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We prove various results that, given a sufficiently rich subgroup $G$ of the group of homeomorphisms on the real line, describe the structure of the other possible actions of $G$ on the real line, and address under which conditions such actions must be semi-conjugate to the natural defining action of $G$. The main assumption is that $G$ should be locally moving, meaning that for every open interval the subgroup of elements whose support is contained in such interval acts on it without fixed points. One example (among many others) is given by Thompson's group $F$. A first rigidity result implies that if $G$ is a locally moving group, every faithful minimal action of $G$ on the real line by $C^1$ diffeomorphisms is conjugate to its standard action. It turns out that the situation is much wilder when considering actions by homeomorphisms: for a large class of groups, including Thompson's group $F$, we describe uncountably many conjugacy classes of minimal faithful actions by homeomorphisms on the real line. To gain insight on such exotic actions, we introduce and develop the notion of $\mathbb{R}$-focal action, a class of actions on the real line that can be encoded by certain actions by homeomorphisms on planar real trees fixing an end. Under a suitable finite generation condition on a locally moving group $G$, we prove that every minimal faithful action of $G$ on the line is either conjugate to the standard action, or it is $\mathbb{R}$-focal and the action on the associated real tree factors via a horofunction onto the standard action of $G$ on the line. Among applications of this result, we show that for a large class of locally moving groups, the standard action is locally rigid, in the sense that every sufficiently small perturbation in the compact-open topology gives a semi-conjugate action. This is based on an analysis of the space of harmonic actions on the line for such groups., 161 pages, 11 figures

Following the recent advances in the study of groups of circle diffeomorphisms, we describe an efficient way of classifying the topological dynamics of locally discrete, finitely generated, virtually free subgroups of the group $\mathsf{Diff}^\omega_+(\mathbb S^1)$ of orientation preserving real-analytic circle diffeomorphisms, which include all subgroups of $\mathsf{Diff}^\omega_+(\mathbb S^1)$ acting with an invariant Cantor set. An important tool that we develop, of independent interest, is the extension of classical ping-pong lemma to actions of fundamental groups of graphs of groups. Our main motivation is an old conjecture by P. R. Dippolito [Ann. Math. 107 (1978), 403--453] from foliation theory, which we solve in this restricted but significant setting: this and other consequences of the classification will be treated in more detail in a companion work., Comment: v3 36 pages, 5 figures; cosmetic changes

This text is an expanded series of lecture notes based on a 5-hour course given at the workshop entitled "Workshop for young researchers: Groups acting on manifolds" held in Teres\'opolis, Brazil in June 2016. The course introduced a number of classical tools in smooth ergodic theory -- particularly Lyapunov exponents and metric entropy -- as tools to study rigidity properties of group actions on manifolds. We do not present comprehensive treatment of group actions or general rigidity programs. Rather, we focus on two rigidity results in higher-rank dynamics: the measure rigidity theorem for affine Anosov abelian actions on tori due to A. Katok and R. Spatzier [Ergodic Theory Dynam. Systems 16, 1996] and recent the work of the main author with D. Fisher, S. Hurtado, F. Rodriguez Hertz, and Z. Wang on actions of lattices in higher-rank semisimple Lie groups on manifolds [arXiv:1608.04995; arXiv:1610.09997]. We give complete proofs of these results and present sufficient background in smooth ergodic theory needed for the proofs. A unifying theme in this text is the use of metric entropy and its relation to the geometry of conditional measures along foliations as a mechanism to verify invariance of measures., Comment: Final version (fixed typos). Main text by A. Brown, with 4 appendices by D. Malicet, D. Obata, B. Santiago and M. Triestino, S.Alvarez and M. Rold\'an. Edited by M. Triestino. To appear in Ensaios Matem\'aticos (by Brazilian Mathematical Society)