# Convolutional Neural Networks and Stokes Response Functions

Rebecca Centeno High Altitude Observatory, National Center for Atmospheric
Research, Boulder, CO Rebecca Centeno<EMAIL_ADDRESS>Natasha Flyer Flyer
Research LLC, Boulder,CO  Lipi Mukherjee  High Altitude Observatory, National
Center for Atmospheric Research, Boulder, CO  Ricky Egeland now at NASA
Johnson Space Center, Houston, TX.  High Altitude Observatory, National Center
for Atmospheric Research, Boulder, CO  Roberto Casini High Altitude
Observatory, National Center for Atmospheric Research, Boulder, CO  Tanausú
del Pino Alemán  Instituto de Astrofísica de Canarias, E-38205, La Laguna,
Tenerife, Spain  Departamento de Astrofísica, Facultad de Física, Universidad
de La Laguna, Tenerife, Spain  Matthias Rempel  High Altitude Observatory,
National Center for Atmospheric Research, Boulder, CO

###### Abstract

In this work, we study the information content learned by a convolutional
neural network (CNN) when trained to carry out the inverse mapping between a
database of synthetic Ca II intensity spectra and the vertical stratification
of the temperature of the atmospheres used to generate such spectra. In
particular, we evaluate the ability of the neural network to extract
information about the sensitivity of the spectral line to temperature as a
function of height. By training the CNN on sufficiently narrow wavelength
intervals across the Ca II spectral profiles, we find that the error in the
temperature prediction shows an inverse relationship to the response function
of the spectral line to temperature, this is, different regions of the
spectrum yield a better temperature prediction at their expected regions of
formation. This work shows that the function that the CNN learns during the
training process contains a physically-meaningful mapping between wavelength
and atmospheric height.

Stokes polarimetry — Chromosphere — Radiative Transfer — Machine Learning

##  1 Introduction

In the rarefied environment of the solar chromosphere the radiation field is
weakly coupled to the local thermodynamic conditions, and common
approximations used to solve the polarized radiative transfer problem in the
photosphere break down. The two main ones are 1) the assumption of local
thermodynamical equilibrium (LTE), and 2) that the source of spectral line
polarization is only due to the Zeeman effect. A more generalized approach
needs to be adopted in the chromosphere, recognizing that the majority of
spectral lines generated there form under non-LTE conditions, and in the
presence of anisotropic excitation mechanisms that lead to the production of
scattering polarization. As a result, the forward modeling of chromospheric
spectral lines can be computationally very expensive, because of the need to
solve the very complex problem of the statistical equilibrium of the atom
under such conditions, and under the joint action of the Hanle and Zeeman
effects.

The possibility to exploit spectropolarimetric observations of the solar
atmosphere to infer its physical (thermodynamic and magnetic) state depends on
our ability to understand and constrain a very complex and generally ill-posed
inverse problem. Spectral-line inversion codes for the photosphere have been
developed and successfully employed for several decades thanks to the
simplifications that are possible in the forward modeling of the formation of
photospheric spectral lines. These codes carry out an “inverse mapping” from
the observations to the physical state of the Sun’s atmosphere. A traditional
solution approach relies on non-linear least squares methods in order to
minimize a merit function that quantifies the differences between the observed
and modeled spectra. This entails solving the forward modeling of the spectral
line many times until the inversion converges. In the case of chromospheric
modeling, this approach adds yet another layer of iterations on top of an
already computationally demanding problem. For this reason, spectral line
inversion codes for chromospheric lines typically adopt different
approximations to make the inversion problem tractable, by either relying on
simplified atmospheres (Asensio Ramos et al., 2008) or assuming that the
spectral line polarization is only due to the Zeeman effect (Socas-Navarro et
al., 2015; de la Cruz Rodríguez & van Noort, 2017; Milić & van Noort, 2018; de
la Cruz Rodríguez et al., 2019). Even after these simplifications, non-LTE
inversion codes are still computationally demanding, have finicky
initialization procedures, and often present unstable convergence behaviors,
requiring a high level of experience to run them.

A different approach to spectral line inversions, which is based on machine
learning (ML), is becoming very popular nowadays. While this approach is not
completely new (see e.g. Carroll & Staude, 2001; Socas-Navarro, 2005), it is
gaining more traction in recent times because of the growing access to large
computing power resources and to open-source ML software packages.

By training a convolutional neural network (CNN) on the inversion results (and
inversion fits) of synthetic spectra, Milić & Gafeira (2020) were able to
speed up the spectral line inversion of these data by 5 orders of magnitude,
compared to the performance of a traditional Levenberg-Marquardt-based
inversion scheme. Following a slightly different philosophy, Asensio Ramos &
Díaz Baso (2019) trained two different CNN architectures on synthetic
spectropolarimetric data from radiative magneto-hydrodynamical MURaM
simulations. Instead of doing this on a pixel-by-pixel basis, these authors
fed the CNNs 2-dimensional patches of the simulated Sun’s surface, taking
advantage of the horizontal continuity of the data. The CNNs so constructed
are able to retrieve the thermodynamical and magnetic properties of the
photosphere as a function of geometric height, as well as to deconvolve the
data from the instrument’s point spread function. This technique was
successfully applied to Hinode/SP observations and the authors predict that it
could invert all the data from the entire mission in just a matter of minutes.
A recent work by Higgins et al. (2021) shows that ML techniques can also be
used to cross-calibrate spectral line inversion results from instruments that
observe the Sun in fundamentally different ways (in their case, Hinode/SP and
SDO/HMI). Sainz Dalda et al. (2019) apply deep learning techniques to the
inversion of chromospheric intensity spectra from the IRIS mission, resulting
in a decrease of 5 to 6 orders of magnitude in the computing time for the
retrieval of the thermodynamical properties of the chromosphere. While the
accuracy of the results is often reported to be good in a qualitative manner,
Gafeira et al. (2021) claim that these kinds of approaches can be used to find
suitable guess atmospheres to feed to traditional non-linear least squares
inversion codes, resulting in a significant reduction in the number of
inversion cycles, and cutting the computing time by a factor of 2 to 4 .

An obvious limitation of the NN approach to spectropolarimetric inversions is
that the atmospheric and magnetic field models that can be inferred through
data inversion are restricted to those contained in the physical parameter
space adopted for the NN training (i.e., NNs cannot be used to extrapolate
solutions outside of the training set). Contrary to how traditional inversion
codes work, the NN inversion does not call an underlying radiative transfer
model. Given enough data, however, it is able to reconstruct a function that
carries out an approximate mapping between the Stokes profiles at the
atmospheric properties that lead to them. This begs the question of whether or
not this mapping function is able to glean physically-meaningful information
from the databases on which it has been trained.

In this work, we conduct a detailed study of the information content that a
CNN is able to infer when trained to carry out the inverse mapping between a
database of synthetic intensity spectra and the atmospheric temperature
profiles used to generate them. In particular, we analyze what the CNN is able
to learn about the sensitivity to temperature of the two lines of Ca II at
3934 Å (K-line) and 8542 Å (IR2), as a function of wavelength and atmospheric
height. This sensitivity is derived from the errors in the temperature
prediction when the CNN is trained to carry out the inverse mapping on narrow
spectral windows within each of the spectral lines. In Section 2, we describe
the numerical setups used to create the databases of atmospheric models and Ca
II spectra. Section 3 presents the CNN architectures used for this work, and
describes the methodology that we follow to quantify the information content
learned by the CNN. A comparison of these results with the response functions
of Ca II K and IR2 to temperature is presented in Section 4, followed by our
concluding remarks and an outlook for future developments.

##  2 Creation of databases of atmospheric models and spectra

In order to compile a sufficiently large database of 1-dimensional (1D) model
atmospheres and spectra on which to train the CNN, we used a snapshot of a 3D
radiative magneto-hydrodynamic simulation from the MURaM code. In particular,
we chose a Quiet Sun (QS) simulation (Rempel, 2014), with a horizontal spatial
domain of 24.5x24.5 Mm2 and a vertical extent of 8.2 Mm ($\sim$2 Mm of which
are above the solar surface). The spatial grid is sampled with a 16 km spacing
in all directions. The simulation was started from thermally relaxed, non-
magnetic convection; successively, a constant, vertical, seed magnetic field
of $10^{-3}$ G was added, unevenly distributed over the horizontal plane. Once
the simulation reached the saturation regime the average magnetic field
strength was 60 G, while the average net vertical flux was negligible.

While this particular MURaM simulation does not produce realistic physical
conditions in the solar chromosphere, the main goal of this work is to
understand the information content learned by the CNN from its training
dataset. The methodology presented here can later be utilized on increasingly
realistic numerical simulations, with the ultimate objective of developing a
functional ML spectral line inversion code that can be applied to solar
observations.

The database of 1D model atmospheres was compiled by extracting individual
columns from the 3D MURaM cube. In order avoid redundancy and correlation in
our atmospheric database, we selected columns on a sparse horizontal grid
(keeping one in every 8, in each spatial direction).

In order to generate Stokes spectra from the different realizations of the
MURaM atmospheres we use the Hanle-RT code del Pino Alemán et al. (2016).
Hanle-RT is a polarized radiative transfer code for 1-D (plane-parallel)
atmospheres. It accepts any multi-term atom (LS-coupling scheme) without
hyperfine structure,111Multi-level atoms with hyperfine structure can also be
modeled with this code. and in the presence of magnetic fields of arbitrary
strength. The code takes into account the effects of non-isotropic radiative
excitation of the atomic system, which are responsible for the manifestation
of scattering polarization. Therefore, the code can model seamlessly all
possible regimes of magnetic-induced effects on the polarization of spectral
lines: the Hanle, Zeeman, and (incomplete) Paschen-Back effects, as well as
the polarization effects associated with level crossing and anti-crossing
interference. The code fully implements the effects of inelastic, elastic, and
depolarizing collisions, and also includes a comprehensive description of
partially coherent scattering (partial redistribution, or PRD) from polarized
atoms (limited to 3-term atoms of the $\Lambda$-type; see Casini & Manso
Sainz, 2016), which is important for modeling deep chromospheric lines.

The atomic model of Ca II used to solve the statistical equilibrium (SE)
includes the five bound levels of the lowest three terms of the ion, as well
as the Ca III ground state. Such a model describes five bound-bound radiative
transitions, namely, the Ca II H & K lines in the near UV and the Ca II IR
triplet. The IR triplet is computed assuming complete frequency redistribution
(CRD), whilst the H & K lines are treated in PRD. Cross-distribution (Raman
scattering) is allowed between these sets of transitions. For computational
expediency, the system is treated as a 5-level system, rather than the actual
3-term atom. Thus, the effects of $J$-level quantum interference (incomplete
Paschen-Back effect, and level crossing and anti-crossing) are not taken into
account in this test model, but they would have a negligible impact on the
Stokes I profile anyway.

