In 1966, M. Kac wrote a famous article asking whether Can one hear the shape of drum?: mathematically speaking, one wants to reconstruct (up to isometries) a domain from the knowledge of the spectrum of its Laplacian.
In his article, M. Kac showed that one can hear the shape of a disk 
- the area
and the perimeter
of a smooth domain
are determined by the eigenvalues
of its Laplacian
via the asymptotics of the trace of the heat operator:
- the isoperimetric inequality says that we can recognise the disk from its perimeter and area: indeed, any smooth domain
satisfies
, and the equality holds if and only if
.
(In particular, a smooth domain 
In a preprint from 2019, Hezari and Zelditch showed that one can also hear the shape of an ellipse 
More concretely, Hezari and Zelditch considered a smooth domain 
- in Section 2 of their paper, it is shown that
with an arc-length parameter
, we can control the
-norms of the derivatives of the curvature
of
where
and, in general,
for a certain constant
and an adequate “universal” polynomial
(here,
is the
-th derivative of
for all
, and Melrose explored this fact to get a pre-compactness bound
for all
,
into
bounds on
for all
;
- in view of a theorem of Avila, de Simoi and Kaloshin, an almost circular domain
, the periodic trajectories with rotation number
in the billiard table determined by
- in Sections 3 to 6 of their paper, it is shown that the portion between
and
coincides with the set
of lengths of periodic trajectories with rotation number
, in an almost circular billiard table
and they form a caustic, so that
A natural question raised by Hezari–Zelditch work is to determine the magnitude of the upper bound 
In this direction, I would like to conclude this short post by noticing that I asked a group of 6 undergraduate students (in their 2nd year) at \’Ecole Polytechnique to follow closely the articles by Avila–de Simoi–Kaloshin and Hezari–Zelditch (while trying to explicitly compute as many implied constants as possible), and, after 6 months of work, they produced this report here (in French) concluding that 
(a feature which is illustrated in this numerical simulation [due to Dyatlov] 
was proved by Baladi–Demers in this article 
forces a rigidity property for periodic orbits, namely, the Lyapunov exponents of all periodic orbits coincide (note that this is a huge number of conditions because the number of periodic orbits of period 
grows exponentially with 
) and, in particular, there are no known examples of Sinai billiards where 
for certain Sinai billiards by computing the Taylor expansion of the derivative of the billiard map at a periodic orbit of period two and checking that its quadratic part is non-degenerate.
centered at the points of a triangular lattice and the distance between adjacent scatterers is 
.
has a probability measure of maximal entropy whenever 
(where the limit case 
corresponds to touching obstacles and the limit case 
produces “infinite horizon”).
and 
.
in the direction 
travels along the line 
until it hits the boundary of the disk of radius 
for the first time 
. By definition, the time 
, i.e.,
where 
and 
, the billiard trajectory starting at 
and, after a specular reflection, it takes the direction 
.
along the following lines. If we adopt the convention in Chernov–Markarian’s book of using arc-length parametrization 
on the obstacle and describing the angle ![{y\in[-\pi/2,\pi/2]}](https://s0.wp.com/latex.php?latex=%7By%5Cin%5B-%5Cpi%2F2%2C%5Cpi%2F2%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
with the normal direction pointing towards the interior of the table using a sign determined by the orientation of the obstacles so that this normal vector stays at the left of the tangent vectors to the obstacles, then the billiard map becomes
![{\arcsin(z)\in[-\frac{\pi}{2},\frac{\pi}{2}]}](https://s0.wp.com/latex.php?latex=%7B%5Carcsin%28z%29%5Cin%5B-%5Cfrac%7B%5Cpi%7D%7B2%7D%2C%5Cfrac%7B%5Cpi%7D%7B2%7D%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
for ![{z\in[-1,1]}](https://s0.wp.com/latex.php?latex=%7Bz%5Cin%5B-1%2C1%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
) and, moreover,
is
, one has
of the Taylor expansion of 
for 
.
in the limit case 
on 
induces an action on the projective space 
. For later use, recall that 
via![\displaystyle \mathbb{T}^1\ni\theta\mapsto [\cos\theta:\sin\theta]=\mathbb{R}\cdot(\cos\theta,\sin\theta)\in X.](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cmathbb%7BT%7D%5E1%5Cni%5Ctheta%5Cmapsto+%5B%5Ccos%5Ctheta%3A%5Csin%5Ctheta%5D%3D%5Cmathbb%7BR%7D%5Ccdot%28%5Ccos%5Ctheta%2C%5Csin%5Ctheta%29%5Cin+X.&bg=ffffff&fg=000000&s=0&c=20201002)
on 
, we can build a 
into 
where 
is chosen accordingly with the law of 
is stabilized by two elements 
, then the random walk starting at 
is not very interesting.
