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The "abelianization" process introduced by Gaiotto, Hollands, Moore, and
Neitzke turns $\operatorname{SL}_K \mathbb{C}$ local systems on a punctured
surface into $\mathbb{C}^\times$ local systems, giving coordinates on the
decorated $\operatorname{SL}_K \mathbb{C}$ character variety that are known to
match Fock and Goncharov's cluster coordinates in the $\operatorname{SL}_2
\mathbb{C}$ case. This paper extends abelianization to $\operatorname{SL}_2
\mathbb{R}$ local systems on compact surfaces, using tools from dynamics to
overcome the technical challenges that arise in the compact case. The approach
taken here seems to complement the one recently used by Bonahon and Dreyer to
arrive at a similar construction in a different geometric setting.

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you can request a copy directly from the author.

... This case is more difficult, since problems of dynamics mix with the combinatorial problems. Some of the necessary tools in case K = 2 have been developed in [14]. ...

... Again see Figure 7 and Figure 11 for illustrative examples. 14 These marked points divide the circle at infinity into 2n + 6 arcs. Call an arc at infinity initial (final) if its two boundary points have the same initial (final) label; there are n + 3 arcs of each type. ...

... Define an abelianization tree compatible with W (P 0 , ϑ) to be a collection of oriented arcs in C, with each arc labeled by a sheet of Σ and a representation of SL(3, R) (either fundamental V or its dual V * ), with the following properties: 14 In comparing the asymptotics in those figures to the description above, one must keep in mind the permutations of sheets which occur when one crosses the branch cuts. For example, in Figure 7, starting from the rightmost marked point just above the branch cut, we see the sequence 31, 21, 23, 13, 12; the next label would ordinarily be 32, but we also cross a branch cut which induces the permutation (123), so the next label is instead 13. 15 Strictly speaking, what [2,3,4] predict is a local coordinate system on a patch of a complexification of • Each arc has two endpoints. ...

We study minimal harmonic maps $g: {\mathbb{C}} \to SO(3) \backslash SL(3,{\mathbb{R}})$, parameterized by polynomial cubic differentials $P$ in the plane. The asymptotic structure of such a $g$ is determined by a convex polygon $Y(P)$ in ${\mathbb{RP}^2}$. We give a conjectural method for determining $Y(P)$ by solving a fixed-point problem for a certain integral operator. The technology of spectral networks and BPS state counts is a key input to the formulation of this fixed-point problem. We work out two families of examples in detail.

We provide a novel proof that the set of directions that admit a saddle
connection on a meromorphic quadratic differential with at least one pole of
order at least two is closed, which generalizes a result of Bridgeland and
Smith, and Gaiotto, Moore, and Neitzke. Secondly, we show that this set has
finite Cantor-Bendixson rank and give a tight bound. Finally, we present a
family of surfaces realizing all possible Cantor-Bendixson ranks. The
techniques in the proof of this result exclusively concern Abelian
differentials on Riemann surfaces, also known as translation surfaces. The
concept of a "slit translation surface" is introduced as the primary tool for
studying meromorphic quadratic differentials with higher order poles.

We explain that spectral networks are a unifying framework that incorporates
both shear (Fock-Goncharov) and length-twist (Fenchel-Nielsen) coordinate
systems on moduli spaces of flat SL(2,C) connections, in the following sense.
Given a spectral network W on a punctured Riemann surface C, we explain the
process of "abelianization" which relates flat SL(2)-connections (with an
additional structure called "W-framing") to flat C*-connections on a covering.
For any W, abelianization gives a construction of a local Darboux coordinate
system on the moduli space of W-framed flat connections. There are two special
types of spectral network, combinatorially dual to ideal triangulations and
pants decompositions; these two types of network lead to Fock-Goncharov and
Fenchel-Nielsen coordinates respectively.