The MURaM simulation provides the thermodynamical and magnetic quantities as a
function of geometric height needed to compute the Stokes spectra:
temperature, gas pressure, line-of-sight (LOS) velocity, and magnetic field
vector. Often times, microturbulent velocity is added to the models for
broadening purposes, in order to increase the similarity between synthetic
spectra and observations (de la Cruz Rodríguez et al., 2012). A direct
comparison with observations, however, is out of the scope of this work, so we
do not see the need to artificially increase the complexity of the model for
this sake.

Before feeding these quantities to Hanle-RT, we set the LOS velocity to 0 km/s
in the interest of preserving the (non-linear) mapping between absolute
wavelength and atmospheric height. If velocities are present, this mapping is
subject to further smearing due to the Doppler shifting of the spectral
lines222The reader interested in the accuracy of the temperature prediction
using CNNs when velocities are included in the spectral line synthesis, is
referred to section B of the Appendix.. The 1D thermodynamical profiles are
then smoothed in the vertical direction by applying a Savitzky-Golay filter
(Savitzky & Golay, 1964) with a 1st-order polynomial and a window size of 17
grid points ($\equiv 256$ km). Then, they are re-sampled onto a 32 km grid,
and the portion of the model that lies below -192 km is discarded (a portion
of atmosphere below 0 km is retained 333height $=0$ km is defined as the point
where the mean continuum optical depth at 500 nm in the MURaM cube is unity
($\tau_{500}=1$). in order to maintain its small contribution to the formation
of the wings of the spectral lines). This vertical smoothing was carried out
in order to reduce the LOS gradients and the inherent fluctuations in the
model atmosphere, while retaining the main trends and the variability across
the different realizations in the atmospheric database. This process yields
36,828 unique 1D model atmospheres, that span from $-$192 km to 1888 km in
height, sampled in steps of 32 km. These model atmospheres are given to Hanle-
RT as an input. Hanle-RT then calculates the Stokes profiles for the 5 Ca II
lines contained in the model atom, assuming an observing geometry of $\mu=1.0$
(where $\mu$ is the cosine of the heliocentric angle). Figure 1 shows an
example of a Ca II K intensity profile as a function of wavelength. The
wavelength sampling is non-equidistant, being much denser in the core of the
line than in the wings (the former is enlarged in the inset of the figure for
better viewing). This is necessary in order to ensure numerical stability
while solving the radiative transfer because of the steeper gradients in the
core of the line compared to the wings. The blue and green boxes delimit
narrow (5 grid-point) wavelength windows, the purpose of which will be
discussed in the next section.

Figure 1: An example of a Ca II K Stokes I profile as a function of
wavelength. The black dots represent the wavelength sampling across the line,
which is non-equidistant. The figure inset zooms into the core of the line.
The vertical blue and green boxes showcase a sliding wavelength window,
covering five wavelength sampling points, used to analyze the sensitivity of
the spectral line to temperature as a function of height (described in Section
3) .

In this way, we create two databases, one of model atmospheres and one of
Stokes spectra, with a known one-to-one correspondence between each individual
1D physical state and each Stokes vector. We can then use these databases to
train a neural network to learn the mapping between the synthetic
“observations” (spectra) and the physical conditions in the Sun’s atmosphere.
In what follows, we will map the Stokes I profiles of the Ca II K and IR2
lines (separately) to the atmospheric temperature as a function of height
using the regression CNN outlined in the next section. The distribution of
temperature as function of height for the 36,828 samples in our atmospheric
database is shown in Figure 2. In the photoshpere ($\sim$0-500 km), the
temperature samples are concentrated in a narrow range of values (4000 K to
6500 K), while as we approach the chromosphere, the distribution spreads,
showing a much larger range of temperatures (typically between 2000 K and 7000
K). As we will see later, this will affect the accuracy of the CNN temperature
retrieval, with the lowest errors found below 500 km.

Figure 2: 3D histogram of the distribution of temperature as a function of
height for the 36,828 samples in the database.

##  3 CNN architecture

The regression CNN developed for this study is programmed in Keras (Chollet et
al., 2015) (see Appendix for the code). As shown in Table 1, it is a 9-layer
CNN with 3 convolutional layers, each followed by an Exponential Linear Unit
(ELU) non-linear activation function with its hyperparameter set to 1. The
choice of activation function relies on the faster learning speeds and higher
accuracy that the ELU confers to deep NNs (Clevert et al., 2016). A mean
absolute percentage error
($100/N\sum_{i=1}^{N}|true_{i}-predicted_{i}|/|true_{i}|$) loss function is
used to calculate the model error from the 60 unit output layer (corresponding
to the temperature sampled at 60 geometric heights) during the optimization
process. The optimization algorithm used is Adam (Kingma & Ba, 2017), a
stochastic first-order gradient method with the learning rate set to 0.01. The
batch size is 50, with $67\%$ of the data used for training, $3\%$ for
validation, and $30\%$ for testing. The network’s batch size was kept
optimally small to achieve higher accuracy without capturing noise (as
illustrated in Keskar et al., 2017). The CNN was trained for 150 epochs. A
min-max scaling was used on the input data (Stokes I) to normalize it between
0 and 1 (Pedregosa et al., 2011).

Layer Type | Description | $\begin{array}[]{c}\textrm{Activations/}\\\ \textrm{Output Shape}\end{array}$ | $\begin{array}[]{c}\textrm{Learnable}\\\ \textrm{Parameters}\end{array}$
---|---|---|---
image_input | 253×1 image (Whole Spectral Line) |  | —
Convolution | 32 3×1 convolutions, stride [1 1] | 253×1×32 | $\begin{array}[]{c}\textrm{Weights}:(3\times 1)\times 32\\\ \textrm{Bias}:32\times 1\end{array}$
ELu | ELU nonlinear activation function | — | —
Convolution | 64 3×1 convolutions, stride [1 1] | 253×1×64 | $\begin{array}[]{c}\textrm{Weights}:(3\times 32)\times 64\\\ \textrm{Bias}:64\times 1\end{array}$
ELu | ELU nonlinear activation function | — | —
Convolution | 64 3×1 convolutions, stride [3 1] | 85×1×64 | $\begin{array}[]{c}\textrm{Weights}:(3\times 64)\times 64\\\ \textrm{Bias}:64\times 1\end{array}$
ELu | ELU nonlinear activation function | — | —
Dense | $\begin{array}[]{c}60\textrm{ unit fully connected output layer}\\\ \textrm{(Temperature output at 60 heights)}\end{array}$ | 60×1 | $\begin{array}[]{c}\textrm{Weights}:(85\times 64)\times 60\\\ \textrm{Bias}:60\times 1\end{array}$
Loss_MAPE | Mean absolute percentage error | — | —
Table 1: CNN architecture for training the Ca II K spectral line (253 values
of the intensity) to map the temperature at 60 geometric heights.

Of special note to this CNN architecture is that no pooling layers are used to
reduce the spatial size of the data. Such layers usually occur in CNN
architectures to downsample the feature maps that are inputted into the next
convolutional layer and thus hopefully retain the most important features in a
map while decreasing the computation in the network. In general, this is
usually warranted when attempting to do classification on high-dimensional
images (e.g. reducing an 128 by 128 image to a vector of 10 for classification
through a softmax function). However, in the presented problem we are doing 1D
convolutions on a vector of length 253 (wavelength samples) with an output of
length 60 (geometric heights), in an attempt to map the spectral wavelength
domain to the temperature-height domain. Instead of pooling layers, we used a
striding approach to perform downsampling in the last convolution layer. By
increasing the stride length of the convolutional filter we can still reduce
the spatial size of the output (layer output = (input - filter size)/stride +
1) while the convolutional filter weights benefit from “learning” on a larger
input feature map than would have occurred if a pooling layer had been used
instead.

This CNN architecture is trained on the Ca II K and IR2 intensity spectra
separately, to carry out the inverse mapping from the spectral profiles to the
atmospheric temperature as a function of height. The black dashed lines in
Fig. 3 show the CNN prediction uncertainty, as a function of height, for the
the CNN trained on Ca II K (left) and IR2 (right). The errors are given in
terms of the L2-norm:

$\text{L2}=\sqrt{\frac{\sum_{i=1}^{N}(t_{i}-p_{i})^{2}}{\sum_{i=1}^{N}t_{i}^{2}}}$
(1)

where $t$ and $p$ represent the true and predicted temperatures, respectively,
and the summation is carried out over all the samples in the database. The
L2-norm is very similar to the root mean square error, but the normalization
is done with respect to the true vector as opposed to the number of samples,
$N$. The advantage is that it gives a relative error that is dimensionless.

The results for the Ca II K and IR2 lines are strikingly similar. As hinted
earlier, the error in the temperature prediction is smaller in the lower parts
of the atmosphere (for both spectral lines) than it is at higher layers.
Interestingly, the error in the temperature prediction at the very top of the
atmosphere (above $\sim 1700$ km) is much larger when the CNN is trained on
IR2 than when it is trained on the K-line. This is likely due to the fact that
the core of the latter forms higher in the atmosphere (see, for instance
Leenaarts et al., 2013), and therefore retains sensitivity for mapping the
temperature at these heights. An analysis of (and comparison with) the Stokes
response function to temperature for both of these lines is presented in
section 4. Figure 4 shows examples of the true (blue) vs. predicted (red)
temperature stratification for a random selection of samples in the testing
set. While the CNN is not able to capture the small-scale variations in the
temperature, it predicts its overall stratification with strikingly good
accuracy.

Figure 3: CNN-derived sensitivity of the spectral line. The results for Ca II
K are shown on the left and those for Ca II 8542 Å are shown on the right. The
colored lines show the L2-error in the temperature prediction when the CNN is
trained on short spectral windows along the spectral line. Dark blue/purple
corresponds to the case when the window is in the far wing of the line and
lightest shade of green corresponds to its placement in the core. The black
dashed line corresponds to the error in the temperature prediction when
training the CNN on the whole spectral line at once.  Figure 4: True (blue)
vs. predicted (red) temperature as a function of height for a selection of 25
random samples from the testing set.

###  3.1 CNN setup for mapping spectral line sensitivity to temperature at
different heights

Given enough data, the CNN-based inversion is able to reconstruct a function
that carries out an approximate mapping between the spectral line intensity
and the temperature stratification. Throughout the remainder of this
manuscript we will try to quantify how much information this CNN inversion
method is able to glean about the sensitivity of different parts of the
spectral line to different heights in the atmosphere, and whether or not this
bears any relation to the temperature response function of the spectral line.
In other words, we are interested in finding out if there is physically-
meaningful information in the mapping function between the wavelength and the
height domains that the CNN learned during its training.