.
on 
such that, for all 
. Here, the convolution of 
on 
on 
is the distribution of points obtained from 
associated to 
-almost every 
whenever 
.
is called Lyapunov exponent.
with 
.
and 
such that
(where 
is the interval of radius 
centered at 
). In particular, 
). More concretely, he proved that:
. In other words, 
is not Zariski dense in 
and 
are upper-triangular).We affirm that there is no decay of Fourier coefficients in this situation. Indeed, recall that if we identify 
via 
, then 
acts on 
as
, 
, and the Fourier coefficients of the stationary measure given by the standard Hausdorff measure on middle-third Cantor set do not decay to zero.In a similar vein, if 
is a real number such that 
is a 
with
(called the Bernoulli convolution of parameter 
describing the distribution of the points 
where 
with probability 
) whose Fourier coefficients do not decay.
, let
as a smooth version of the characteristic function of an interval 
, we see that 
is “counting random products 
”. In this context, Guivarch and Le Page established the following renewal theorem:
.
of the elements of 
(whenever (i) is satisfied).
is the 
, then there exists 
(depending only on the dimension of 
.
, consider the transfer operator
(with 
small enough). Then,we have the following spectral gap property: there exists 
such that the spectral radius of 
satisfies
.
satisfies
Theorem 
;
, the 
th Fourier coefficient of the Furstenberg measure is
is reduced to the study of
, we see that the size of the integral above depends on the “number of random products 
with norm 
in a given interval”, and the answer to this kind of “counting problem” is encoded in the asymptotic property of the renewal operator 
grows polynomially with 
with 
is a compact subset of reductive, real linear algebraic group 
generated by 
is Zariski dense in 
and the Jordan projections 
associated to 
, an object baptised “joint spectrum of 
is isomorphic to 
.
(with 
a maximal compact subgroup of 
for an unique 
where 
is a choice of Weyl chamber in the Lie algebra 
of 
. The interior of the Weyl chamber 
.
, we can take 
in 
, so that 
.
.
, where 
are the singular values of 
is defined in terms of the Jordan-Chevalley decomposition. For 
with 
with 
unipotent, 
, and 
has eigenvalues 
where 
are the eigenvalues of 
of distinguished representations such that the components of the vectors 
, resp. 
, resp. 
. In particular, the usual formula for the spectral radius implies that 
as 
of 
, 
, have the property that the eigenvalue of 
with the largest modulus is 
.
where 
is the dimension of the center 
of 
is called the semi-simple rank of ![{G/[G,G]}](https://s0.wp.com/latex.php?latex=%7BG%2F%5BG%2CG%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
.
, and the representations 
is “truly” distinguished and the determinant representation 
comes from the center.
of a representation 
of 
is a weight whenever
if and only if 
for all 
, and any irreducible representation 
(and, as it turns out, 
is one-dimensional).In this context, the distinguished representations 
form a family of representations whose maximal weights 
provide a basis of 
.
is proximal when its projective action on 
possesses an attracting fixed point 
and a repulsive hyperplane 
. Also, an element 
(or, equivalently, 
).
-proximal whenever 
is proximal, 
, and 
for all 
, 
(where 
–proximal if and only if the matrices 
of 
such that for all 
, one can find a finite subset 
with the property that for any 
with 
called the joint spectrum of 
.
;
;
such that 
for all 
are 
and 
such that 
for all 
of largest moduli of proximal matrices 
converge to a projection 
on 
parallel to 
as 
. Also, an analogous statement is valid for 
, one has
such that 
is not nilpotent: in fact, this happens because 
and, by continuity, there exists 
with
for all 
.
so that for any 
we can find 
such that any 
for some 
.
