# Exploring Eco-Friendly Gas Mixtures for Resistive Plate Chambers: A
Comprehensive Study on Performance and Aging

The RPC ECOgas@GIF++ Collaboration: L. Quaglia M. Abbrescia G. Aielli R. Aly
M. C. Arena M. Barroso L. Benussi S. Bianco D. Boscherini F. Bordon A. Bruni
S. Buontempo M. Busato P. Camarri R. Cardarelli L. Congedo D. De Jesus Damiao
M. De Serio A. Di Ciaccio L. Di Stante P. Dupieux J. Eysermans A. Ferretti G.
Galati M. Gagliardi R. Guida G. Iaselli B. Joly S.A. Juks K.S. Lee B. Liberti
D. Lucero Ramirez B. Mandelli S.P. Manen L. Massa A. Pastore E. Pastori D.
Piccolo L. Pizzimento A. Polini G. Proto G. Pugliese D. Ramos G. Rigoletti A.
Rocchi M. Romano A. Samalan P. Salvini R. Santonico G. Saviano M. Sessa S.
Simone L. Terlizzi M. Tytgat E. Vercellin M. Verzeroli N. Zaganidis

###### Abstract

Resistive Plate Chambers (RPCs) are gaseous detectors widely used in high
energy physics experiments, operating with a gas mixture primarily containing
Tetrafluoroethane (C2H2F4), commonly known as R-134a, which has a global
warming potential (GWP) of 1430. To comply with European regulations and
explore environmentally friendly alternatives, the RPC EcoGas@GIF++
collaboration, involving ALICE, ATLAS, CMS, LHCb/SHiP, and EP-DT communities,
has undertaken intensive R&D efforts to explore new gas mixtures for RPC
technology.

A leading alternative under investigation is HFO1234ze, boasting a low GWP of
6 and demonstrating reasonable performance compared to R-134a. Over the past
few years, RPC detectors with slightly different characteristics and
electronics have been studied using HFO and CO2-based gas mixtures at the CERN
Gamma Irradiation Facility. An aging test campaign was launched in August
2022, and during the latest test beam in July 2023, all detector systems
underwent evaluation. This contribution will report the results of the aging
studies and the performance evaluations of the detectors with and without
irradiation.

###### keywords:

Resistive Plate Chambers , eco-friendly gas mixtures , aging studies

††journal: Nuclear Inst. and Methods in Physics Research, A

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##  1 Introduction

Resistive Plate Chambers (RPCs) are gaseous detectors with planar geometry and
resistive electrodes (either made out of bakelite or glass). Thanks to their
relatively low-cost and $\approx$ns time resolution they are widely employed
in the muon trigger/identification systems of the LHC experiments [1, 2, 3].
These RPCs are operated in avalanche mode, with a gas mixture containing a
high fraction ($>$ 90%) of C2H2F4 and SF6 (plus a fraction of i-C4H10 as
photon quencher) and, although this mixture satisfies all the performance
requirements, it contains a high fraction of C2H2F4 and SF6, which are
classified as fluorinated greenhouse gases (F-gases/GHGs).

Starting from 2014, new European Union regulations [4] have imposed a
progressive phase-down in the production and usage of these compounds, leading
to an increase of cost and reduction in availability. For this reason, CERN
has adopted a policy of F-gases reduction and, since RPCs represent a
significant fraction of the total GHG-gases emission of the LHC
experiments[5], it is of the utmost importance to search for more eco-friendly
RPC gas mixtures. The first efforts have been concentrated on the replacement
of C2H2F4 using its industrial replacement, the tetrafluoropropene (C3H2F4 or
simply HFO, in its -ze isomer) diluted with other gases to lower the detector
working voltage [6, 7, 8, 9]. Promising gas mixtures, where C2H2F4 has been
replaced with different fractions of HFO/CO2 have been identified and now a
more complete characterization of those mixtures in controlled data-taking
environments (such as beam tests), as well as the study of the detectors long-
term behavior (aging studies) when operated with these new gas mixtures is
needed.

To this aim, the RPC ECOgas@GIF++ collaboration (including researchers from
ALICE, ATLAS, CMS, LHCb/SHiP and the CERN EP-DT group) was created to join
forces among RPC experts of the different LHC experiments, sharing knowledge
and manpower. Each group has provided a detector prototype, which has been
installed on a common mechanical support and both beam tests as well as aging
studies are being carried out using HFO-based gas mixtures. This contribution
will provide an overview of the latest results obtained from the ongoing aging
studies.