To analyze how different wavelengths of the Ca II K and IR2 spectral lines are
sensitive to the temperature at different atmospheric heights, we train a two-
layer CNN (given in Table 2) on a short wavelength window composed of five
consecutive wavelength samples in the wing of Stokes I (such a wavelength
window is represented by the left-most blue box in Fig. 1). The window, always
of five wavelength samples, is then slid across the spectral line with a
stride of 5 wavelength positions and the CNN is trained again on the new
wavelength range (indicated by the second blue box in Fig. 1). This process is
repeated all the way to the central wavelength of the spectral line (right-
most green box in the inset of Fig. 1), and the L2 errors for the temperature
prediction as a function of height are computed for each position of the short
spectral window. Only half of the spectral line is considered because of the
(near perfect) symmetry properties of Stokes I in the absence of velocity
gradients. Notice that the window width (in wavelength units) narrows as it
moves from the wing to the core of the line due to the non-equidistant
wavelength sampling. We decided to work with this uneven sampling because the
information density in the core of the line is much larger than in the wings.

Layer Type | Description | $\begin{array}[]{c}\textrm{Activations/}\\\ \textrm{Output Shape}\end{array}$ | $\begin{array}[]{c}\textrm{Learnable}\\\ \textrm{Parameters}\end{array}$
---|---|---|---
image_input | 5×1 image (Window of Spectral Line) |  | —
Convolution | 32 3×1 convolutions, stride [1 1] | 5×1×32 | $\begin{array}[]{c}\textrm{Weights}:(3\times 1)\times 32\\\ \textrm{Bias}:32\times 1\end{array}$
ELu | ELU nonlinear activation function | — | —
Convolution | 64 3×1 convolutions, stride [1 1] | 5×1×64 | $\begin{array}[]{c}\textrm{Weights}:(3\times 1)\times 64\\\ \textrm{Bias}:64\times 1\end{array}$
ELu | ELU nonlinear activation function | — | —
Dense | $\begin{array}[]{c}60\textrm{ unit fully connected output layer}\\\ \textrm{(Temperature output at 60 heights)}\end{array}$ | 60×1 | $\begin{array}[]{c}\textrm{Weights}:(5\times 32)\times 60\\\ \textrm{Bias}:60\times 1\end{array}$
Loss_MAPE | Mean absolute percentage error | — | —
Table 2:  Two-layer CNN for training on a short wavelength window within the
spectral line to carry out the mapping between the intensity and the
temperature as a function of height.

The colored lines in Fig. 3 show the L2 prediction errors derived from
training the two-layer CNN on these short spectral windows, for Ca II K (left)
and for IR2 (right). The hue of the lines in the plots reflects the
positioning of the 5-sample wavelength windows along the spectral line, with
dark blue corresponding to the far wing and light green to the line core (the
results for specific wavelength windows are highlighted by different symbols
in the plots).

Examine the results for the K-line (left). At the lowest heights ($<160$ km),
the best prediction is achieved by the darkest blue line (also represented
with squares), which corresponds to the errors of the CNN trained in the far
wing of the spectral line (more than 1.2 nm away from the line center,
$\lambda_{0}$). As the spectral window is slid inwards (and the color of the
lines in the plot becomes a lighter shade of blue), the temperature prediction
improves for the mid and high photospheric layers (200-500) km. The window
that includes the “knee” of the spectral line (represented with triangles)
yields the best temperature predictions just above the photosphere-
chromosphere interface. Here, the “knee” of the spectral line is taken to be
the apparent wing-core boundary, marked by the vertical orange lines in Fig. 6
(which for Ca II K lies at $\lambda-\lambda_{0}\sim 0.011$ nm). The “knee” of
the spectral line seems to mark the separation between two differentiated
regimes in terms of the height of maximum sensitivity (HMS) of the spectral
line as a function of wavelength. Above 750 km, the best predictions are
attained by the three curves with the lightest shade of green, which include
the wavelength ranges in the core of the line proper (from
$\sim\lambda_{0}-0.01$ nm all the way to the line center). It seems as if this
wavelength range contained the vast majority of the information pertaining to
the mid-to-upper chromosphere. The line with the solid circles (which
corresponds to the spectral window that includes $\lambda_{0}$) intersects all
other lines and delivers the lowest error of all at the top of the atmosphere.
These results are qualitatively compatible with what we know about spectral
line formation: the wings of the line form in the photosphere while the core
forms in the high chromosphere (see e.g. Bjørgen et al., 2018).

The results for Ca II 8542 Å (right) are qualitatively similar, with the core
of the line yielding better predictions at higher atmospheric layers and the
wing of the line performing better at lower heights. The main difference lies
in the prediction error at the very top of the atmosphere (above $\sim 1700$
km), which is significantly worse for IR2 than for the K-line. This is likely
due to the fact that the latter forms higher in the atmosphere than the
former.

The dashed line represents the errors derived from training the three-layer
CNN on the whole spectral line at once. Not surprisingly, it gives the lower
bound on the error of the temperature prediction across all spectral windows,
and in some height ranges, it performs better than any/all of the individual
windows. It is possible that no single narrow wavelength interval contains
enough information to capture the LOS integration due to the photon mean-free-
path. Also, note that the CNN used to train the whole spectral line uses a
larger input and is deeper than the two-layer CNN used to train on short
spectral windows, and thus it may have a larger learning ability, leading to
slightly smaller errors overall.

The L2-error metric appears to be inversely related to the sensitivity of the
spectral line to temperature as a function of height across all of the short
wavelength intervals. In the following section we will compare this CNN-
derived sensitivity to the numerical response function of the spectral lines
to temperature.

##  4 Numerical response functions to temperature

All spectral lines form over a range of heights in the Sun’s atmosphere.
Response functions (RF; see, e.g. del Toro Iniesta & Ruiz Cobo, 2016) are
defined as the change in the intensity and polarization of a spectral line in
response to a small perturbation of an atmospheric parameter in a narrow range
of optical depths. RFs provide information about the sensitivity of the Stokes
profiles to the atmospheric properties as a function of height, and can help
us determine whether a spectral line carries or not (useful) information to
diagnose a particular physical parameter at such height.

In this section, we calculate the response functions of Ca II K and IR2 to
temperature following the strategy described in Quintero Noda et al. (2016).
Instead of working on an optical depth grid, we work directly on a geometric
height grid in order to compare the RFs to the CNN-derived sensitivity
obtained for each spectral line in the previous section.

RFs are model dependent (del Toro Iniesta & Ruiz Cobo, 2016). This means that
each 1D model atmosphere extracted from the MURaM simulation will result in a
slightly different response function to temperature. In order to produce
meaningful RFs for the whole dataset, we computed the average atmosphere from
the entire MURaM atmospheric database and used that as the reference model for
the calculation. Fig. 5 shows the horizontally averaged atmospheric quantities
as a function of height. The left panel represents the average temperature
($\overline{T}$), which drops fast from the bottom of the photosphere towards
the temperature minimum (around 700 km), and then rises into the chromosphere.
Note that this particular MURaM simulation does not achieve realistic
temperatures at the base of the corona, which has an impact on the heights of
formation of the spectral lines. The middle panel shows the average density
($\overline{\rho}$), which drops exponentially with height; and lastly, the
panel on the right contains the averages of the 3 magnetic field components
($B_{\parallel}$, $B_{x}$ and $B_{y}$, where $\parallel$ refers to the
vertical direction, and $x$ and $y$ are in the perpendicular plane). This
quiet-Sun MURaM simulation is set up to have a zero vertical net flux
($\overline{B}_{\parallel}=0$) and small, yet fluctuating averages of the
horizontal components. In this figure, $\overline{B}_{\parallel}$ is not
strictly zero. This is likely due to the fact that our MURaM dataset comprises
just over $\sim 1\%$ of the original snapshot (only 1 in every 8 grid points
in each horizontal direction was used to construct the atmospheric database).

Figure 5: Physical properties of the spatially averaged MURaM quiet Sun run.

For each height $z_{i}$, we apply a perturbation of $\Delta T=+25$ K to the
temperature of the base model and calculate the emergent spectra $I^{+}_{i}$.
We repeat this exercise by applying a negative perturbation of $\Delta T=-25$
K to the base model and computing the corresponding $I^{-}_{i}$. From the
emergent intensity spectra, the response of Stokes I to temperature for each
wavelength and height is given by:

$\mathbf{R}_{T}(\lambda,z_{i})=\frac{I^{+}_{i}-I^{-}_{i}}{2I_{C}\Delta T}$ (2)

where $\mathbf{R}_{T}(\lambda,z_{i})$ is the response function, normalized by
the local continuum intensity $I_{C}$.

Figure 6: Response functions of Ca II K (left) and Ca II 8542 Å (right) to
temperature in the average MURaM atmosphere. The x-axis represents wavelength
and the y-axis geometric height in the mean atmospheric model. A positive
response (teal) means that the intensity increases in response to a positive
temperature perturbation. The average Stokes I (in blue) is overplotted for
reference. The vertical orange line marks the approximate position of the
”knee” of the spectral lines.

Fig. 6 shows the response functions of of Ca II K (left) and IR2 (right) to
temperature as a function of wavelength on the x-axis and height on the
y-axis. A positive response (teal) means that the intensity increases in
response to a positive temperature perturbation. As expected, the wings of
both spectral lines are sensitive to changes in temperature at photospheric
heights, with a very slow dependence on wavelength. It is not until we reach
the knee of the spectral line (marked with an orange vertical line), that a
sharp increase in the HMS begins. This change in behavior takes place around
$\sim 500-700$ km for both lines. It is interesting to note that, in this
particular atmospheric model, the sensitivity of the core of the K-line to
temperature extends to the very top (and beyond) of the model atmosphere,
while the sensitivity of the IR2 line drops off around $\sim 1700$ km. This is
qualitatively and quantitatively compatible with the results presented in the
previous section, where the L2 error for the CNN trained on IR2 shows a sharp
increase above this height. In this context, the L2-error metric of the CNN
performance shows an inverse relationship to the response function of the
spectral line to temperature.

Figure 7 shows a quantitative comparison between these two metrics of spectral
line sensitivity. These plots show, in teal, the wavelength that delivers the
maximum value of the RF for any given height, or in other words, the height at
which the RF peaks as a function of wavelength. We can extract a similar
relationship from the L2 curves presented in Fig. 3: at each height in the
atmosphere we find the L2 curve (and the central $\lambda$ of the
corresponding wavelength interval) that delivers the lowest error in the
temperature prediction. This is shown by the orange squares. Note that the
orange curve shows a lot more degeneracy than the one derived from the RF,
owing to the size of the wavelength intervals on which the CNN was trained.
For this reason, a single wavelength interval can yield the lowest L2-error
for a range of heights. However, the curves follow each other very well, and
seem to tell the same story: the HMS shows a monotonic rise as we move from
the far wing to the knee of the spectral line; at this wavelength, a dramatic
shift in behavior takes place, leading to a sudden increase of this quantity.