. Hence, given 
, we have that for all 
sufficiently large, there are 
with
with 
and 
). Furthermore, by Lemma 
, and 
.
, by taking 
) and by letting 
(so that 
), we see that
. This completes the proof of Theorem 
is 
. (Recall that 
are the singular values of 
is 
.
is a 
is 
for all 
.
there are 
is a 
. It is not hard to see that if 
is 
, then 
.
decomposition, we can assume that 
. Given ![{[v]\in\Omega}](https://s0.wp.com/latex.php?latex=%7B%5Bv%5D%5Cin%5COmega%7D&bg=ffffff&fg=000000&s=0&c=20201002)
and ![{d([gv], [gv+\delta ge_1])<\varepsilon d([v],[v+\delta e_1])}](https://s0.wp.com/latex.php?latex=%7Bd%28%5Bgv%5D%2C+%5Bgv%2B%5Cdelta+ge_1%5D%29%3C%5Cvarepsilon+d%28%5Bv%5D%2C%5Bv%2B%5Cdelta+e_1%5D%29%7D&bg=ffffff&fg=000000&s=0&c=20201002)
and ![{d([gv], [gv+\delta ge_2])<\varepsilon d([v],[v+\delta e_2])}](https://s0.wp.com/latex.php?latex=%7Bd%28%5Bgv%5D%2C+%5Bgv%2B%5Cdelta+ge_2%5D%29%3C%5Cvarepsilon+d%28%5Bv%5D%2C%5Bv%2B%5Cdelta+e_2%5D%29%7D&bg=ffffff&fg=000000&s=0&c=20201002)
. These inequalities imply the desired fact that 
-Lipschitz on a neighborhood of any fixed 
, 
for all 
for all 
. For this sake, we use a result of 
, i.e.,
and 
such that 
for all 
(here, 
is the Grassmannian of hyperplanes of 
);
and 
for all 
(here, 
denotes the left shift dynamics 
);
dominates the strongest expansion along 
, that is, there are 
such that 
, the equivalence between 
depends only on 
and 
depends only on 
for all 
for all 
. Indeed, this statement can be easily obtained by contradiction: if 
for some 
and 
, then 
has the property that 
and 
. Hence, 
, a contradiction with the splitting condition above.
. In fact, given 
be the periodic sequence obtained by infinite concatenation of the word 
and 
, and 
for 
included in a sufficiently small neighborhood of 
is 
is continuous at 
by isometries of the hyperbolic metric. Consider 
, where 
and 
on 
to 
and from 
to 
. We assume that the endpoints of the axes 
as 
, and we denote by 
and 
the translation lengths of 
and, for this sake, we need to understand 
where 
is a word of length 
denotes the distance between the axes of 
, then ![\displaystyle J(S)=[\tau_a/2, \tau_b/2].](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+J%28S%29%3D%5B%5Ctau_a%2F2%2C+%5Ctau_b%2F2%5D.&bg=ffffff&fg=000000&s=0&c=20201002)
is a loxodromic element whose axis stays between the axes of ![{[x_a^-, x_b^-]}](https://s0.wp.com/latex.php?latex=%7B%5Bx_a%5E-%2C+x_b%5E-%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
to a point in ![{[x_b^+, x_a^+]}](https://s0.wp.com/latex.php?latex=%7B%5Bx_b%5E%2B%2C+x_a%5E%2B%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
, and the translation length of 
.
is a word obtained from 
. In fact, by performing a conjugation if necessary, we can assume that 
and 
, so that 
and 
.
and we successively replace 
, then we see from the claim in the previous paragraph that ![{[\tau_a/2, \tau_b/2]}](https://s0.wp.com/latex.php?latex=%7B%5B%5Ctau_a%2F2%2C+%5Ctau_b%2F2%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
as ![{J(S)=[\tau_a/2,\tau_b/2]}](https://s0.wp.com/latex.php?latex=%7BJ%28S%29%3D%5B%5Ctau_a%2F2%2C%5Ctau_b%2F2%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
. 
as above and fix 
where 
is the rotation by 
.
of 
in the plane with axis 
and 
is a triangle with vertex at 
, intersecting the ![{[\log\lambda_1(a), \log\lambda_1(b)]}](https://s0.wp.com/latex.php?latex=%7B%5B%5Clog%5Clambda_1%28a%29%2C+%5Clog%5Clambda_1%28b%29%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
, and the side opposite to the vertex 
. Indeed, one eventually get this description of 
, 
thanks to the fact that 
commutes with 
.