The text is divided as follows: Section 2 contains a description of the
experimental setup and the methodology used in the data-taking/analysis,
Section 3 reports the main results obtained from the ongoing aging campaign
where mixtures with different HFO/CO2 ratios are being studied and, lastly,
Section 4 is dedicated to the conclusion and to possible outlooks for the
future of this work.

##  2 Experimental setup and methodology

The experimental setup of the RPC ECOgas@GIF++ collaboration is installed at
the Gamma Irradiation Facility (GIF++)[10], located on the H4 secondary SPS
beam line at CERN. This facility is equipped with a high activity 137Cs source
($\approx$12.5 TBq), which can be used to induce a high background radiation
on the detectors under test, allowing one to simulate long operation periods
in a much shorter time-span (years in $\approx$months). The radiation from the
137Cs source can be modulated by means of a set of 3$\times$3 lead attenuation
filters, leading to 27 possible irradiation intensities. Moreover, in
dedicated periods, the facility is also traversed by a high-energy (150 GeV)
muon beam, allowing one to periodically test the performance of the detectors
under study.

Table 1 reports the main features of each detector of the RPC ECOgas@GIF++
collaboration. Note that the CMS RE11 RPC is the only one having a double gas
gap: top (T in Table 1, divided in two: top-wide (TW)) and bottom (B in Table
1 or BOT for short). Moreover, it is the only one with a trapezoidal shape
while all the others are rectangles.

Name | Gaps | Gap (mm) | Electrode (mm) | Area (cm2)
---|---|---|---|---
ALICE | 1 | 2 | 2 | 2500
ATLAS | 1 | 2 | 2 | 550
CMS RE11 | 2 | 2 | 2 | 3627(T)/4215(B)
EP-DT | 1 | 2 | 2 | 7000
LHCb/SHiP | 1 | 1.6 | 1.6 | 7000
Table 1: Features of the RPC ECOgas@GIF++ collaboration detectors

Figure 1 shows a sketch of the experimental setup currently installed at the
GIF++. The gas mixing and distribution system allows the users to mix up to
four gases, by means of the same amount of mass flow controllers (MFCs). Two
more MFCs are used to regulate the relative humidity of the mixture by
changing the amount of gas flowing in a humidifier tank. The high voltage (HV)
to power the detectors is provided by means of a CAEN SY1527
mainframe111https://www.caen.it/subfamilies/mainframes/ and two CAEN A1526
boards222https://www.caen.it/products/a1526/ (one with positive and one with
negative polarity). A monitoring software is employed to continuously store
all the relevant parameters (i.e. environmental conditions, gas mixture
composition as well as current absorbed by the detectors and applied high
voltage) on a dedicated database for later analysis. During the beam test
campaigns, a scintillator-based trigger system is also installed and the data
from the detectors are acquired by the readout system (which is detector
specific, as detailed in [11]). For what concerns the aging studies, a
dedicated software [12] (referred to as webdcs), specifically developed for
the needs of the CMS collaboration studies at the GIF++, has been adapted for
the RPC ECOgas@GIF++ collaboration studies and it is employed as an online
Detector Control System (DCS) for the studies described in the following.

Figure 1: Sketch of the experimental setup installed in GIF++ (a detailed
description of each component is reported in the text)

###  2.1 Methodology

During the ongoing aging studies (which began in July 2022), the stability in
time of the absorbed current has been studies. In particular, the HV applied
to the detectors is set at a fixed value of effective high voltage (i.e.
corrected for temperature and pressure variations) and the absorbed current is
sampled with a frequency of 1 measurement every 30 s.

Moreover, once a week, the 137Cs source is fully shielded (source-off in the
following) and a measurement of the current absorbed without any background is
carried out as a function of the high voltage (dark current scan). The aim of
this operation is two-fold: on the one hand it is used to monitor the
stability of the dark current over time (an increase of this quantity could be
a sign of potential detector aging) and, on the other, it is used to more
accurately estimate the charge integrated by the RPCs during the aging.
Indeed, during aging studies, the integrated charge (defined as the integral
of the current in time) is used as a proxy for the age of the detector. Figure
2 shows an example of a dark current scan from the EP-DT detector. One can see
that, for voltages below the gas multiplication threshold ($\approx$7-8 kV for
a 2 mm gas gap detector), a non-zero current is flowing through the detector
and, given the voltage range, this current is not passing through the gas but
rather through other conductive paths in the RPCs and, therefor, it is not
directly contributing to the gas-induced aging of the detectors. By looking at
Figure 2, one can see that in the range 0-5 kV the current increases linearly
(for this reason this is usually referred to as Ohmic dark current) with the
HV and, by carrying out a linear interpolation in this range, one can estimate
the Ohmic dark current at the irradiation voltage (working point) and subtract
it from the total current absorbed under irradiation, to get a more realistic
estimate of the actual current flowing through the gas and use this to
calculate the integrated charge.