The agreement between the information contained in the L2-error metric and the
RF of the spectral line indicates that the non-linear function learned by the
CNN contains physically-meaningful information about the mapping between
spectral wavelength and atmospheric height.

Figure 7: Quantitative comparison between the spectral line response function
and the L2-error metric of the CNN temperature prediction (Ca II K on the left
and IR2 on the right). In teal we show the wavelength that yields the maximum
value of the RF at each height. The orange squares show which wavelength
window (represented by its central wavelength) delivers the lowest L2-error
for the temperature prediction at each height. The average Stokes I (in grey)
is over-plotted for reference.

##  5 Conclusions

In this work, we analyze the information content learned by a neural network
when trained to carry out the inverse mapping between the intensity spectra of
two Ca II lines (separately) and the atmospheric temperature as a function of
height. The numerical setup comprised 36,828 1D model atmospheres extracted
from a radiative magneto-hydrodynamic MURaM simulation, and an equal number of
Ca II K and IR2 Stokes I spectra computed with the Hanle-RT code.

First, a three-layer CNN was trained to map the whole spectral line to the
atmospheric temperature as a function of height, and its performance was
evaluated using an L2-error metric to capture the uncertainty in the
temperature prediction (dashed lines in Fig. 3). As expected, this prediction
error is partially a reflection of the spread of the temperature at each
height in the model, but it also captures information about the sensitivity of
the spectral line to temperature at different atmospheric heights. This is
epitomized by the fact that IR2 leads to much larger prediction errors than
the K-line above 1700km, which speaks to the average formation heights of the
cores of these two spectral lines in this particular MURaM model.

Then, a two-layer CNN was trained on short wavelength intervals that were slid
across the spectral line, from the far wing to the line core. For each
spectral window, the CNN is trained to retrieve the temperature as a function
of height, and the L2-error metric is collected in each case (the latter is
shown by the colored lines in Fig. 3). The performance of the CNN trained on
these short spectral windows shows a monotonic trend in the height that
delivers the lowest prediction error. For the lowest atmospheric heights, the
window positioned in the far wing of the spectral line delivers the best
result (i.e. the smallest error). As we progressively move to higher
atmospheric layers, the lowest errors are delivered by spectral windows that
are increasingly closer to the core of the line. The best prediction at the
top of the atmosphere is achieved only for the window that contains the
central wavelength, $\lambda_{0}$.

Lastly, we compute the numerical response functions to the temperature in
order to evaluate the sensitivity of the two Ca II lines to thermal
perturbations in the model. The response function gives us a quantitative
measure of the change in Stokes I as a function of wavelength when the
temperature is perturbed at any given height in the atmosphere. We can draw a
one-to-one correspondence between the L2-error metric of the CNN performance
(trained on short wavelength intervals) with the response function of the
spectral line to temperature. By comparing the wavelength dependence of the
height of maximum sensitivity of the RF and the height of minimum L2-error of
the CNN temperature prediction, we see a one-to-one mapping between both
curves.

This work shows that a NN trained to carry out the inverse mapping between
Stokes I profiles and the atmospheric temperature used to generate them, is
able to glean information about the wavelength-height sensitivity of the
spectral line in a physically meaningful way.

This material is based upon work supported by the National Center for
Atmospheric Research, which is a major facility sponsored by the National
Science Foundation under Cooperative Agreement No. 1852977. We would like to
acknowledge high-performance computing support from Cheyenne
(doi:10.5065/D6RX99HX) provided by NCAR’s Computational and Information
Systems Laboratory, sponsored by the National Science Foundation.

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##  Appendix A CNN architecture

Below is the Keras CNN model trained to carry out the mapping between Stokes I
and the atmospheric temperature as a function of geometric height. Note that
the 2-layer CNN used to train on short spectral intervals has a very similar
architecture but is missing the last convolution layer.

      import numpy as np
      import tensorflow as tf
      from sklearn.preprocessing import MinMaxScaler
      from sklearn.model_selection import train_test_split
      from tensorflow.keras.layers import Conv2D
      from keras.models import Sequential
      from keras.layers import Dense, Flatten, Activation
      from keras.optimizers import Adam
      from sklearn.model_selection import train_test_split

      scaler = MinMaxScaler()
      % X is the input vector, Stokes I and
      % Y is the output vector which is the temperature at corresponding geometric heights
      scaler.fit(X)
      X = scaler.transform(X)

      Xx = X.reshape(X.shape[0], X.shape[1], 1, 1)
      Yy = Y.reshape(Y.shape[0], Y.shape[1], 1)
      n_inputs, n_outputs = X.shape[1], Y.shape[1],

      model = Sequential()
      model.add(Conv2D(32, (3,1), strides=(1, 1),activation="elu",padding=’same’, input_shape=(n_inputs,1,1)))
      model.add(Conv2D(64, (3,1), strides=(1, 1),activation="elu",padding=’same’))
      model.add(Conv2D(64, (3,1), strides=(3, 1),activation="elu",padding=’same’))
      model.add(Flatten())
      model.add(Dense(n_outputs))
      opt=Adam(lr=0.01, epsilon=None, decay=0.0, amsgrad=False)
      model.compile(loss="mape", optimizer=opt)
      batch_size = 50
      xtr, xte, ytr, yte=train_test_split(Xx, Yy, test_size=0.30, random_state=42)
      model.fit(xtr, ytr, batch_size=batch_size,validation_split=0.03,epochs=150)

##  Appendix B CNN performance in the presence of LOS velocities

Figure 8: Relative L2 error of the temperature retrieval from the CNN
inversion of Ca II K, with and without LOS velocities.

The inversion of spectral lines generated in static atmospheres is of limited
practical value. Below we show preliminary results in the accuracy of the
temperature retrieval when the Ca ii K synthetic spectra are computed in the
presence of LOS velocities from the original MURaM simulation. Figure 8 shows
the relative L2 temperature prediction error in the presence of LOS velocities
(blue), which should be compared to the case without velocities (black; this
is identical to the black dashed line in the left panel of Fig. 3). The two
curves present error values within 5% of each other up to $\sim 1600$km,
height above which the L2-error metric for the case with velocities shoots up.
This is work in progress and we believe there is room for improvement.


# Non-degenerate Birkhoff functions

Mauricio Garay and Duco van Straten

###### Abstract.

It is known after the works of Mahler, Kleinbock, Margulis, Sprindžuk that
very well approximated numbers on a manifold form a zero measure set, assuming
non-degeneracy conditions. These non-degeneracy conditions are, in many
applications, difficult to check. We propose here a setting in which these are
automatic. Rather than functions, we consider correspondences which have a
better behaviour.

##  1\. Introduction

In 1942, Siegel proved that a biholomorphic germ

$f:(\mathbb{C},0)\longrightarrow(\mathbb{C},0),\ z\mapsto\alpha z+o(|z|)$

can be linearised, assuming an arithmetic condition on the coefficient
$\alpha$ [20]. Ever since, the field of Diophantine approximation has been of
relevance for dynamical systems and this led Arnold, Margulis and Pyartli and
others to reconsider the classical works on Diophantine approximation and
extend their scope [14, 17]. Nowadays, the work of Kleinbock and Margulis
finds application in KAM theory in order to prove that in certain situations
the quasiperiodic solutions make up a set of positive measure in phase space
[18]. To apply these results, one needs to assume certain non-degeneracy
conditions, introduced in the works of Mahler, Sprindžuk, Margulis and
Kleinbock [14, 15, 22, 23]. To our knowledge the weakest of these conditions
was introduced by Kleinbock:

######  Definition 1.1 ([13]).

Let $U$ be an open set in $\mathbb{R}^{d}$. A $C^{k}$-map
$B:U\longrightarrow\mathbb{R}^{e}$ is called $KMS$-non-degenerate (for
Kleinbock-Margulis-Sprindžuk) in an affine space ${\mathcal{L}}$ if:

  1. (1)

$B(U)\subset{\mathcal{L}}$.

  2. (2)

For some $l\leq k$ the partial derivatives at any point of $U$ of order $l$
span the vector space associated to ${\mathcal{L}}$.

It is not difficult to see that real analytic maps are always KMS non-
degenerate (see 3.3) and this will constitute our starting point. A simple
example of a KMS-degenerate function is given by the flat function:

$\mathbb{R}\longrightarrow\mathbb{R},\ x\mapsto e^{-1/x^{2}}$

Assuming the KMS non-degeneracy condition, one may estimate the preimage under
the map $B$ of points which are badly approximated by the integer lattice
$\mathbb{Z}^{e}\subset\mathbb{R}^{e}$ in various ways. For instance, under the
KMS non-degeneracy condition, Kleinbock-Margulis and Kleinbock proved that the
preimages of points satisfying an adequate Diophantine condition form a full
measure set ([14, Theorem A], [13, Theorem 1.2], see also [7]).

Unfortunately the verification of non-degeneracy conditions for the frequency
map of a dynamical system often rely on tedious computations and are
notoriously difficult to establish. For instance, it took many years before
Féjoz and Herman proposed a proof of the non-degeneracy condition for the
$N$-body problem initiated by Arnol’d in the sixties ([2, 6], see also [4]).
In this paper, we will show that for the type of normal forms introduced by
the authors in [8, 9, 10] the KMS-non degeneracy conditions are essentially
automatic. So let us now explain what type of generalisation of the usual
setting is needed by considering an instructive example.

###  1.1. The Poincaré correspondences

In Les Méthodes Nouvelles de la Mécanique Céleste [16, § 119]. Poincaré
considered the series

$s=1+\frac{x}{1+y}+\frac{x^{2}}{1+2y}+\ldots=\sum_{k=0}^{\infty}\frac{x^{k}}{1+ky}$

as a prototype of the kind of series that appear in perturbation theory (see
also the paper [19] of Sauzin for the resurgence properties). In fact, similar
expressions play a role in many areas of mathematics, for example in the
theory of $q$-analogs of classical series (logarithm, exponential,
hypergeometric), see for instance [21].

