arXiv subjects
Publications and source records attributed to Erwan Lanneau.
We show that for $g\ge 2$, all integers $1 \le d \le 3g-3$ arise as trace field degrees of pseudo-Anosov mapping classes in the Torelli group of the closed orientable surface of genus $g$. Our method uses the Thurston-Veech construction of pseudo-Anosov maps, and we provide examples where the stretch factor has algebraic degree any even number between two and $6g-6$. This validates a claim by Thurston from the 1980s.
Fix $g \geq 2$. Let $\mathsf{t}(g)$ be the maximal order of the translation group among all genus-$g$ abelian differentials. By work of Schlage-Puchta and Weitze-Schmithüsen, $\mathsf{t}(g) \leq 4(g - 1)$. They also classify the $g$ attaining this bound. We assume $g$ is outside this class. We first prove that either $\mathsf{t}(g) = (2(m + 1) / m) (g - 1)$ for some $m \in \mathbb{N} \setminus \{0\}$, when regular genus-$g$ origamis exist, or $\mathsf{t}(g) = 2(g - 1)$, when they do not exist. In the former case, only some values of $m > 1$ are realizable; $m = 5$ is the smallest. The resulting set of genera, those satisfying $\mathsf{t}(g) = (12/5)(g - 1)$, contains infinitely long arithmetic progressions. The same holds for any odd prime $m$ congruent to $2$ modulo $3$. In the latter case, "many" strata of the form $\mathcal{H}(g - 1, g - 1)$, $\mathcal{H}(2k^q)$ or $\mathcal{H}(k^{2q})$, where $k \geq 1$ is an integer and $q$ is prime, contain no regular origamis; we derive a complete classification. As an application, we exhibit infinite families of genera $g$ for which $\mathsf{t}(g) = 2(g - 1)$: $g = p + 1$ for prime $p \geq 5$; $g = p^2 + 1$ for prime, but not Sophie Germain prime, $p$; and $g = pq + 1$, for distinct primes $p, q \geq 5$.
We show that the stretch factor $λ(f)$ of an orientation-reversing fully-punctured pseudo-Anosov map $f$ on a finite-type orientable surface $S$, with $-χ(S) \geq 4$ and having at least two puncture orbits, satisfies the inequality $λ(f)^{-χ(S)} \geq σ^2$, where $σ=1+\sqrt{2}$ is the silver ratio. We provide examples showing that this bound is asymptotically sharp. This extends previous results of Hironaka and the third author to orientation-reversing maps.
We show that any real number in [0,1) is a diffusion rate for the wind-tree model with rational parameters. We will also provide a criterion in order to describe the shape of the Lyapunov spectrum of cocycles obtained as suspension of a representation. As an application, we exhibit an infinite family of wind-tree billiards for which the interior of the Lyapunov spectrum is a big as possible: this is the full square (0,1)^2. To the best of the knowledge of the authors, these are the first complete descriptions where the interior of the Lyapunov spectrum is known explicitly in dimension two, even for general Fuchsian groups.
We prove that every even number $2 \le 2d \le 2g$ is realised as the degree of a Thurston-Veech pseudo-Anosov stretch factor in every connected component of every stratum of the moduli space of Abelian differentials.
We study the function $$\mbox{KVol} : (X,ω)\mapsto \mbox{Vol} (X,ω) \sup_{α,β} \frac{\mbox{Int} (α,β)}{l_g (α) l_g (β)}$$ defined on the moduli spaces of translation surfaces. More precisely, let $\mathcal T_n$ be the Teichmüller discs of the original Veech surface $(X_n,ω_n)$ arising from right-angled triangle with angles $(π/2,π/n,(n-2)π/2n)$ by the unfolding construction for $n\geq 5$. For $n \equiv 1 \mod 2$ and any $(X,ω)\in \mathcal T_n$, we establish the (sharp) bounds $$ \frac{n}{2} \cot \fracπ{n} \leq \mbox{KVol}(X,ω) \leq \frac{n}{2} \cot \fracπ{n} \cdot \frac1{\sin \frac{2π}{n}}.$$ The lower bound is uniquely realized at $(X_n,ω_n)$.
In the current note we extend results by Marmi, Moussa and Yoccoz about cohomological equations for interval exchange transformations to irreducible linear involutions.
The Ruelle resonances of a dynamical system are spectral data describing the precise asymptotics of correlations. We classify them completely for a class of chaotic two-dimensional maps, the linear pseudo-Anosov maps, in terms of the action of the map on cohomology. As applications, we obtain a full description of the distributions which are invariant under the linear flow in the stable direction of such a linear pseudo-Anosov map, and we solve the cohomological equation for this flow.
We prove that for each discriminant $D \equiv 0,1 \mod 4, D \not\in\{4,9\}$, the corresponding Prym eigenform locus discovered by McMullen in the stratum $\mathcal{H}(6)$ is connected. Thus, the projection of any of those loci in the moduli space is a single Teichmüller curve. Along the way, we obtain a classification of primitive square-tiled surfaces in the locus $\mathrm{Prym}(6)$ of Prym forms in $\mathcal{H}(6)$.
