For negative-torsion maps on the annulus we show that on every $ \mathscr{C}^1 $ essential curve there is at least one point of zero torsion. As an outcome we deduce that the Hausdorff dimension of the set of points of zero torsion is greater or equal 1. As a byproduct we obtain a Birkhoff's Theorem-like result for $ \mathscr{C}^1 $ essential curves in the framework of negative-torsion maps.

<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M1">\begin{document}$ a < b $\end{document}</tex-math></inline-formula> be multiplicatively independent integers, both at least <inline-formula><tex-math id="M2">\begin{document}$ 2 $\end{document}</tex-math></inline-formula>. Let <inline-formula><tex-math id="M3">\begin{document}$ A,B $\end{document}</tex-math></inline-formula> be closed subsets of <inline-formula><tex-math id="M4">\begin{document}$ [0,1] $\end{document}</tex-math></inline-formula> that are forward invariant under multiplication by <inline-formula><tex-math id="M5">\begin{document}$ a $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M6">\begin{document}$ b $\end{document}</tex-math></inline-formula> respectively, and let <inline-formula><tex-math id="M7">\begin{document}$ C : = A\times B $\end{document}</tex-math></inline-formula>. An old conjecture of Furstenberg asserted that any planar line <inline-formula><tex-math id="M8">\begin{document}$ L $\end{document}</tex-math></inline-formula> not parallel to either axis must intersect <inline-formula><tex-math id="M9">\begin{document}$ C $\end{document}</tex-math></inline-formula> in Hausdorff dimension at most <inline-formula><tex-math id="M10">\begin{document}$ \max\{\dim C,1\} - 1 $\end{document}</tex-math></inline-formula>. Two recent works by Shmerkin and Wu have given two different proofs of this conjecture. This note provides a third proof. Like Wu's, it stays close to the ergodic theoretic machinery that Furstenberg introduced to study such questions, but it uses less substantial background from ergodic theory. The same method is also used to re-prove a recent result of Yu about certain sequences of sums.</p>

We prove a pointwise ergodic theorem for quasi-probability-measure-preserving (quasi-pmp) locally countable measurable graphs, equivalently, Schreier graphs of quasi-pmp actions of countable groups. For ergodic graphs, the theorem gives an increasing sequence of Borel subgraphs with finite connected components over which the averages of any $L^1$ function converges to its expectation. This implies that every (not necessarily pmp) locally countable ergodic Borel graph on a standard probability space contains an ergodic hyperfinite subgraph. A consequence of this is that every ergodic treeable equivalence relation has an ergodic hyperfinite free factor. The pmp case of the main theorem was first proven by R. Tucker-Drob using a deep result from probability theory. Our proof is different: it is self-contained and applies more generally to quasi-pmp graphs. Among other things, it involves introducing a graph invariant concerning asymptotic averages of functions and a method of tiling a large part of the space with finite sets with prescribed properties. The non-pmp setting additionally exploits a new quasi-order called visibility to analyze the interplay between the Radon--Nikodym cocycle and the graph structure, providing a sufficient condition for hyperfiniteness.

<p style='text-indent:20px;'>We study the regularity of a conjugacy between an Anosov automorphism <inline-formula><tex-math id="M1">\begin{document}$ L $\end{document}</tex-math></inline-formula> of a nilmanifold <inline-formula><tex-math id="M2">\begin{document}$ N/\Gamma $\end{document}</tex-math></inline-formula> and a volume-preserving, <inline-formula><tex-math id="M3">\begin{document}$ C^1 $\end{document}</tex-math></inline-formula>-small perturbation <inline-formula><tex-math id="M4">\begin{document}$ f $\end{document}</tex-math></inline-formula>. We say that <inline-formula><tex-math id="M5">\begin{document}$ L $\end{document}</tex-math></inline-formula> is <i>locally Lyapunov spectrum rigid</i> if this conjugacy is <inline-formula><tex-math id="M6">\begin{document}$ C^{1+} $\end{document}</tex-math></inline-formula> whenever <inline-formula><tex-math id="M7">\begin{document}$ f $\end{document}</tex-math></inline-formula> is <inline-formula><tex-math id="M8">\begin{document}$ C^{1+} $\end{document}</tex-math></inline-formula> and has the same volume Lyapunov spectrum as <inline-formula><tex-math id="M9">\begin{document}$ L $\end{document}</tex-math></inline-formula>. For <inline-formula><tex-math id="M10">\begin{document}$ L $\end{document}</tex-math></inline-formula> with simple spectrum, we show that local Lyapunov spectrum rigidity is equivalent to <inline-formula><tex-math id="M11">\begin{document}$ L $\end{document}</tex-math></inline-formula> satisfying both an irreducibility condition and an ordering condition on its Lyapunov exponents.