Assume now that we want to solve the equation

$0=f(x,y):=-y+\frac{x}{1+y}+\frac{x^{2}}{1+2y}++\frac{x^{3}}{1+3y}+\ldots.$

If we truncate the series at $x^{n}$ we obtain the rational functions

$f_{n}(x,y)=-y+\sum_{k=1}^{n}\frac{x^{k}}{1+ky}=\frac{P_{n}(x,y)}{D_{n}(y)}$

which determines a polynomial $P_{n}(x,y)$ in the variables $x$ and $y$:

$P_{1}(x,y)=-(1+y)y+x,\;\;P_{2}(x,y)=-(1+y)(1+2y)y+(1+2y)x+x^{2},\ldots$
$D_{1}(y)=1+y,\;\;\;D_{2}(y)=(1+y)(1+2y),\;\;\;D_{3}(y)=(1+y)(1+2y)(1+3y),\ldots$

These polynomials define algebraic curves

$V_{n}:=\\{(x,y)\in\mathbb{C}^{2}:P_{n}(x,y)=0\\}\subset\mathbb{C}\times\mathbb{C}$

of degree $(n+1)$ in $y$ and degree $n$ in $x$. These curves exhibit a
peculiar bending behaviour caused by the poles, located on horizontal lines,
defined by the denominators $D_{n}(y)$ that were cleared out.

Such a polynomial relation defines $y$ as a multivalued function of $x$,
sometimes called a correspondence. In a neighbourhood of $x=0,y=0$, the $n$-th
curve $V_{n}$ is the graph of a holomorphic function $b_{n}$ that can be
expanded in power series:

$\begin{array}[]{rcl}b_{3}(x)&=&x+2x^{5}-14x^{6}+64x^{7}+\ldots\\\
b_{4}(x)&=&x+x^{4}-3x^{5}+7x^{6}-19x^{7}+\ldots\\\
b_{5}(x)&=&x+x^{4}-2x^{5}+x^{6}+12x^{7}+\ldots\\\
b_{6}(x)&=&x+x^{4}-2x^{5}+2x^{6}+5x^{7}+\ldots\\\ \end{array}$

It is not difficult to see that $b_{n+1}(x)=b_{n}(x)+o(x^{n})$, so these
series converge coefficientwise to a formal power series solution
$b(x)\in\mathbb{C}[[x]]$ to the equation $f(x,y)=0$

$b(x)=x+x^{4}-2x^{5}+2x^{6}+6x^{7}-35x^{8}+86x^{9}+\ldots$

We call it the Birkhoff expansion of the series $f$ and its truncation are
called the Birkhoff polynomials of the series.

Correspondences in the Poincaré example.

The domain of definition of the $b_{n}$ shrinks as $n\longrightarrow\infty$
due to the peculiar winding, which provides a geometrical reason for the
divergence of the series $b(x)$. This is an illustration of a general
divergence theorem for such expressions, that we intend to publish elsewhere
[12]. On the other hand, the pictures strongly suggest that the part of curves
in the part $y>0$ nicely converge to the graph of a $C^{\infty}$-function $B$.

We may construct such a $C^{\infty}$ function as follows: start with closed
disks

$D_{\rho}:=\\{x\in\mathbb{C}:|x|\leq\rho\\},\
D_{\eta}:=\\{y\in\mathbb{C}:|y|\leq\eta\\},\ $

where $\rho<1$ and remove from the disk $D_{\eta}$ small disks
$D_{k}:=\\{y\in\mathbb{C}:|y+\frac{1}{k}|<r_{k}\\}$ around the poles $y=-1/k$.
The distance between two nearby poles being $1/k(k+1)$, if we choose
$r_{k}=k^{-\alpha},\ \alpha>2$, the complement of the disks $D_{k}$ contains
the origin. We have in the complement of $D_{k}$ an obvious estimate:

$\left|\frac{x^{k}}{1+ky}\right|\leq k^{\alpha-1}|x^{k}|\leq
k^{\alpha-1}\rho^{k}$

So for $\rho<1$, we see that the sequence obtained by restricting the rational
functions $f_{n}(x,y)$ to the closed set

$X_{\infty}:=D_{\rho}\times\left(D_{\eta}\setminus\cup_{k}D_{k}\right)$

converges in the $C^{0}$-topology to a limit function, that we denote in the
same way:

$f:X_{\infty}\longrightarrow\mathbb{C}$

###  1.2. Whitney differentiability

The closed set $X_{\infty}$ does contain $0$, but it is not an interior point
as the poles accumulate at the origin. However, the formal power series $b$
that solves the equation $f(x,b(x))=0$ is the Taylor expansion of a function,
$C^{\infty}$ in a sense originally discovered by Borel and now called Whitney
differentiability [3] Let us explain this property.

If $X\subset\mathbb{R}^{k}$ is a locally closed subset, then a function

$g:X\longrightarrow\mathbb{R}$

is called Whitney differentiable if there exists a map

$L:X\longrightarrow Hom(\mathbb{R}^{k},\mathbb{R})$

such that

$g(y)=g(x)+L(x)(y-x)+o(\|x-y\|).$

The main difference with usual derivatives is the requirement that the
variables $x,y$ play a more symmetric role. One defines similarly higher
derivatives and Whitney $C^{\infty}$ functions. Holomorphicity in the Whitney
setting is defined by requiring that the map $L$ is $\mathbb{C}$-linear
($\mathbb{C}^{d}=\mathbb{R}^{2d}$) and in that case the function was called
monogenic by Borel.

The Whitney extension theorem states that Whitney differentiable functions are
restrictions of usual differentiable functions and the Whitney derivative
commutes with restriction [24]. Fefferman showed that the extension operator
can be chosen to be bounded linear for any finite level of differentiability
[5]. In particular, the space of functions with bounded Whitney derivatives up
to order $k$ form a Banach space, like for ordinary differentiability.

Let us now show that our function $f$ are indeed Whitney differentiable on
$X_{\infty}$. The proof is elementary but is worthwhile understanding. We will
keep the notation $f_{n}$ for the restriction of $f_{n}$ to $X_{\infty}$.

We first create a second sequence $X^{\prime}_{n}$ by starting on a slightly
bigger disc $D_{\rho+\varepsilon}$, $\rho+\varepsilon<1$, and by removing from
$D_{\eta}$ disks $D^{\prime}_{k}\subset D_{k}$ of radius smaller than $r_{k}$,
say for instance $r_{k}/2$, for $k\leq n$, so that $X_{n}\subset
X^{\prime}_{n}$. The Cauchy inequalities give an estimate for the derivatives
of a holomorphic function on $X^{\prime}_{n}$ restricted to $X_{n}$:

$\sup_{(x,y)\in
X_{n}}|\partial_{x}^{i}\partial_{y}^{j}g(x,y)|\leq\frac{2^{j}i!j!}{r_{k}^{j}\varepsilon^{i}}\sup_{(x,y)\in
X^{\prime}_{n}}|g(x,y)|$

Using the estimate on the complement of $D^{\prime}_{k}$:

$\left|\frac{x^{k}}{1+ky}\right|<2k^{\alpha-1}|x^{k}|<2k^{\alpha-1}(\rho+\varepsilon)^{k},$

and the Cauchy estimates, one easily deduces that, for any $k\geq 0$, the
sequence $(f_{n})$ is a Cauchy sequence in the $C^{k}$-Whitney holomorphic
sense and therefore the sum function $f$ is Whitney $C^{\infty}$ in the
holomorphic sense.

By the Whitney extension theorem, the function $f$ is the restriction to
$X_{\infty}$ of a standard $C^{\infty}$ function

$F:\mathbb{C}^{2}\longrightarrow\mathbb{C}$

The set $F=0$ serves as $C^{\infty}$-version of the limit correspondence
defined by $f=0$. As $\partial_{y}F(0)=\partial_{y}f(0)=1\neq 0$, the usual
$C^{\infty}$ implicit function theorem shows that in a neighbourhood of the
origin the set $F=0$ is the graph of a $C^{\infty}$ function $B$, that we call
a $C^{\infty}$ Birkhoff-function attached to $f(x,y)$. Clearly, the Taylor-
series of $B$ is $b$.

Somehow we proved more than expected, as the function $B$ is now also defined
for $y<0$. This may look surprising, if we think of the way the real parts of
the curves $V_{n}$ have multiple components which stretch along the horizontal
lines near the $x$-axis. The resolution of this paradox is that by removing
the tubes $D_{\rho}\times D_{k}$, we removed the bendings of the curve at the
same time.

###  1.3. Content of the paper

If one wants to apply the Kleinbock-Margulis theorem, one should be able to
define and establish the KMS non-degeneracy condition for the correspondence
before going to the Birkhoff series and doing the Whitney extension. The
purpose of this paper is to show that this is indeed the case.

That the Birkhoff series $b$ does not see any of the bendings may explain
heuristically why establishing the KMS non-degeneracy condition based on
considerations of $b$ alone turns out to be very difficult. Our approach
consists of working as much as possible with the correspondences lying on the
product space, rather than with the implicit functions they define. So most of
the paper concerns the definition and analysis of non-degeneracy conditions
for correspondences.

In the last part, we explain the application of the results of this paper to
the Herman conjecture. We also hope that our result, elementary as it may
look, is of interest independently of the context in which it originally
emerged, and that the deep connection between number theory and dynamical
systems can be further strengthened, in the spirit of Siegel, Arnold and
Margulis.

##  2\. Statement of the theorem

###  2.1. Birkhoff functions, series and polynomials

We consider the space $\mathbb{C}^{N}=\mathbb{C}^{d}\times\mathbb{C}^{e}$ with
coordinates $x=(x_{1},\dots,x_{d})$, $y=(y_{1},\dots,y_{e})$ and projections:

$\textstyle{\mathbb{C}^{N}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\scriptstyle{p}$$\textstyle{\mathbb{C}^{d}}$$\textstyle{\mathbb{C}^{e}}$

We then consider a decreasing chain $X=(X_{n})$ of locally closed subsets of
$\mathbb{C}^{N}$:

$X_{0}\supset X_{1}\supset X_{2}\supset\ldots\supset
X_{\infty}:=\cap_{n}X_{n}.$

and assume that the sets $X_{n}$ are equal to the closure of their interior:
$\overline{X_{n}^{\circ}}=X_{n}$. We also assume that $X_{n}$ is of the form

$X_{n}={\mathcal{U}}\times{\mathcal{V}}_{n}$

where ${\mathcal{U}}\subset\mathbb{C}^{d}$ is a neighbourhood of the origin.

Such sequences arise often in constructions where $X_{n+1}$ is obtained from
$X_{n}$ by removing some open subset. Note that the closed set $X_{\infty}$ in
general could have empty interior. We also assume to be given a formal series
(in multi-index notations):

$f(x,y)=\sum_{I\in\mathbb{N}^{d}}a_{I}(y)x^{I},\ a_{I}\in\mathbb{C}\\{y\\}$

where $\mathbb{C}\\{y\\}$ denotes the ring of convergent power series in
$y=(y_{1},\dots,y_{e})$. We identify the coefficients of the series with
functions of $z=(x,y)$ depending only on $y$. We use the multi-index notation
and consider the Whitney $C^{\alpha}$-norms:

$\|a\|^{\alpha}_{U}=\sup_{z\in U\cap
X_{\infty}}|\partial^{\alpha}_{y}a(z)|,\alpha\in\mathbb{N}^{e}$

where the partial derivatives are taken only in the $y$-direction as the
functions involved as coefficients of the series only depend on $y$.