, where 
denotes the rotation by 
. We affirm that 
for all 
and, a fortiori, 
is discontinuous at 
(because 
as 
). In fact, given 
. Therefore,
is a 
.
. In order to respect the Zariski-density constraint, we fix 
and we set 
where 
is a small neighborhood of the identity. In this way, the monoid generated by 
whose convex hull is 
where 
is a finite set with sufficiently many points so that the monoid generated by 
, 
, and a subset 
of 
of cardinality 
. The corresponding set of ellipsephic integers 
is
and
. Despite their sparseness, it was proved by 
and 
such that
, 
, 
and 
. The “main term” 
comes from the case 
, so that our task consists into estimating the “error term”. For this sake, one has essentially to study
. Observe that
are controlled thanks to the following lemma (giving some saving over the trivial bound 
for all 
):
. For any 
.
and 
, one has
such that for all 
there exists 
with the property that
. 
such that
stands for the number of prime factors of 
in the last theorem of the previous section and, in this direction, it is possible to show that
, then one can take
(and 
) for 
,
(and 
) for 
, …,
for 
, and
as 
, 
, then one can take 
.
(of ellipsephic primes) in Dartyge–Mauduit theorem above. This was recently accomplished by 
and 
, then
digits in basis 
with two prime factors.
and for all 
of the form 
whose largest prime factor is 
.
such that
(which is close to one for 
of the form:
for all 
), Kirsty Briggs showed that for 
prime and 
, the trivial bound 
on the number of solutions of the Vinogradov system
can be improved into 
whenever 
. (In particular, this result is saying that in the case 
, the main contribution to the number of ellipsephic solutions of the corresponding Vinogradov system comes from the trivial solutions 
.)
be the power of a prime number 
and denote by 
a 
of 
over 
. In this way, we can represent a number 
as
with 
, the associated subset of ellipsephic numbers is
![{f(x)\in\mathbb{F}_q[x]}](https://s0.wp.com/latex.php?latex=%7Bf%28x%29%5Cin%5Cmathbb%7BF%7D_q%5Bx%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
, we can study the set of its ellipsephic values via the set
is described by the following theorem:
, then
. Finally, the reader can consult the 
and Jordan projection 
were defined in the previous post via the Cartan decomposition 
with 
elliptic, 
unipotent, and 
“hyperbolic” conjugated to 
.
where 
, 
a 
is a base of simple roots.
induces a 
satisfying 
and 
for all 
, where 
is the Lie subalgebra of 
and 
is a fixed extension of the 
of ![{A\cap [G,G]}](https://s0.wp.com/latex.php?latex=%7BA%5Ccap+%5BG%2CG%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
to the Lie algebra 
becomes an orthogonal decomposition.
, 
is a choice of 
-invariant norm with 
diagonalisable in an orthonormal basis and 
stable under the adjoint operation, and 
denotes the top eigenvalue of a matrix 
. In particular, the Cartan projection 
![{d([x],[y]):=\frac{\|x\wedge y\|}{\|x\|\cdot\|y\|}}](https://s0.wp.com/latex.php?latex=%7Bd%28%5Bx%5D%2C%5By%5D%29%3A%3D%5Cfrac%7B%5C%7Cx%5Cwedge+y%5C%7C%7D%7B%5C%7Cx%5C%7C%5Ccdot%5C%7Cy%5C%7C%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
be the 
of a finite-dimensional real vector space 
.Given 
is a 
and 
;
;
for all 
with 
and 
.
such that
.For each 
is a Zariski dense monoid. Then, there exists 
with the property: given 
, 
such that 
for all 
and 
.