We start with a mini-survey on some problems of pseudoperiodic topology. In the main part of the paper we consider analogs of irrational winding lines on a torus for arbitrary Riemann surfaces. These analogs are leaves of folia-tions defined by closed differential 1-forms. We study asymptotic topological dynamics of the winding lines. We take long pieces of leaves of the foliation and consider the behavior of cycles obtained by joining the endpoints of each piece by short segments. We prove that generically there is a flag of subspaces V 1 ⊂ V 2 ⊆ · · · ⊆ Vg ⊆ V ⊂ H 1 (M 2 g ; R) in the first homology group with the following properties. The 1-dimensional subspace V 1 is spanned by the asymptotic cycle. Deviation of a cycle representing a long piece of leaf from the subspace V j is of order l ν j+1 , j = 1, . . . , g − 1, where l is the length of corresponding piece of leaf. The bound is uniform with respect to choice of leaf and position of the piece of leaf on it. The deviation of any leaf from the subspace V is uniformly bounded by a constant. "Universal constants" 0 ≤ ν j < 1 are represented in terms of Lyapunov exponents of the Teichmüller geodesic flow on the corresponding moduli space of Abelian differentials. This statement is a corollary of an analogous statement for interval ex-change transformations. Structure of the paper In the first part of the paper we present a mini-survey on some problems of pseudoperiodic topology. It is independent from the remaining part of the paper. In section 2 we consider foliations on Riemann surfaces defined by closed 1-forms. We show why the interesting topological dynamics of such foliations can be represented by a class of 1-forms obtained as real parts of Abelian differentials. In section 3 we formulate the principal results. In section 4 we reformulate the problem and the principal results in the language of interval exchange transformations. Then we prove the main theorem using the properties of a discrete analog of the Teichmüller geodesic flow on the space of interval exchange transformations. In Appendix A we discuss irreducibility of the corresponding cocycle. In Appendix B we prove irreducibility for some particular case.

Let G be a split semisimple algebraic group over Q with trivial center. Let S be a compact oriented surface, with or without boundary. We define positive representations of the fundamental group of S to G(R), construct explicitly all positive representations, and prove that they are faithful, discrete, and positive hyperbolic;
the moduli space of positive representations is a topologically trivial open domain in the space of all representations. When
S have holes, we defined two moduli spaces closely related to the moduli spaces of G-local systems on S. We show that they
carry a lot of interesting structures. In particular we define a distinguished collection of coordinate systems, equivariant
under the action of the mapping class group of S. We prove that their transition functions are subtraction free. Thus we have
positive structures on these moduli spaces. Therefore we can take their points with values in any positive semifield. Their
positive real points provide the two higher Teichmller spaces related to G and S, while the points with values in the tropical
semifields provide the lamination spaces. We define the motivic avatar of the Weil–Petersson form for one of these spaces.
It is related to the motivic dilogarithm.

This article is based on a talk delivered at the RIMS--OCAMI Joint International Conference on Geometry Related to Integrable Systems in September, 2007. Its aim is to review a recent progress in the Hitchin integrable systems and character varieties of the fundamental groups of Riemann surfaces. A survey on geometric aspects of these character varieties is also provided as we develop the exposition from a simple case to more elaborate cases. Comment: 21 pages

We show that grafting any fixed hyperbolic surface defines a homeomorphism from the space of measured laminations to Teichmuller space, complementing a result of Scannell-Wolf on grafting by a fixed lamination. This result is used to study the relationship between the complex-analytic and geometric coordinate systems for the space of complex projective ($\CP^1$) structures on a surface. We also study the rays in Teichmuller space associated to the grafting coordinates, obtaining estimates for extremal and hyperbolic length functions and their derivatives along these grafting rays.

We compute the E-polynomials of the moduli spaces of representations of the fundamental group of a complex curve of genus g = 3 into SL(2, ℂ), and also of the moduli space of twisted representations. The case of genus g = 1, 2 has already been done in [12]. We follow the geometric technique introduced in [12], based on stratifying the space of representations, and on the analysis of the behaviour of the E-polynomial under fibrations.

For Hölder cocycles over a Lipschitz base transformation, possibly non-invertible, we show that the subbundles given by the Oseledets Theorem are Hölder-continuous on compact sets of measure arbitrarily close to 1. The results extend to vector bundle automorphisms, as well as to the Kontsevich–Zorich cocycle over the Teichmüller flow on the moduli space of abelian differentials. Following a recent result of Chaika–Eskin, our results also extend to any given Teichmüller disk.

For a closed surface S, the Hitchin component Hit_n(S) is a preferred
component of the character variety consisting of group homomorphisms from the
fundamental group pi_1(S) to the Lie group PSL_n(R). We construct a
parametrization of the Hitchin component that is well-adapted to a maximal
geodesic lamination on the surface. This is a natural extension of Thurston's
parametrization of the Teichmueller space of S by shear coordinates associated
to a maximal geodesic lamination, corresponding to the case n=2. However,
significantly new ideas are needed in this higher dimensional case. The article
concludes with a few applications.

