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Register a new userBurnside problem for measure preserving groups of toral homeomorphisms and for 2-groups of toral homeomorphisms
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A group $G$ is said to be periodic if for any $gin G$ there exists a positive integer $n$ with $g^n=id$. We prove that a finitely generated periodic group of homeomorphisms on the 2-torus that preserves a measure $mu$ is finite. Moreover if the group consists in homeomorphisms isotopic to the identity, then it is abelian and acts freely on $mathbb{T}^2$. In the Appendix, we show that every finitely generated 2-group of toral homeomorphisms is finite.
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A group $Gamma$ is said to be periodic if for any $g$ in $Gamma$ there is a positive integer $n$ with $g^n=id$. We first prove that a finitely generated periodic group acting on the 2-sphere $SS^2$ by $C^1$-diffeomorphisms with a finite orbit, is finite and conjugate to a subgroup of $mathrm{O}(3,R)$ and we use it for proving that a finitely generated periodic group of spherical diffeomorphisms with even bounded orders is finite. Finally, we show that a finitely generated periodic group of homeomorphisms of any orientable compact surface other than the 2-sphere or the 2-torus (which is the purpose of a previous paper of the authors) is finite.
We propose a method for proving that a toral partition into polygons is a Markov partition for a given toral $mathbb{Z}^2$-rotation, i.e., $mathbb{Z}^2$-action defined by rotations on a torus. If $mathcal{X}_{mathcal{P},R}$ denotes the symbolic dynamical system corresponding to a partition $mathcal{P}$ and $mathbb{Z}^2$-action $R$ such that $R$ is Cartesian on a sub-domain $W$, we express the 2-dimensional configurations in $mathcal{X}_{mathcal{P},R}$ as the image under a $2$-dimensional morphism (up to a shift) of a configuration in $mathcal{X}_{widehat{mathcal{P}}|_W,widehat{R}|_W}$ where $widehat{mathcal{P}}|_W$ is the induced partition and $widehat{R}|_W$ is the induced $mathbb{Z}^2$-action on the sub-domain $W$. The induced $mathbb{Z}^2$-action extends the notion of Rauzy induction of IETs to the case of $mathbb{Z}^2$-actions where subactions are polytope exchange transformations. This allows to describe $mathcal{X}_{mathcal{P},R}$ by a $S$-adic sequence of 2-dimensional morphisms. We apply the method on one example and we obtain a sequence of 2-dimensional morphisms which is eventually periodic leading to a self-induced partition. We prove that its substitutive structure is the same as the substitutive structure of the minimal subshift $X_0$ of the Jeandel-Rao Wang shift computed in an earlier work by the author. As a consequence, we deduce the equality of the two subshifts and it implies that the partition is a Markov partition for the associated toral $mathbb{Z}^2$-rotation since $X_0$ is a shift of finite type. It also implies that $X_0$ is uniquely ergodic and is isomorphic to the toral $mathbb{Z}^2$-rotation $R_0$ which can be seen as a generalization for 2-dimensional subshifts of the relation between Sturmian sequences and irrational rotations on a circle. Batteries included: the algorithms and the code to reproduce the proofs are provided.
We show in prime dimension that for two non-commuting totally irreducible toral automorphisms the set of points that equidistribute under the first map but have non-dense orbit under the second has full Hausdorff dimension. In non-prime dimension the argument fails only if the automorphisms have strong algebraic relations.
We construct different types of quasiperiodically forced circle homeomorphisms with transitive but non-minimal dynamics. Concerning the recent Poincare-like classification for this class of maps of Jaeger-Stark, we demonstrate that transitive but non-minimal behaviour can occur in each of the different cases. This closes one of the last gaps in the topological classification. Actually, we are able to get some transitive quasiperiodically forced circle homeomorphisms with rather complicated minimal sets. For example, we show that, in some of the examples we construct, the unique minimal set is a Cantor set and its intersection with each vertical fibre is uncountable and nowhere dense (but may contain isolated points). We also prove that minimal sets of the later kind cannot occur when the dynamics are given by the projective action of a quasiperiodic SL(2,R)-cocycle. More precisely, we show that, for a quasiperiodic SL(2,R)-cocycle, any minimal strict subset of the torus either is a union of finitely many continuous curves, or contains at most two points on generic fibres.
We study the polynomial entropy of the wandering part of any invertible dynamical system on a compact metric space. As an application we compute the polynomial entropy of Brouwer homeomorphisms (fixed point free orientation preserving homeomorphisms of the plane), and show in particular that it takes every real value greater or equal to 2.
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