. It follows that 
for all 
for 
with 
, there exists 
so that 
is 
is 
, it follows from the triangular inequality that
and we write 
, we can use the Euclidean division 
, 
to obtain an element of 
via the formula
, 
, of irreducible representations of 
and 
, 
, of points and hyperplanes in 
, there is an element 
such that
are non-empty and Zariski open in 
from a finite subset of 
and 
, there exists 
, 
, 
with
is 
.
and the dominated hyperplane 
for the actions of the proximal matrices 
on 
, say 
, such that
.
is attracted towards 
. By rendering this argument slightly more quantitative (with the aid of the so-called Tits proximality criterion), Breuillard and Sert proved in 
is 
is a finite subset such that
and 
such that for all 
and 
is 
-proximal for all 
.
, we can select 
is 
.
with 
and
.
as 
.
and a constant 
such that for all 
with 
and
, we have that 
is bounded and, a fortiori,
, as desired. 
with 
is realized by a single sequence 
in the sense that 
.
with the property that 
is a Schottky family in the sense that 
is 
and 
for all 
where 
.
so that
by
has the form 
where 
and 
is a prefix of 
. Observe that
. Moreover, 
makes that 
is a 
-proximal element with
converges to 
converges to 
-plane can be written as the sum of three terms; for the 
-norm of Beltrami differentials with essentially disjoint supports; finally, this 
, and we dispose of good upper bounds for 
and all of its derivatives in terms of the distance of 
using the techniques of these articles 
are small whenever 
is a polynomial function on ![{[0,1]}](https://s0.wp.com/latex.php?latex=%7B%5B0%2C1%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
whose degree and 
and 
with 
, there exists a constant 
with the following property.If 
denotes the product of the lengths of the short geodesics of a hyperbolic surface 
.
with 
. If we write 
, where 
is the usual hyperbolic plane and 
of Riemann surfaces of genus 
of harmonic 
induced by the Hermitian inner product
and 
is the hyperbolic area form on 
and 
are Beltrami differentials, then 
is a function on 
and 
is an operator related to the 
on 
.
.
and 
span a 
and 
such that 
, 
, and 
is orthonormal. Then, Bochner showed that the 
is a positive operator on 
with unit norm. Finally, 
. (Here, 
stands for the hyperbolic distance on 
and 
whenever 
of a 
into its real and imaginary parts, say 
, then we see that![\displaystyle \begin{array}{rcl} \textrm{Re}(1\overline{2}, 1\overline{2}) - (1\overline{2}, 2\overline{1}) &=& \left[\int_X f\,\mathcal{D}f \, dA - \int_X h\, \mathcal{D}h \, dA\right] - \left[\int_X f\,\mathcal{D}f \, dA + \int_X h\, \mathcal{D}h \, dA\right] \\ &=& -2\int_X h\, \mathcal{D}h \, dA. \end{array}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cbegin%7Barray%7D%7Brcl%7D+%5Ctextrm%7BRe%7D%281%5Coverline%7B2%7D%2C+1%5Coverline%7B2%7D%29+-+%281%5Coverline%7B2%7D%2C+2%5Coverline%7B1%7D%29+%26%3D%26+%5Cleft%5B%5Cint_X+f%5C%2C%5Cmathcal%7BD%7Df+%5C%2C+dA+-+%5Cint_X+h%5C%2C+%5Cmathcal%7BD%7Dh+%5C%2C+dA%5Cright%5D+-+%5Cleft%5B%5Cint_X+f%5C%2C%5Cmathcal%7BD%7Df+%5C%2C+dA+%2B+%5Cint_X+h%5C%2C+%5Cmathcal%7BD%7Dh+%5C%2C+dA%5Cright%5D+%5C%5C+%26%3D%26+-2%5Cint_X+h%5C%2C+%5Cmathcal%7BD%7Dh+%5C%2C+dA.+%5Cend%7Barray%7D+&bg=ffffff&fg=000000&s=0&c=20201002)
and, a fortiori,
. Therefore,
, so that it follows from 
.
at this stage: indeed, 
would force a case of equality in Cauchy-Schwarz inequality and this is not possible in our context because 
for the Lyapunov exponents of the Teichmüller geodesic flow. In fact, after some computations with variational formulas for the so-called Hodge norm, Forni establishes that 
and 
with the following property. If we have an almost equality
and 
in 
and 
[implied by 
. In order to illustrate this point, let us now show Theorem 
such that
do not belong to the cusp region 
of 
(equipped with the hyperbolic metric 
).
