ is defined over $\mathbb{Q}$ and $\mathbb{Q}$-split. Our analysis also yields the following: there is a compact $K \subset G/\Gamma$ which intersects every $T$-orbit. If $\mathbb{Q} {\rm -rank}(G)<\mathbb{R}{\rm -rank}(G)$, there are no divergent orbits for $T$.

Let $f: [0, +\infty) \longrightarrow (0, +\infty)$ be a sufficiently smooth convex function, vanishing at infinity. Consider the planar domain $Q$ delimited by the positive $x$-semiaxis, the positive $y$-semiaxis, and the graph of $f$. Under certain conditions on $f$, we prove that the billiard flow in $Q$ has a hyperbolic structure and, for some examples, that it is also ergodic. This is done using the cross section corresponding to collisions with the dispersing part of the boundary. The relevant invariant measure for this Poincaré section is infinite, whence the need to surpass the existing results, designed for finite-measure dynamical systems.