######  Definition 2.1.

We say the series $\sum_{I\in\mathbb{N}^{d}}a_{I}(y)x^{I}$ is $X$-convergent
if there is a neighbourhood $U$ of the origin such that for any
$\alpha\in\mathbb{N}^{e}$, the power series

$\sum_{I\in\mathbb{N}^{d}}\|a_{I}\|^{\alpha}_{U}x^{I}\in\mathbb{C}[[x]]$

is convergent in $p(U)$.

An $X$-convergent series in $U\cap X_{\infty}$ defines a Whitney $C^{\infty}$
function that we denote by the same name:

$f:U\cap X_{\infty}\longrightarrow\mathbb{C},\
(x,y)\mapsto\sum_{I\in\mathbb{N}^{d}}a_{I}(y)x^{I}$

Note that for $(x_{0},y_{0})\in U\cap X_{\infty}$, the map-germ

$(\mathbb{C}^{d},x_{0})\longrightarrow\mathbb{C},\ x\mapsto f(x,y_{0})$

is an ordinary holomorphic map-germ.

We use the notation:

$V(f):=\\{(x,y)\in U\cap X_{\infty}\;|\;f(x,y)=0\\}$

where $f=(f_{1},\dots,f_{k})$ are $X$-convergent series and we let $(V(f),0)$
be the germ of this set at the origin.

######  Definition 2.2.

A Birkhoff function-germ associated to $X$-convergent series
$f=(f_{1},\dots,f_{e})$, where $\partial_{y}f(0)$ has maximal rank, is a
$C^{\infty}$-function germ:

$B:(\mathbb{C}^{d},0)\longrightarrow(\mathbb{C}^{e},0)$

whose graph (denoted $\Gamma_{B}$) satisfies:

$(\Gamma_{B}\cap X_{\infty},0)=(V(f),0).$

Birkhoff polynomials and Birkhoff series are defined in a similar way by
considering infinitesimal and formal neighbourhoods instead of germs at the
origin.

Note that the Taylor expansion of a Birkhoff function $B$ at the origin is the
Birkhoff series of $f$. Indeed by the Whitney-Fefferman theorem ([5]), we may
choose for any $k>0$ and any neighbourhood $U$ of the origin a (real)
continuous linear extension operator111We consider independently the real and
imaginary parts when doing the extension. :

$T:C^{k}(U\cap X_{\infty},\mathbb{C})\longrightarrow C^{k}(U,\mathbb{C})$

The series

$F(x,y)=\sum_{I\in\mathbb{N}^{d}}T(a_{I}(y))x^{I}$

defines a $C^{\infty}$ function-germ. We have

$F(x,b_{k}(x))=o(\|x\|^{k})$

where $b_{k}$ is the degree $k$ Birkhoff polynomial of $f$. The manifolds
$\\{y=B(x)\\}$, $\\{F=0\\}$ and $\\{y=b_{k}(x)\\}$ are therefore tangent at
order $k$ at the origin. Hence the degree $k$ Taylor polynomial of the
function $B$ is the degree $k$ Birkhoff polynomial.

###  2.2. Non-degeneracy theorem

If $f=\sum_{I\in\mathbb{N}^{d}}a_{I}(y)x^{I}$ is an $X$-convergent series then
we may, by Whitney’s theorem, extend $f$ to a $C^{\infty}$ function-germ $F$.
By the implicit function theorem, the zero locus of $F$ is then the graph of a
Birkhoff function-germ. So Birkhoff functions always exists. The aim of this
paper is to show that the Birkhoff function can be chosen to be KMS non-
degenerate:

######  Theorem 2.3.

Assume that the formal series:

$f(x,y)=\sum_{I\in\mathbb{N}^{d}}a_{I}(y)x^{I}$

is $X$-convergent and that the matrix
$\partial_{y}f(0,0):=(\partial_{y_{j}}f_{i}(0,0))$ is of maximal rank. Then
the series $f$ admits KMS non-degenerate Birkhoff function germs, i.e., there
exists Whitney $C^{\infty}$ germs:

$B:(\mathbb{C}^{d},0)\longrightarrow(\mathbb{C}^{e},0)\text{ with
}(\Gamma_{B}\cap X_{\infty},0)=(V(f),0).$

It is useful to work over the field of complex numbers but of course our
constructions commute with complex conjugation, in particular if the $a_{I}$
are real analytic and $X$ invariant under complex conjugation then the
function $B$ can be chosen to be real, i.e.:

$B(\bar{x},\bar{y})=\overline{B(x,y)}$

Our construction is going to be explicit and not just an existence theorem.

##  3\. Maps, non-degeneracy and the Kleinbock lemma

###  3.1. The span

For a subset $X\subset\mathbb{C}^{e}$, we denote by $[X]$ the span of $X$,
defined as the smallest affine subspace containing $X$ .

Clearly, if $X\subset Y$, then $[X]\subset[Y]$. For a germ $(X,p)$ of a set we
define its span as

$[X,p]:=\bigcap_{U\in{\mathcal{V}}(p)}[U\cap X],$

where $U$ runs over the set ${\mathcal{V}}(p)$ of open neighbourhoods of the
point $p$. We always can find a sufficiently small representative $U$ for the
germ $(X,p)$, so that $[X,p]=[U]$.

For an open set $U\subset\mathbb{C}^{d}$ and map
$g:U\longrightarrow\mathbb{C}^{e}$ we also define its span as $[g(U)]$, and we
similarly define the span of a map germ
$g:(\mathbb{C}^{d},\alpha)\longrightarrow(\mathbb{C}^{e},\beta)$ at $\alpha$
as the affine space

$[g,\alpha]:=\cap_{U}[g(U),\beta],$

where the intersection runs over all representatives of the germ $g$.

######  Lemma 3.1.

Let $g:U\longrightarrow\mathbb{C}^{e}$ be an analytic map and $U(\alpha)$ the
connected component of $U$ containing $\alpha$. Then $g(U(\alpha))$ is
contained in the span of the germ of $g$ at $\alpha$:

$g(U(\alpha))\subset[g,\alpha].$

Consequently,

$[g(U(\alpha))]=[g,\alpha].$

###  3.2. The jet-span of a map

Given a multi-index $I\in\mathbb{N}^{d}$, we denote by

$\partial^{I}g(\alpha):=\frac{\partial^{I}g}{\partial
x^{I}}(\alpha)\in\mathbb{C}^{e}$

the vector of $I$-th derivative of $g$ at $\alpha$. We will also use the
notation

$|I|=i_{1}+\dots+i_{d},\ I=(i_{1},\dots,i_{d}).$

We define the $m$-jet span of a Whitney $C^{\infty}$ map

$g:X\longrightarrow\mathbb{C}^{e}$

at a point $\alpha\in X$ as the affine space:

$[g,\alpha]_{m}:=[\\{\partial^{I}g(\alpha):I\in\mathbb{N}^{d},\ |I|\leq m\\}]$

The $m$-jet spans form a increasing chain, so become stable, at some minimal
index $\tau=\tau(g,\alpha)$:

$[g,\alpha]_{\tau}=[g,\alpha]_{\tau+1}=\ldots$

We call $\tau(g,\alpha)$ the torsion index of the map $g$ at the point
$\alpha$. The condition $[g,\alpha]_{\tau}=[g,\alpha]$ corresponds to the KMS
non-degeneracy condition.

As an illustration, consider the germ

$g:(\mathbb{C},0)\longrightarrow(\mathbb{C}^{4},0):x\mapsto(1+x,x^{3},x^{100},x+x^{3}+x^{100})=(y_{1},\dots,y_{4})$

We have

$[g,0]_{0}\subset[g,0]_{1}=[g,0]_{2}\subset[g,0]_{3}=[g,0]_{4}=\ldots=[g,0]_{99}\subset[g,0]_{100}=[g,0]_{101}=\ldots.$

The span $[g,0]$ is the affine hyperplane
$[g,0]_{100}=\\{y_{4}=y_{1}+y_{2}+y_{3}+1\\}$ and the map-germ has torsion
index $\tau(g,0)=100$ at the origin. It is KMS non-degenerate, as any
holomorphic map.

A flat $C^{\infty}$ map-germ like

$g:(\mathbb{C},0)\longrightarrow(\mathbb{C},0),\
x\mapsto\left\\{\begin{matrix}e^{-1/x^{2}}\text{ if Re }x>0\\\ 0\text{
otherwise}\end{matrix}\right.$

has torsion index equal to zero at the origin but it is KMS degenerate.
Indeed, the span of this map germ is $[g,0]=\mathbb{C}$, but
$[g,0]_{n}=\\{0\\}$ for any $n$ therefore:

$[g,0]_{n}\neq[g,0]$

###  3.3. A lemma of Kleinbock

The following observation due to Kleinbock was formulated originally in the
real analytic setting:

######  Lemma 3.2 ([13]).

If $g:(\mathbb{C}^{d},\alpha)\longrightarrow(\mathbb{C}^{e},\beta)$ is a
holomorphic map germ, then it is KMS non degenerate.

###### Proof.

For simplicity assume that $\beta=0$ so that the span is a vector space. Let
$g:U\longrightarrow\mathbb{C}^{e}$ be a connected representative of the germ
$(g,\alpha)$ The sequence of spans $[g,\alpha]_{n}$ generated by the vectors
$\partial^{I}g(\alpha)$, $|I|\leq n$ stabilises at level $\tau$. Clearly
$[g,\alpha]_{\tau}\subset[g(U)]$. To show the other inclusion, let $\ell$ be
an affine linear form vanishing on $[g,\alpha]_{\tau}$. Then the Taylor series
of $\ell\circ g$ at $\alpha$ vanishes identically and therefore the
holomorphic function $\ell\circ g$ vanishes on all of $U$. As this holds for
all $\ell$ defining $[g,\alpha]_{\tau}$, we find
$[g(U)]\subset[g,\alpha]_{\tau}$, hence we have equality. ∎

As we saw previously, in the $C^{\infty}$ case, we may very well have a proper
inclusion:

$\bigcup_{k}[g,\alpha]_{k}\subset[g,\alpha]\text{ but
}\bigcup_{k}[g,\alpha]_{k}\neq[g,\alpha].$

##  4\. Correspondences and non-degeneracy

###  4.1. Maps and correspondences

Many naturally appearing expressions like $x^{2}+y^{2}=1$ define $y$ only as a
multivalued function $y=\sqrt{1-x^{2}}$. Especially in the complex domain,
insisting on working with a monovalent function is cumbersome and not very
enlightening. Therefore, in algebraic geometry one is used to working with
correspondences, which provide a convenient extension of the usual strict
concept of function. The Poincaré example shows that this point of view is
suited to the type of objects we are considering. So our purpose will be to
adapt the previous notions defined for maps to correspondences.