Figure 2: Example of dark current scan, obtained from the EP-DT detector. The
linear interpolation to estimate the Ohmic dark current at the irradiation
voltage is also shown in red in the figure and the blue line represents the
irradiation voltage

##  3 Results

Two mixtures have been aging-tested by the RPC ECOgas@GIF++ collaboration
(referred to as ECO1 and ECO2 in the following). Their composition is reported
in Table 2, together with the currently employed gas mixture (STD), which is
taken as a reference to which the eco-friendly alternatives have to be
compared.

Name | C2H2F4 (%) | HFO (%) | CO2 (%) | i-C4H10 (%) | SF6 (%)
---|---|---|---|---|---
STD | 95.2 | 0 | 0 | 4.5 | 0.3
ECO1 | 0 | 45 | 50 | 4 | 1
ECO2 | 0 | 35 | 60 | 4 | 1
Table 2: Composition of the gas mixtures used in the aging studies

Some preliminary studies using the STD gas mixture, described for example in
[13], have shown that the data-taking procedure and apparatus work as expected
hence a first eco-friendly alternative was tested (ECO1). The working point of
this mixture was found to be $\approx$2 kV higher than the STD gas mixture
(11.6 kV rather than 9.6 kV for the 2 mm gap detectors). Moreover, it was
observed that, given the same background radiation level, the current absorbed
by the detectors operated with the ECO1 mixture, was $\approx$1.7 times higher
than with the STD gas mixture. Following the integration of $\approx$10 mC/cm2
all the detectors of the collaboration started to show signs of current
instability (both in the dark as well as in the one absorbed under
irradiation); moreover, the greater absorbed current could also be related to
a higher impurity production under irradiation. These two observations lead to
discard this mixture from further studies and the ECO2 mixture, with a reduced
HFO content (and a lower working point) was introduced. Using this mixture the
currents were still higher than with the STD one (by a factor $\approx$1.5)
but the detectors behaved in a more stable manner and a longer irradiation
campaign was launched.

The latter started in July 2022 (concurrently with a beam test, used to set
the baseline performance of the RPCs) and is still ongoing at the time of
writing. For most of the irradiation period, the background radiation has been
$\approx$500 Hz/cm2 and, to limit the current absorbed by the detectors, it
was decided to operate them at $\approx$70% efficiency.

Figure 3 shows the main results obtained during the aging studies, for the
LHCb/SHiP detector. Rather than as a function of time, the data are shown as a
function of the integrated charge (calculated with the subtraction of the
Ohmic dark current, as explained in Section 2). The data-points in blue in
Figure 3 represent the total current density (in nA/cm2) absorbed by detector
(as measured by the HV module) while the ones in green represent the total
current density with the subtraction of the Ohmic dark current (following the
procedure highlighted in Section 2).

It has to be noted that the current is shown independently of the irradiation
condition and the portions of the figure where the current is lower then 5
nA/cm2 represent periods of time without irradiation while changing HV values
correspond to the dark current scans mentioned earlier.

The HV applied to the LHCb/SHiP RPC has been increased in steps (from 8.5 up
to 9.8 kV), being the only detector of the collaboration to reach full
efficiency under irradiation. By looking at Figure 3 a few comments can be
made: in the 0-20 mC/cm2 range both the current under irradiation as well as
the dark current (both Ohmic and not) were completely stable. Increasing the
voltage (20-40 mC/cm2) the total dark current shows a slight growth, which
continued when the voltage was further risen (although in both cases the Ohmic
dark current did not increase). Towards the 100 mC/cm2 mark (corresponding to
full efficiency), a more significant increase of the absorbed current can be
observed. Finally, the detector was moved closer to the 137Cs source and a
lower HV was applied to further limit the absorbed current. Nonetheless, the
current steadily increased and, this time, the Ohmic dark current increased as
well. As a follow-up to these observations, it was decided to investigate
whether a significant variation in electrode resistivity was taking place
since this could possibly explain the absorbed current variations.

Figure 3: Trend of current and effective high voltage as a function of the
integrated charge for the LHCb/SHiP RPC

A somewhat different behavior can be observed by looking at Figure 4, which
shows the results obtained with the BOT gap of the CMS RE11 detector.