We construct Bratteli-Vershik models for minimal interval exchange transformations. We use this to show that the interval exchange transformations over quadratic fields, recently studied by Boshernitzan and Carroll, actually are (conjugate to) substitution minimal systems. We also prove a partial converse to this. Furthermore, these systems are orbit equivalent to Sturmian systems.

We compute the E-polynomials of the moduli spaces of representations of the
fundamental group of a complex curve of genus g=3 into $ SL(2,C), and also of
the moduli space of twisted representations. The case of genus g=1,2 has
already been done in [http://arxiv.org/abs/1106.6011]. We follow the geometric
technique introduced in [http://arxiv.org/abs/1106.6011], based on stratifying
the space of representations, and on the analysis of the behaviour of the
E-polynomial under fibrations.

A theory is developed for infinite products in a noncommutative Banach algebra. Sufficient conditions for the convergence of such a product are given. Conditions for analyticity of the product are also given in the case when the factors depend on a complex parameter.

Given an Anosov representation $\rho \colon \pi_1(S) \to
\PSL_{n}(\mathbb{R})$ and a maximal geodesic lamination $\lambda$ in a surface
$S$, we construct shear deformations along the leaves of the geodesic
lamination $\lambda$ endowed with a certain flag decoration, that is provided
by the associated flag curve $\mathcal{F}_\rho\colon \Sinf \to
\mathrm{Flag}(\mathbb{R}^n)$ of the Anosov representation $\rho$; these
deformations generalize to Labourie's Anosov representations Thurston's
cataclysms for hyperbolic structures on surfaces. A cataclysm is parametrized
by a transverse $n$--twisted cocycle for the orientation cover $\La$ of
$\lambda$. In addition, we establish various geometric properties for these
deformations. Among others, we prove a variation formula for the associated
length functions $\ell^i_\rho$ of the Anosov representation $\rho$.

We introduce new geometric objects called spectral networks. Spectral
networks are networks of trajectories on Riemann surfaces obeying certain local
rules. Spectral networks arise naturally in four-dimensional N=2 theories
coupled to surface defects, particularly the theories of class S. In these
theories spectral networks provide a useful tool for the computation of BPS
degeneracies: the network directly determines the degeneracies of solitons
living on the surface defect, which in turn determine the degeneracies for
particles living in the 4d bulk. Spectral networks also lead to a new map
between flat GL(K,C) connections on a two-dimensional surface C and flat
abelian connections on an appropriate branched cover Sigma of C. This
construction produces natural coordinate systems on moduli spaces of flat
GL(K,C) connections on C, which we conjecture are cluster coordinate systems.

We study existence of non-uniform continuous -valued cocycles over uniquely ergodic dynamical systems. We present a class of subshifts over finite alphabets on which every locally constant cocycle is uniform. On the other hand, we also show that every irrational rotation admits non-uniform cocycles. Finally, we discuss characterizations of uniformity.RésuméOn étudie l'existence de cocycles non uniformes à valeurs dans , pour les systèmes dynamiques uniquement ergodiques. On présente une classe de sous-shifts à alphabets finis pour lesquels tout cocycle localement constant est uniforme. Par ailleurs, on montre que toute rotation irrationelle admet des cocycles non-uniformes. Enfin, nous présentons des caractérisations de l'uniformité.

We consider the question of uniform convergence in the multiplicative ergodic theorem lim 1/n·log||A(Tn-1χ)⋯A(χ)||=Λ(A) n→∞ for continuous function A : X → GLd(ℝ), where (X,T) is a uniquely ergodic system. We show that the inequality lim supn→∞ n-1 log||A(Tn-1χ)⋯A(χ)|| ≤ Λ(A) holds uniformly on X, but it may happen that for some exceptional zero measure set E ⊂ X of the second Baire category: lim infn→∞ n-1 · log||A(Tn-1χ)⋯A(χ)|| < Λ(A). We call such A a non-uniform function. We give sufficient conditions for A to be uniform, which turn out to be necessary in the two-dimensional case. More precisely, A is uniform iff either it has trivial Lyapunov exponents, or A is continuously cohomologous to a diagonal function. For equicontinuous system (X, T), such as irrational rotations, we identify the collection of non-uniform matrix functions as the set of discontinuity of the functional Λ on the space C(X, GL2(ℝ)), thereby proving, that the set of all uniform matrix functions forms a dense Gδ-set in C(X, GL2(ℝ)). It follows, that M. Herman's construction of a non-uniform matrix function on an irrational rotation, gives an example of discontinuity of Λ on C(X, GL2(ℝ)).