In this paper we will use the following terminology:

######  Definition 4.1.

A complex analytic variety
$V\subset\mathbb{C}^{N}=\mathbb{C}^{d}\times\mathbb{C}^{e}$ is said to define
a regular correspondence at $\omega=(\alpha,\beta)$, if the following
conditions are verified:
1) the set $V$ is smooth of dimension $d$ at $\omega$.
2) Under the projection onto the first factor, the tangent space $T_{\omega}V$
maps isomorphically onto $\mathbb{C}^{d}$.

Of course, by the implicit function theorem, we then can find a local
parametrisation
$g:(\mathbb{C}^{d},\alpha)\longrightarrow(\mathbb{C}^{e},\beta),x\mapsto
y=g(x)$ so that $(x,g(x))\in V$, or $(\Gamma_{g},\omega)=(V,\omega)$, where
$\Gamma_{g}$ denotes the graph of $g$.

###  4.2. The relative span

For a subset $X\subset\mathbb{C}^{N}:=\mathbb{C}^{d}\times\mathbb{C}^{e}$, we
denote by $\langle X\rangle$ the relative span of $X$, defined as the smallest
affine subspace of the form $\mathbb{C}^{d}\times F$ containing $X$ where
$F\subset\mathbb{C}^{e}$ is an affine subspace. Clearly, one has

$\langle X\rangle=q^{-1}(q([X])),$

where $q:\mathbb{C}^{d}\times\mathbb{C}^{e}\longrightarrow\mathbb{C}^{e}$ is
the projection on the second factor.

Of course, the notions of span of a set and of a map germ are related:

######  Lemma 4.2.

Let $\Gamma_{g}\subset\mathbb{C}^{d}\times\mathbb{C}^{e}=\mathbb{C}^{N}$
denote the graph of $g:U\longrightarrow\mathbb{C}^{e}$ and let
$q:\mathbb{C}^{N}\longrightarrow\mathbb{C}^{e}$ be the projection onto the
second factor. Then

$[g(U)]=q(\langle\Gamma_{g}\rangle)$

###### Proof.

Let $\ell:\mathbb{C}^{e}\longrightarrow\mathbb{C}$ be an affine function
vanishing on $[g(U)]$ then

$\ell^{\prime}:\mathbb{C}^{d}\times\mathbb{C}^{e}\longrightarrow\mathbb{C},\
(x,y)\mapsto\ell(y)$

is an affine function vanishing on $\Gamma_{g}$ and conversely. ∎

We define the relative span at a point $\omega\in X$ as the affine space

$\langle X,\omega\rangle=\cap_{U}\langle X\cap U,\omega\rangle,$

where the intersection runs over all neighbourhoods of
$\omega\in\mathbb{C}^{N}$ and get an analog of Lemma 3.1:

######  Lemma 4.3.

Let $V\subset\mathbb{C}^{N}$ be an analytic set and $V(\omega)$ the connected
component of $V$ that contains the point $\omega\in V$. Then $V(\omega)$ is
contained in the span of the germ $(V,\omega)$:

$V(\omega)\subset\langle V,\omega\rangle$

and consequently:

$\langle V(\omega)\rangle=\langle V,\omega\rangle.$

###### Proof.

Let $\ell$ be an affine function vanishing on $\langle V,\omega\rangle$. As a
holomorphic function vanishing on the germ $(V,\omega)$, it vanishes on a
neighbourhood of $V$ containing $\omega$. By the identity theorem of analytic
functions, $\ell$ vanishes on the connected component $V(\omega)$. ∎

###  4.3. The relative jet-span of an analytic set

For a germ of an analytic set $(V,\omega)$ we define the $n$-jet span

$\langle V,\omega\rangle_{n}$

as the smallest affine subspace of the form $\mathbb{C}^{d}\times F$ that
contains the $n$-th infinitesimal neighbourhood of the fibre at $\omega\in V$
of the projection $\mathbb{C}^{N}\longrightarrow\mathbb{C}^{d}$. It is
alternatively defined by the kernel of the affine linear functions contained
in the ideal $I+M_{\omega,x}^{n+1}$ which depend only on the $y$-variables.
Here $y_{1},\dots,y_{e}$ denote coordinates on $\mathbb{C}^{e}$,
$I\subset{\mathcal{O}}_{\mathbb{C}^{N},p}$ is the ideal defining $(V,\omega)$
and $M_{\omega,x}\subset{\mathcal{O}}_{\mathbb{C}^{N},\omega}$ the maximal
ideal of holomorphic functions vanishing at $x=\alpha$,
$\omega=(\alpha,\beta)$.

Clearly, one has

$\langle V,\omega\rangle_{1}\subset\langle V,\omega\rangle_{2}\subset\langle
V,\omega\rangle_{3}\subset\ldots,$

so this chain stabilises for a certain index $\tau:=\tau(V,p)$, that we call
the torsion index of $V$ at $p$:

$\langle V,\omega\rangle_{\tau}=\langle V,\omega\rangle_{\tau+1}=\ldots$

######  Lemma 4.4.

For any holomorphic map-germ
$g:(\mathbb{C}^{d},\alpha)\longrightarrow(\mathbb{C}^{e},\beta)$ we have

$q(\langle\Gamma_{g},\omega\rangle_{m})=[g,\alpha]_{m}$

for $\omega=(\alpha,\beta)$ and any $m\in\mathbb{N}$.

###### Proof.

Write $\langle\Gamma_{g},\alpha\rangle_{m}=\\{\ell=0\\}$ where

$\ell:\mathbb{C}^{d}\times\mathbb{C}^{e}\longrightarrow\mathbb{C}^{r},\
(x,y)\mapsto\ell(y)$

is some affine map depending only on the $y$-variables. By the Taylor formula,
we have

$\ell(g(x))=\sum_{|I|\leq
m}\frac{\partial^{I}g(\alpha)}{|I|!}(x-\alpha)^{I}+o(\|x\|^{m})$

and therefore the conditions $\ell(g(x))=o(\|x\|^{m})$ and
$\partial^{I}g(\alpha)=0$ for $|I|\leq m$ are equivalent. ∎

###  4.4. A variant of the Kleinbock lemma

The following result is the analog of the Kleinbock lemma for analytic spaces:

######  Lemma 4.5.

Let $(V,\omega)\subset(\mathbb{C}^{N},0)$ be a germ of an analytic space with
torsion index $\tau$ at $\omega$ then:

$\langle V,\omega\rangle=\langle V,\omega\rangle_{\tau}$

###### Proof.

Clearly, $\langle V,\omega\rangle_{\tau}\subset\langle V,\omega\rangle$. For
the converse, note that we have $\cap_{n}M_{p}^{n}=(0)$, so that a function
that vanishes on all infinitesimal neighbourhoods of $\omega$ vanishes on
$(V,\omega)$, from which we obtain the reverse inclusion. ∎

###  4.5. $n$-Correspondences and the finite determinacy lemma

We now consider truncated sums as we did in the introduction for the Poincaré
example:

######  Definition 4.6.

A correspondence $V$ is called an $n$-correspondence at $\omega$, if it is
defined by $d$ equations of the form

$f(x,y)=\sum_{|I|\leq n,\
I\in\mathbb{N}^{d}}a_{I}(y)x^{I}\in\mathbb{C}\\{y\\}[x]$

The following lemma is a refinement of Lemma 4.5:

######  Lemma 4.7.

If $V$ is an $n$-correspondence at $\omega$, then

$\tau(V,\omega)\leq n$

that is: $\langle V,\omega\rangle_{n}=\langle V,\omega\rangle$.

###### Proof.

Consider an affine function $\ell$ vanishing on $\langle V,\omega\rangle_{n}$
depending only on the $y$-variables. Let $v\in\mathbb{C}^{d}$ be an arbitrary
vector and consider the function

$\gamma:(\mathbb{C},0)\longrightarrow(\mathbb{C}^{e},0),\ t\mapsto
f(\alpha+tv,y)$

By definition of the $n$-span, the polynomial $\ell\circ\gamma$ in the
variable $t$ has a zero of multiplicity $\geq n+1$ at the origin:

$(\ell\circ\gamma)^{(k)}(0)=0,\ k=0,1,2,\ldots,n.$

As $(V,\omega)$ is an $n$-correspondence, the equation $\ell\circ\gamma(t)=0$
has degree $\leq n$ therefore $\ell\circ\gamma$ vanishes identically. As the
vector $v$ is arbitrary, we deduce that $\ell$ vanishes on $(V,\omega)$. This
inclusion being true for any hyperplane containing $\langle
V,\omega\rangle_{n}$, we get that $\langle V,\omega\rangle\subset\langle
V,\omega\rangle_{n}$. This proves the lemma. ∎

###  4.6. The spans stabilise

We return to the situation of subsection 2.2 and consider a decreasing chains
of subsets of $\mathbb{C}^{N}=\mathbb{C}^{d}\times\mathbb{C}^{e}$:

$X_{0}\supset X_{1}\supset X_{2}\supset\ldots\supset
X_{\infty}:=\cap_{n}X_{n}.$

with $\overline{X_{n}^{\circ}}=X_{n}$ and consider a formal power series

$f(x,y)=\sum_{I\in\mathbb{N}^{d}}a_{I}(y)x^{I}\in\mathbb{C}\\{y\\}[[x]],\
f(0)=0$

and define

$f_{n}(x,y)=\sum_{|I|\leq n,\
I\in\mathbb{N}^{d}}a_{I}(y)x^{I}\in\mathbb{C}\\{y\\}[x]$

We denote their zero-sets by

$V_{n}:=\\{(x,y)\in X_{n}\;|\;f_{n}(x,y)=0\\}\subset X_{n}$

We also have to consider

$V_{\infty}:=\\{(x,y)\in X_{\infty}\;|\;f(x,y)=0\\}\subset X_{\infty}$

######  Lemma 4.8.

One has

$\langle V_{n},0\rangle\subset\langle V_{n+1},0\rangle.$

###### Proof.

We have

$\langle V_{n},0\rangle=\langle V_{n},0\rangle_{n}=\langle
V_{n+1},0\rangle_{n}\subset\langle V_{n+1},0\rangle_{n+1}=\langle
V_{n+1},0\rangle.$

For the first and last equalities we used that $V_{n}$ is an
$n$-correspondence (for $n$ and $n+1$), the second equality is due to the fact
that

$f_{n+1}(x,y)=f_{n}(x,y)+O(\|x\|^{n+1})$

The middle inclusion is by the definition of jet-span. ∎

As a consequence, the sequence $\langle V_{n},0\rangle$ stabilises, say at
level $\rho$ and we define

$W_{\infty}:=\langle V_{n},0\rangle,n\geq\rho.$

######  Lemma 4.9.

For $n\geq\rho$, we have:

$\langle V_{\infty},0\rangle\subset\langle V_{n},0\rangle=W_{\infty}$

###### Proof.

Take $(x_{0},y_{0})\notin W_{\infty}$. By definition of $W_{\infty}$, this
means that for any $n\geq\rho$, the point $(x_{0},y_{0})$ does not belong to
$\langle V_{n},0\rangle$ and therefore is not in $V_{n}$, so we have
$f_{n}(x_{0},y_{0})\neq 0$ for $n\geq\rho$. Consequently, for $n\geq\rho$, the
holomorphic map-germs

$\phi_{n}:(\mathbb{C}^{d},x_{0})\longrightarrow\mathbb{C}^{e},\ x\mapsto
f_{n}(x,y_{0})=\sum_{|I|\leq n}a_{I}(y_{0})x^{I}$

are non-zero at $x=x_{0}$.