We propose a general framework for studying pseudo-Anosov homeomorphisms on translation surfaces. This new approach, among other consequences, allows us to compute the systole of the Teichmueller geodesic flow restricted to the hyperelliptic connected components, settling a question of Farb. We stress that all proofs and computations are performed without the help of a computer. As a byproduct, our methods give a way to describe the bottom of the lengths spectrum of the hyperelliptic components.
The minimal stratum in Prym loci have been the first source of infinitely many primitive, but not algebraically primitive Teichmueller curves. We show that the stratum Prym(2,1,1) contains no such Teichmueller curve and the stratum Prym(2,2) at most 92 such Teichmueller curves. This complements the recent progress establishing general -- but non-effective -- methods to prove finiteness results for Teichmueller curves and serves as proof of concept how to use the torsion condition in the non-algebraically primitive case.
We show that in any non-arithmetic rank 1 orbit closure of translation surfaces, there are only finitely many Teichmüller curves. We also show that in any non-arithmetic rank 1 orbit closure, any completely parabolic surface is Veech.
The Teichmueller polynomial of a fibered 3-manifold plays a useful role in the construction of mapping class having small stretch factor. We provide an algorithm that computes this polynomial of the fibered face associated to a pseudo-Anosov mapping class of a disc homeomorphism. As a byproduct, our algorithm allows us to derive all the relevant informations on the topology of the different fibers that belong to the fibered face.
This paper is devoted to the classification of connected components of Prym eigenform loci in the strata H(2,2)^odd and H(1,1,2) in the Abelian differentials bundle in genus 3. These loci, discovered by McMullen are GL^+(2,R)-invariant submanifolds (of complex dimension 3) that project to the locus of Riemann surfaces whose Jacobian variety has a factor admitting real multiplication by some quadratic order Ord_D. It turns out that these subvarieties can be classified by the discriminant D of the corresponding quadratic orders. However there algebraic varieties are not necessarily irreducible. The main result we show is that for each discriminant D the corresponding locus has one component if D is congruent to 0 or 4 mod 8, two components if D is congruent to 1 mod 8, and is empty otherwise. Our result contrasts with the case of Prym eigenform loci in the strata H(1,1) (studied by McMullen) that is connected for every discriminant D.
This paper is devoted to the classification of GL^+(2,R)-orbit closures of surfaces in the intersection of the Prym eigenform locus with various strata of quadratic differentials. We show that the following dichotomy holds: an orbit is either closed or dense in a connected component of the Prym eigenform locus. The proof uses several topological properties of Prym eigenforms, which are proved by the authors in a previous work. In particular the tools and the proof are independent of the recent results of Eskin-Mirzakhani-Mohammadi. As an application we obtain a finiteness result for the number of closed GL^+(2,R)-orbits (not necessarily primitive) in the Prym eigenform locus Prym_D(2,2) for any fixed D that is not a square.
This paper deals with Prym eigenforms which are introduced previously by McMullen. We prove several results on the directional flow on those surfaces, related to complete periodicity (introduced by Calta). More precisely we show that any homological direction is algebraically periodic, and any direction of a regular closed geodesic is a completely periodic direction. As a consequence we draw that the limit set of the Veech group of every Prym eigenform in some Prym loci of genus 3,4, and 5 is either empty, one point, or the full circle at infinity. We also construct new examples of translation surfaces satisfying the topological Veech dichotomy. As a corollary we obtain new translation surfaces whose Veech group is infinitely generated and of the first kind.
This paper is devoted to the classification of the infinite families of Teichmuller curves generated by Prym eigenforms of genus 3 having a single zero. These curves were discovered by McMullen. The main invariants of our classification is the discriminant D of the corresponding quadratic order, and the generators of this order. It turns out that for D sufficiently large, there are two Teichmueller curves when D=1 modulo 8, only one Teichmueller curve when D=0,4 modulo 8, and no Teichmueller curves when D=5 modulo 8. For small values of D, where this classification is not necessarily true, the number of Teichmueller curves can be determined directly. The ingredients of our proof are first, a description of these curves in terms of prototypes and models, and then a careful analysis of the combinatorial connectedness in the spirit of McMullen. As a consequence, we obtain a description of cusps of Teichmueller curves given by Prym eigenforms. We would like also to emphasis that even though we have the same statement compared to, when D=1 modulo 8, the reason for this disconnectedness is different. The classification of these Teichmueller curves plays a key role in our investigation of the dynamics of SL(2,R) on the intersection of the Prym eigenform locus with the stratum H(2,2), which is the object of a forthcoming paper.
We prove that the dilatation of any pseudo-Anosov homeomorphism on a translation surface that belong to a hyperelliptic component is bounded from below uniformly by sqrt{2}. This is in contrast to Penner's asymptotic. Penner proved that the logarithm of the least dilatation of any pseudo-Anosov homeomorphism on a surface of genus g tends to zero at rate 1/g (as g goes to infinity). We also show that our uniform lower bound sqrt{2} is sharp. More precisely the least dilatation of a pseudo-Anosov on a genus g>1 translation surface in a hyperelliptic component belongs to the interval ]sqrt{2},sqrt{2}+2^{1-g}[. The proof uses the Rauzy-Veech induction.