We study a diophantine property for translation surfaces, defined in term of saddle connections and inspired by the classical theorem of Khinchin. We prove that the same dichotomy holds as in Khinchin' result, then we deduce a sharp estimation on how fast the typical Teichmuller geodesic wanders towards infinity in the moduli space of translation surfaces. Finally we prove some stronger result in genus one. Comment: 37 pages

We define a diophantine condition for interval exchange transformations (i.e.t.). When the number of intervals is two, that is for rotations on the circle, our condition coincides with the one in the classical Khinchin theorem, modulo the identification of a rotation with its rotation number. We prove that for i.e.t.s we have the same dichotomy of Khinchin theorem. Comment: 49 pages

We consider BPS states in a large class of d=4, N=2 field theories, obtained
by reducing six-dimensional (2,0) superconformal field theories on Riemann
surfaces, with defect operators inserted at points of the Riemann surface.
Further dimensional reduction on S^1 yields sigma models, whose target spaces
are moduli spaces of Higgs bundles on Riemann surfaces with ramification. In
the case where the Higgs bundles have rank 2, we construct canonical Darboux
coordinate systems on their moduli spaces. These coordinate systems are related
to one another by Poisson transformations associated to BPS states, and have
well-controlled asymptotic behavior, obtained from the WKB approximation. The
existence of these coordinates implies the Kontsevich-Soibelman wall-crossing
formula for the BPS spectrum. This construction provides a concrete realization
of a general physical explanation of the wall-crossing formula which was
proposed in 0807.4723. It also yields a new method for computing the spectrum
using the combinatorics of triangulations of the Riemann surface.

The dynamics of billiard flows in rational polygons of dynamical systems

- J Smillie

J. Smillie, "The dynamics of billiard flows in rational polygons of
dynamical systems," in Dynamical Systems, Ergodic Theory and
Applications, Y. Sinai, ed., ch. 11. 2000.

MH 7750: Set theoretic topology

- G Gruenhage

G. Gruenhage, " MH 7750: Set theoretic topology. "
http://www.auburn.edu/~gruengf/fall14.html, 2014.

Potentially cluster-like coordinates from dense spectral networks https://www.ma.utexas.edu/users/afenyes/writing.html. Poster presented at Positive Grassmannians: Applications to integrable systems and super Yang-Mills scattering amplitudes

- A Fenyes

A. Fenyes, " Potentially cluster-like coordinates from dense spectral
networks. " https://www.ma.utexas.edu/users/afenyes/writing.html.
Poster presented at Positive Grassmannians: Applications to integrable
systems and super Yang-Mills scattering amplitudes. July 2015, CRM.

Moduli spaces of local systems and higher Teichmüller theory

- V Fock
- A Goncharov

V. Fock and A. Goncharov, "Moduli spaces of local systems and higher
Teichmüller theory," arXiv:math/0311149 [math.AG].

- L Hollands
- A Neitzke

L. Hollands and A. Neitzke, "Spectral networks and Fenchel-Nielsen
coordinates," arXiv:1312.2979 [math.GT].

Hitchin characters and geodesic laminations

- F Bonahon
- G Dreyer

F. Bonahon and G. Dreyer, "Hitchin characters and geodesic
laminations," arXiv:1410.0729v2 [math.GT].

of Cambridge studies in advanced mathematics

- M Viana

M. Viana, Lectures on Lyapunov Exponents, vol. 145 of Cambridge studies
in advanced mathematics. Cambridge University Press, 2014.

Poster presented at Positive Grassmannians: Applications to integrable systems and super Yang-Mills scattering amplitudes

- A Fenyes

A. Fenyes, "Potentially cluster-like coordinates from dense spectral
networks." https://www.ma.utexas.edu/users/afenyes/writing.html.
Poster presented at Positive Grassmannians: Applications to integrable
systems and super Yang-Mills scattering amplitudes. July 2015, CRM.