&lt;p style='text-indent:20px;'&gt;We extend the notion of Rauzy induction of interval exchange transformations to the case of toral &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ \mathbb{Z}^2 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-rotation, i.e., &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ \mathbb{Z}^2 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-action defined by rotations on a 2-torus. If &lt;inline-formula&gt;&lt;tex-math id="M3"&gt;\begin{document}$ \mathscr{X}_{\mathscr{P}, R} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; denotes the symbolic dynamical system corresponding to a partition &lt;inline-formula&gt;&lt;tex-math id="M4"&gt;\begin{document}$ \mathscr{P} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; and &lt;inline-formula&gt;&lt;tex-math id="M5"&gt;\begin{document}$ \mathbb{Z}^2 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-action &lt;inline-formula&gt;&lt;tex-math id="M6"&gt;\begin{document}$ R $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; such that &lt;inline-formula&gt;&lt;tex-math id="M7"&gt;\begin{document}$ R $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; is Cartesian on a sub-domain &lt;inline-formula&gt;&lt;tex-math id="M8"&gt;\begin{document}$ W $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, we express the 2-dimensional configurations in &lt;inline-formula&gt;&lt;tex-math id="M9"&gt;\begin{document}$ \mathscr{X}_{\mathscr{P}, R} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; as the image under a &lt;inline-formula&gt;&lt;tex-math id="M10"&gt;\begin{document}$ 2 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-dimensional morphism (up to a shift) of a configuration in &lt;inline-formula&gt;&lt;tex-math id="M11"&gt;\begin{document}$ \mathscr{X}_{\widehat{\mathscr{P}}|_W, \widehat{R}|_W} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; where &lt;inline-formula&gt;&lt;tex-math id="M12"&gt;\begin{document}$ \widehat{\mathscr{P}}|_W $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; is the induced partition and &lt;inline-formula&gt;&lt;tex-math id="M13"&gt;\begin{document}$ \widehat{R}|_W $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; is the induced &lt;inline-formula&gt;&lt;tex-math id="M14"&gt;\begin{document}$ \mathbb{Z}^2 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-action on &lt;inline-formula&gt;&lt;tex-math id="M15"&gt;\begin{document}$ W $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;.&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;We focus on one example, &lt;inline-formula&gt;&lt;tex-math id="M16"&gt;\begin{document}$ \mathscr{X}_{\mathscr{P}_0, R_0} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, for which we obtain an eventually periodic sequence of 2-dimensional morphisms. We prove that it is the same as the substitutive structure of the minimal subshift &lt;inline-formula&gt;&lt;tex-math id="M17"&gt;\begin{document}$ X_0 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; of the Jeandel–Rao Wang shift computed in an earlier work by the author. As a consequence, &lt;inline-formula&gt;&lt;tex-math id="M18"&gt;\begin{document}$ {\mathscr{P}}_0 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; is a Markov partition for the associated toral &lt;inline-formula&gt;&lt;tex-math id="M19"&gt;\begin{document}$ \mathbb{Z}^2 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-rotation &lt;inline-formula&gt;&lt;tex-math id="M20"&gt;\begin{document}$ R_0 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;. It also implies that the subshift &lt;inline-formula&gt;&lt;tex-math id="M21"&gt;\begin{document}$ X_0 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; is uniquely ergodic and is isomorphic to the toral &lt;inline-formula&gt;&lt;tex-math id="M22"&gt;\begin{document}$ \mathbb{Z}^2 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-rotation &lt;inline-formula&gt;&lt;tex-math id="M23"&gt;\begin{document}$ R_0 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; 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 code to reproduce the proofs are provided.&lt;/p&gt;