We now use the classical Hurwitz lemma (see e.g. [1, Chapter 5, Theorem 2]):
Consider a sequence of holomorphic functions $\phi_{n}$ converging in the
$C^{0}$-topology and assume that the limit $\phi_{\infty}$ is not the zero-
function. Then: if the functions $\phi_{n}$ are non-zero at a point, then the
limit $\phi_{\infty}$ is also non-zero at this point 222This Hurwitz lemma,
sometimes called the Hurwitz theorem, is a straightforward consequence of the
Cauchy integral formula..

As $f_{n}(x_{0},y_{0})\neq 0$, the coefficients $a_{I}$ are not all equal to
zero and therefore the holomorphic map-germ

$\phi_{\infty}:(\mathbb{C}^{d},x_{0})\longrightarrow\mathbb{C}^{e},\
x\mapsto\sum_{I\in\mathbb{N}^{d}}a_{I}(y_{0})x^{I}$

is not identically zero. From the Hurwitz lemma we conclude that
$(x_{0},y_{0})\notin V_{\infty}$ and therefore

$(V_{\infty},z_{0})\subset W_{\infty},\ z_{0}=(x_{0},y_{0})$

∎

###  4.7. Proof of Theorem 2.3

We keep the same notations and let

$g:({\mathcal{U}}\times W_{\infty},0)\longrightarrow(\mathbb{C}^{e},0)$

be the restriction of $f$ to $W_{\infty}$. By Lemma 4.9, we have

$(V(f),0)=(V(g),0).$

We choose a Whitney extension $G$ of $g$. By the implicit function theorem the
zero locus of $G$ is the graph of a function-germ

$\beta:(\mathbb{C}^{d},0)\longrightarrow(W_{\infty},0),\ x\mapsto\beta(x)$

and we may extend the image to $\mathbb{C}^{e}$ to get a Birkhoff function

$B:(\mathbb{C}^{d},0)\longrightarrow(\mathbb{C}^{e},0),\ x\mapsto\beta(x)$

The Taylor expansion of $B$ at the origin is the Birkhoff series whose partial
derivatives generate $W_{\infty}$ by definition. Hence the function $B$ is KMS
non-degenerate. This concludes the proof of the theorem.

##  5\. Application to the Herman conjecture

We give a brief outlook of our proof of the Herman invariant tori conjecture,
details will be published elsewhere, the reader may consult [10] for further
details.

###  5.1. The frequency map

In KAM theory, the invariant tori of a Hamiltonian system are parametrised by
the frequencies of motions which are not very well approximated by the
standard integral lattice $\mathbb{Z}^{e}$. In order to measure how a
frequency vector $y\in\mathbb{C}^{e}$ is approximated by the lattice, one
considers the collection of resonant hyperplanes $(\Sigma_{\alpha})$ with

$\Sigma_{\alpha}=\\{y\in\mathbb{C}^{e}:(y,\alpha)=0\\},\
\alpha\in\mathbb{Z}^{e}$

where

$(y,\alpha)=\sum_{j=1}^{e}y_{j}\alpha_{j}$

is the complexification of the Euclidean scalar product. We then fix a real
positive sequence $(\delta_{\alpha}),\ \alpha\in\mathbb{Z}^{e}$, which
converges to zero not too fast and consider tubular neighbourhoods of
hyperplanes

$T_{\alpha}=\\{y\in\mathbb{C}^{e}:|(y,\alpha)|<\delta_{\alpha}\\},\
\alpha\in\mathbb{Z}^{e}$

We now define the sequence $(X_{n})$ by:

$X_{n}=\mathbb{C}^{e}\setminus\bigcup_{\|\alpha\|\leq n}T_{\alpha}$

and this set parametrises the invariant tori, in the sense that there is a
Birkhoff function

$B:U\longrightarrow\mathbb{C}^{e}$

called the frequency map whose fibres above points of
$X_{\infty}=\bigcap_{n}X_{n}$ are (complexifications) of invariant tori. In
[8, 10], we constructed $B$ as a map obtained from a correspondence as we now
explain.

###  5.2. The small denominators ring

We consider the polynomial ring
$\mathbb{C}[\omega]=\mathbb{C}[\omega_{1},\omega_{2},\ldots,\omega_{d}]$ and
construct a ring $SD_{\alpha}$ of small denominators at
$\alpha\in\mathbb{C}^{d}$. The ring $SD_{\alpha}$ is a subring of the field
$\mathbb{C}(\omega)=\mathbb{C}(\omega_{1},\omega_{2},\ldots,\omega_{d})$ of
rational functions defined by localisation of
$\mathbb{C}[\omega]=\mathbb{C}[\omega_{1},\omega_{2},\ldots,\omega_{d}]$ with
respect with the multiplicative subset $S$ generated by all linear polynomials
$(\alpha+\omega,J),J\in\mathbb{Z}^{n}\setminus\\{0\\}$:

$SD_{\alpha}:=\mathbb{C}[\omega]_{S}:=\mathbb{C}[\omega,\frac{1}{(\alpha+\omega,J)},J\in\mathbb{Z}^{n}\setminus\\{0\\}]\subset\mathbb{C}(\omega).$

The elements of this ring may have poles along the resonance hyperplanes

$H_{J}:=\\{\omega\in\mathbb{C}^{d}:(\alpha+\omega,J)=0\\}$

We will construct a normal form inside the algebra

$R:=SD_{\alpha}[[\tau,p,q]]\subset\mathbb{C}[[\omega,\tau,p,q]],$

a subalgebra of the power series in $4d$ variables

$\omega_{1},\omega_{2},\ldots,\omega_{d},\tau_{1},\ldots,\tau_{d},q_{1},\ldots,q_{d},p_{1},\ldots,p_{d}.$

We provide $R$ with the standard Poisson bracket, so that the Poisson centre
of $R$ is the ring

$R_{0}:=SD_{\alpha}[[\tau]]\subset\mathbb{C}[[\omega_{1},\ldots,\omega_{d},\tau_{1},\ldots,\tau_{d}]].$

We endow this ring of quasi-homogenous filtration with weights

$w(q_{i})=w(p_{i})=1,\ w(\tau_{i})=2,\ w(\omega_{i})=0$

We use the notations $[f]_{i}^{j}$ for the sum of terms of weighted degree
$\geq i$ and $<j$ and omit the indices if $i=0$ or $j=+\infty$. We also use
the notation $O(n)$ for terms of degree $\geq n$.

###  5.3. The Hamiltonian normal form iteration

Starting from a Hamiltonian

$H=\sum_{i=0}^{d}\alpha_{i}p_{i}q_{i}+O(3),$

we first form the $\omega$-extension of $H$:

$F_{0}=\sum_{i=0}^{d}(\alpha_{i}+\omega_{i})p_{i}q_{i}+O(3),$

Then we solve a homological equation of the form:

$v_{0}(A_{0})=[F_{0}]_{3}^{4}+t_{0},\;\;\;t_{0}\in R_{0}+I^{2},$

where $I$ is the ideal generated by the $p_{i}q_{i}-\tau_{i}$ and $v_{0}$ is a
Poisson derivation. The definition might seem involved but it produces a
sequence of biholomorphic maps $e^{-v_{i}}$ which are defined globally and not
"blocked by resonances."

Exponentiating the Poisson derivation $v_{0}$, we produce a Poisson
automorphism $e^{-v_{0}}$. The application of $e^{-v_{0}}$ transforms $F_{0}$
into $F_{1}$, where the cubic term is removed modulo an element of
$R_{0}+I^{2}$ ; we put $A_{1}=A_{0}+t_{0}$. For the $n=0$ case, it turns out
that $t=0$ but at the next level we have to solve

$v_{1}(A_{1})=[F_{1}]_{4}^{6}+t_{1},\;\;\;t_{1}\in R_{0}+I^{2}$

for the degree $4$ and $5$ part of $F_{1}$ on $A_{1}$ and, as a general rule
$t_{1}\neq 0$.

Then the application of $e^{-v_{1}}$ to $F_{1}$ produces $F_{2}$, where now
these terms of degree $4$ and $5$ are removed, but certain terms in
$R_{0}+I^{2}$ are introduced. These remaining terms we add to $A_{1}$ and
obtain $A_{2}$. Next we solve the homological equation for the terms of degree
$6,7,8,9$ of $F_{2}$, but now on $A_{2}$, etc. Thus we obtain a by iteration a
sequence of triples

$(F_{n},A_{n},v_{n}),\;\;\;n=0,1,2,\ldots$

We have correspondences defined by the rational functions

$[e^{-v_{n}}\dots e^{-v_{0}}\omega_{i}]^{2^{n}}=\sum_{\alpha,|\alpha|\leq
2^{n}}a_{\alpha}(\omega)\tau^{\alpha},\ a_{\alpha}\in SD_{\alpha}$

precisely of the kind envisioned by Poincaré! This power series is
$X$-convergent, the partial sums define complex variety and the limit variety
parametrises the pairs consisting of invariant tori and associated
frequencies.

The theorem of this paper shows that the limiting correspondence contains the
graph of a KMS non-degenerate map. Using an adequate variant of the Kleinbock-
Margulis theorem ([7]), one deduces the fact that the invariant tori form a
set of positive measure. The convergence of the iteration scheme proposed is
subtle, but it turns out that it can be streamlined using the theory of Banach
functors developed in [11]. In that set-up, the non-degeneracy argument is
easy and, in fact, as this paper shows, “almost trivial”.

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