Figure 4: Trend of current density and effective high voltage as a function of
the integrated charge for the CMS RE11 BOT gap

The increase of separation between the blue and green curves (in the 40-100
mC/cm2 range) is a sign of an increase in absorbed Ohmic dark current, which
seems to fade after the $\approx$100 mC/cm2 mark. The total dark current seems
to follow a similar trend. Nonetheless the current for all the three gaps of
the CMS RE11 detector shows a more stable behavior over time.

As it was anticipated, a significant change in the resistivity of the
electrodes could explain the observed trend. Indeed, using the Ar method
(described in [14]), one can calculate this quantity and periodic resistivity
measurements are being carried out to monitor its evolution. Figure 5 shows
the trend of the electrode resistivity over time for the LHCb/SHiP detector.
The value seems to be quite stable over time, hence the change in absorbed
current cannot be attributed to a simple change in electrode resistivity and a
more in-depth investigation is needed to uncover the root cause of this
behavior. Similar observations can be made for the BOT gap of the CMS RE11
detector. Indeed, its resistivity shows an increasing trend over time and this
cannot explain the observed behavior of the absorbed current.

Figure 5: Trend of the electrode resistivity (measured with the Ar method)
during the aging test time period for the SHiP/LHCb detector (the 0 values
represent measurements where the detector was not included)

Figures 6(a) and 6(b) show the trend of the integrated charge over time for
all the detectors of the collaboration. The latter figure refers to the three
gaps of the CMS RE11 detector while the former to all the other RPCs. On
average, a total of $\approx$100 mC/cm2 has been accumulated by all the RPCs
(the slight difference among them can be attributed to the different
efficiency at the irradiation voltage as well as different distances from the
irradiation source).

(a) Image A

(b) Image B

Figure 6: This is a figure with two subfigures

In July 2023 a second beam test campaign was carried out, in order to compare
the RPC performance to the baseline values obtained in July 2022. This
comparison has, so far, only been carried out for the ALICE RPC and the
preliminary results are shown in Figure 7. The top panel of said figure shows
the comparison between the source-off efficiency in July 2022 (black curve)
and July 2023 (red curve) while the bottom panel shows (with the same color
convention of the top panel) the dark current measured in the two beam tests.

The efficiency curves have been interpolated using a logistic function (as
explained in [11]) to extract the working point, defined as the high voltage
knee (HVeff where the efficiency is 95% of its plateau value) + 150 V. The
comparison of the two curves shows no appreciable decrease of the maximum
efficiency, although the working point is shifted to higher voltages by
$\approx$250 V while the slope of the 2023 curve seems to be decreasing, with
respect to the 2022 data. The comparison of the dark current curves shows an
increase between the two beam tests and further studies are ongoing to
understand if the working point shift could be justified by the observed
current increase. Moreover, the analyses of the data from other collaboration
members will show if similar effects are observed in other detectors as well.

Figure 7: Top panel: comparison of source-off efficiency curves between July
2022 and 2023, obtained with the ECO2 gas mixture. Bottom panel: comparison of
dark current curves between July 2022 and 2023, obtained with the ECO2 gas
mixture

##  4 Conclusions and outlook

The search for an eco-friendly alternative gas mixture for RPC detectors is a
hot-topic in the gaseous detector community, especially due to the new EU
regulations which foresee a progressive phase down in the production and usage
of F-gases (which represent more than 90% of the currently employed gas
mixture).

Several eco-friendly alternatives, where C2H2F4 is replaced by different
concentrations of HFO/CO2 have been identified using cosmic muons and,
currently, a long-term aging campaing is ongoing. Two different eco-friendly
alternatives (ECO1 and ECO2) have been tested. For both gas mixtures the
detector working point is $\approx$2 and 1 kV higher than the one of the
currently employed gas mixture respectively.

ECO1 was discarded due to current instabilities in all detectors (observed
after the integration of only $\approx$10 mC/cm2) as well as due to its higher
working point, possibly leading to a greater production of impurities. ECO2
has shown a less unstable behavior, for some detectors more than for others
and more in depth analysis (and hardware upgrades) are ongoing to clarify this
observation. In general, a total of $\approx$100 mC/cm2 has been integrated up
to now by all the detectors of the ECOgas collaboration.

A preliminary comparison of the RPC performance before and after the
irradiation campaign (using the muon beam) has been carried out for the ALICE
detector and it has shown that the maximum efficiency reached has not
appreciably decreased, although a shift of the WP of $\approx$150-250 V
(depending on the mixture) has been observed and further analysis is ongoing
to possibly correlate it with the observed increase in absorbed current.