<p style='text-indent:20px;'>We address the problem of computing the critical regularity of groups of homeomorphisms of the interval. Our main result is that if <inline-formula><tex-math id="M1">\begin{document}$ H $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ K $\end{document}</tex-math></inline-formula> are two non-solvable groups then a faithful <inline-formula><tex-math id="M3">\begin{document}$ C^{1,\tau} $\end{document}</tex-math></inline-formula> action of <inline-formula><tex-math id="M4">\begin{document}$ H\times K $\end{document}</tex-math></inline-formula> on a compact interval <inline-formula><tex-math id="M5">\begin{document}$ I $\end{document}</tex-math></inline-formula> is <i>not overlapping</i> for all <inline-formula><tex-math id="M6">\begin{document}$ \tau&gt;0 $\end{document}</tex-math></inline-formula>, which by definition means that there must be non-trivial <inline-formula><tex-math id="M7">\begin{document}$ h\in H $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M8">\begin{document}$ k\in K $\end{document}</tex-math></inline-formula> with disjoint support. As a corollary we prove that the right-angled Artin group <inline-formula><tex-math id="M9">\begin{document}$ (F_2\times F_2)*\mathbb{Z} $\end{document}</tex-math></inline-formula> has critical regularity one, which is to say that it admits a faithful <inline-formula><tex-math id="M10">\begin{document}$ C^1 $\end{document}</tex-math></inline-formula> action on <inline-formula><tex-math id="M11">\begin{document}$ I $\end{document}</tex-math></inline-formula>, but no faithful <inline-formula><tex-math id="M12">\begin{document}$ C^{1,\tau} $\end{document}</tex-math></inline-formula> action. This is the first explicit example of a group of exponential growth which is without nonabelian subexponential growth subgroups, whose critical regularity is finite, achieved, and known exactly. Another corollary we get is that Thompson's group <inline-formula><tex-math id="M13">\begin{document}$ F $\end{document}</tex-math></inline-formula> does not admit a faithful <inline-formula><tex-math id="M14">\begin{document}$ C^1 $\end{document}</tex-math></inline-formula> overlapping action on <inline-formula><tex-math id="M15">\begin{document}$ I $\end{document}</tex-math></inline-formula>, so that <inline-formula><tex-math id="M16">\begin{document}$ F*\mathbb{Z} $\end{document}</tex-math></inline-formula> is a new example of a locally indicable group admitting no faithful <inline-formula><tex-math id="M17">\begin{document}$ C^1 $\end{document}</tex-math></inline-formula> action on <inline-formula><tex-math id="M18">\begin{document}$ I $\end{document}</tex-math></inline-formula>.

We show that the number of square-tiled surfaces of genus $ g $, with $ n $ marked points, with one or both of its horizontal and vertical foliations belonging to fixed mapping class group orbits, and having at most $ L $ squares, is asymptotic to $ L^{6g-6+2n} $ times a product of constants appearing in Mirzakhani's count of simple closed hyperbolic geodesics. Many of the results in this paper reflect recent discoveries of Delecroix, Goujard, Zograf, and Zorich, but the approach considered here is very different from theirs. We follow conceptual and geometric methods inspired by Mirzakhani's work.

We study the near action of the group $ \mathrm{PC} $ of piecewise continuous self-transformations of the circle. Elements of this group are only defined modulo indeterminacy on a finite subset, which raises the question of realizability: a subgroup of $ \mathrm{PC} $ is said to be realizable if it can be lifted to a group of permutations of the circle.We prove that every finitely generated abelian subgroup of $ \mathrm{PC} $ is realizable. We show that this is not true for arbitrary subgroups, by exhibiting a non-realizable finitely generated subgroup of the group of interval exchanges with flips.The group of (oriented) interval exchanges is obviously realizable (choosing the unique left-continuous representative). We show that it has only two realizations (up to conjugation by a finitely supported permutation): the left and right-continuous ones.

We consider the flow in direction $ \theta $ on a translation surface and we study the asymptotic behavior for $ r\to 0 $ of the time needed by orbits to hit the $ r $-neighborhood of a prescribed point, or more precisely the exponent of the corresponding power law, which is known as hitting time. For flat tori the limsup of hitting time is equal to the Diophantine type of the direction $ \theta $. In higher genus, we consider a generalized geometric notion of Diophantine type of a direction $ \theta $ and we seek for relations with hitting time. For genus two surfaces with just one conical singularity we prove that the limsup of hitting time is always less or equal to the square of the Diophantine type. For any square-tiled surface with the same topology the Diophantine type itself is a lower bound, and any value between the two bounds can be realized, moreover this holds also for a larger class of origamis satisfying a specific topological assumption. Finally, for the so-called Eierlegende Wollmilchsau origami, the equality between limsup of hitting time and Diophantine type subsists. Our results apply to L-shaped billiards.

We look at interval exchange transformations defined as first return maps on the set of diagonals of a flow of direction $ \theta $ on a square-tiled surface: using a combinatorial approach, we show that, when the surface has at least one true singularity both the flow and the interval exchange are rigid if and only if $ \tan\theta $ has bounded partial quotients. Moreover, if all vertices of the squares are singularities of the flat metric, and $ \tan\theta $ has bounded partial quotients, the square-tiled interval exchange transformation $ T $ is not of rank one. Finally, for another class of surfaces, those defined by the unfolding of billiards in Veech triangles, we build an uncountable set of rigid directional flows and an uncountable set of rigid interval exchange transformations.