6533b828fe1ef96bd1288a39

RESEARCH PRODUCT

groups acting on the line and the circle with at most N fixed points

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A classical theme in dynamical systems is that the first fundamental information comes from the understanding of periodic orbits. When studying group actions, this means that we want to understand the fixed points of elements of the group, and a natural question that emerges from that is: Which groups of homeomorphisms can act on a 1-manifold having all non-trivial elements with at most N fixed points? Our main objective in this work is to approach that question and understand what properties can such dynamical hypothesis induces to the group.For the case N=0, a classical result from O. Hölder implies that such group of homeomorphisms acting on the line is always semi-conjugate to a subgroup of translations and that such group of homeomorphisms acting on the circle is always is semi-conjugate to a subgroup of rotations. Now, for N>0 there are two classical examples for that question, the action of the affine group on the line, with N=1, and the action of the protective linear group on the circle, with N=2, and if by one hand a result from V. V. Solodov shows that we have a similar classification for group actions on the line which states that if N=1 then the group is either elementary or semi-conjugate to a subgroup of the affine group, by the other hand, N. Kovačević presented new examples of group actions on the circle with N=2 which are not semi-conjugate to any subgroup of the protective linear group, which proves that a similar statement doesn't hold for group actions on the circle.In this work we show that Solodov's result holds even for N=2 and that once included the hypothesis of non-discreteness a similar classification also holds for group action on the circle with N=2. Moreover, inspired by some of the ideas of Kovačević we introduced the concept of amalgamated product of actions of the circle by considering the blow-up of two distinct groups actions and rearranging them so that the minimal invariant set of one group action is included in the complement of the minimal invariant set of the other. This concept proves to be a great tool to create new examples of group actions on the circle which are not semi-conjugate to any subgroup of the protective linear group, and such that every non-trivial element has at most N fixed points, and it also leads to the construction of a second family of examples of group actions where every non-trivial element has at most N fixed points, which are HNN-extensions of actions.Finally, we present examples with high regularity, that cannot be obtained directly by the amalgamated product of actions, of finitely generated groups of diffeomorphisms of the circle where every non-trivial element fixes at most 2 points and which are not semi-conjugate (and even not isomorphic) to any subgroup of the protective linear group. Therefore, we can conclude that only increase the regularity doesn't give us a classification theorem.