, before explaining the extra ingredient needed to treat the general case.
for a constant 
to be chosen later. In this regime, our goal is to show that 
is “small”, so that 
is necessarily “big”.
(cf. 
(where 
and 
are the positive and negative parts of the real part 
, then we obtain that
. Thus, if 
for all 
. It follows that
, i.e., 
. By plugging this information into the previous inequality, we obtain the estimate
(cf. 
for 
. Since 
for 
bound on 
can be converted into a 
and replacing Beltrami differentials 
and 
(namely, quadratic differentials), we are led to study quartic differentials 
. By Cauchy integral formula on 
and the cusp region 
is 
. Thus, there exists an universal constant 
such that
, we have that the previous inequality implies the following 
:
. This proves Theorem 
is chosen correctly, then the Cauchy integral formula and the Schwarz lemma can be used to prove that
.
and this allows us to repeat the arguments of the case 
, 
, etc. without any extra difficulty: see Section 5 of Wolpert’s preprint for more details.
and 
(cf. Section 
of a short simple closed geodesic 
.
in terms of the behaviours of simple geodesic arcs connecting 
be the shortest geodesic connecting 
permits to check that it suffices to study the passages of 
giving a “contribution” to 
depending only on the topology of 
-diffeomorphism 
of the circle 
has a well-defined 
(which can be defined using the cyclic order of its orbits, for instance);
(i.e., 
for a surjective continuous map 
) whenever its rotation number 
has irrational rotation number 
(i.e., there is an homeomorphism 
such that 
);
for some 
, 
, and all 
, then there exists 
conjugating 
such that 
for all 
conjugated to 
. (The nomenclature is motivated by 
, the 
g.i.e.t.s 
of restricted Roth type such that 
-submanifold of codimension 
where 
and 
intervals.
, so that 
and also 
. Hence, Marmi–Moussa–Yoccoz theorem says that its local conjugacy class of 
regardless of the differentiability scale 
grows linearly with the differentiability scale 
g.i.e.t. 
are 
sending a finite partition (modulo zero) 
of 
into closed subintervals 
disposed accordingly to a bijection 
to a finite partition (modulo zero) 
of 
disposed accordingly to a bijection 
(via 
), an elementary step of the Rauzy–Veech algorithm produces a new 
by taking the first return map of 
where 
, resp. 
whenever 
, resp. 
(and 
).
can be iterated indefinitely. This nomenclature is partly justified by the fact that Yoccoz generalized the proof of Poincaré’s theorem in order to establish that a 
for all 
, 
).
. By definition, the logarithm 
of the slope of 
taking a constant value 
on each continuity interval 
with bi-infinite 
at a point 
with orbit 
for all 
is defined as 
, resp. 
for 
.
for all 
with bi-infinite 
given by the first return map of 
.
associated to an elementary step 
to a function 
, 
, where 
stands for the first return time to 
.
is constant on each 
whenever 
is constant on each 
. In particular, the restriction of 
. The family of matrices obtained from the successive iterates of the Rauzy–Veech algorithm provides a concrete description of the so-called Kontsevich–Zorich cocycle.
and 
of genera 
. Among their several curious features, we would like to point out that the following fact proved by 
or 
intervals (resp.) associated to the first return map of the translation flow 
, 
and a 
-dimensional vector subspace 
such that 
is an equivariant decomposition with respect to the matrices of the Kontsevich–Zorich cocycle with the following properties:
;
;
are usually called the tautological exponents of the Kontsevich–Zorich cocycle. In this terminology, the third item above is saying that all non-tautological Lyapunov exponents of the Kontsevich–Zorich associated to 
of 
(see, e.g., Section 3.4 of 
under the Rauzy–Veech algorithm actually coincides with 
, then the item (c) from Subsection 
for all 
and this allows to select a bounded subsequence of 
thanks to the repetition of certain words in the prefix-suffix decomposition.
such that 
, when 
always lead to piecewise affine i.e.t.s which are not 
for some 
is a 
is a solution of the 
are bounded and, by continuity of 
of 
belongs to the weak stable space of the Kontsevich–Zorich cocycle (compare with Remark 3.9 of Marmi–Moussa–Yoccoz paper).
Disquisitiones Mathematicae 
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