## References

  * [1] ALICE collaboration, _ALICE Technical Design Report of the Dimuon Forward Spectrometer_ , CERN/LHCC 99-22 (1999)
  * [2] CMS collaboration, _The CMS muon project: Technical Design Report_ , CERN/LHCC 97-32 (1997)
  * [3] ATLAS collaboration, _ATLAS muon spectrometer: Technical Design Report_ , CERN/LHCC 97-022 (1997)
  * [4] Council of European Union, _Council regulation (EU) no 517/2014_ (2014)
  * [5] Mandelli, B. _et al._ , _Strategies for reducing the use of greenhouse gases from particle detectors operation at the CERN LHC Experiments_ , Journal of Physics: Conference Series (2022
  * [6] Proto, G., _Study of the performance of the RPC detector with new eco-friendly gas mixtures_ , Il nuovo Cimento C (2021)
  * [7] Bianchi, A. _et al._ , _Studies on tetrafluoropropene-based gas mixtures with low environmental impact for Resistive Plate Chambers_ , JINST 15 C04039 (2020)
  * [8] Rigoletti, G. _et al._ , _Characterization of RPC detectors with LHC-like background radiation and new eco-friendly gas mixtures_ , JINST 15 C11003 (2020)
  * [9] Benussi, L. _et al._ , _A study of HFO-1234ze (1,3,3,3-Tetrafluoropropene) as an eco-friendly replacement in RPC detectors_ , INFN-REPORT-INFN-14-14-LNF (2015)
  * [10] Pfeiffer, D. _et al._ , _The radiation field in the Gamma Irradiation Facility GIF++ at CERN_ , NIM A, vol. 866 91-103 (2017)
  * [11] L. Quaglia _et al._ , _Preliminary results on the long term operation of RPCs with eco-friendly gas mixtures under irradiation at the CERN Gamma Irradiation Facility_ , submitted to EPJ Plus (2023)
  * [12] Fagot, A. (2020), _Consolidation and Extension of the CMS Resistive Plate Chamber System in View of the High-Luminosity LHC Upgrade_ , PhD thesis defended at Universiteit Gent. Faculteit Wetenschappen (2020)
  * [13] L. Quaglia, _Development of eco-friendly gas mixtures for Resistive Plate Chambers_. PhD thesis the University of Turin Apr 26th, (2023)
  * [14] G. Aielli _et al._ , _Fluoride production in RPCs operated with F-compound gases_ , Nuclear Physics B, 158 143–148 (2006)


# On the orbits of plane automorphisms and their stabilizers

###### Abstract.

Let $\Bbbk$ be a perfect field with algebraic closure $\overline{\Bbbk}$. If
$H$ is a subgroup of plane automorphisms over $\Bbbk$ and
$p\in\overline{\Bbbk}^{2}$ is a point, we describe the subgroup consisting of
plane automorphisms which stabilize the orbit of $p$ under $H$, when this
orbit has irreducible closure in $\overline{\Bbbk}^{2}$. As an application, we
treat the case where $H$ is cyclic and the closure of the orbit of $p$ is an
arbitrary (non-necessarily irreducible) curve.

Alvaro Rittatore and Iván Pan111Research of both authors was partially
supported by CSIC (Udelar), ANII and PEDECIBA, of Uruguay.

##  1\. Introduction

In [BlSt2015] the authors study the group of automorphisms of the affine plane
preserving some given curve over an arbitrary field $\Bbbk$. As an application
of their main results, they propose a classification of the group of
automorphisms stabilizing an arbitrary set
$\Delta\subset\overline{{\mathbb{k}}}^{2}$ — here $\overline{{\mathbb{k}}}$
denotes the algebraic closure of $\Bbbk$. However, this classification (see
[BlSt2015, Prop. 3.11]) is incomplete, since the authors assume implicitly
that $\Delta$ is such that the Zariski closure of $\Delta$ in
$\overline{{\mathbb{k}}}^{2}$ coincides with the zero locus of
$I_{\mathbb{k}}(\Delta)$, the ideal of the ${\Bbbk}$–polynomials vanishing at
$\Delta$.

Inspired by the aim of completing the classification mentioned above, we have
been led to consider the following problem, which is interesting in its own
right and whose study is the objective of this paper:

> If $H$ is a subgroup of plane automorphisms over a perfect field
> ${\mathbb{k}}$ and $p\in\overline{{\mathbb{k}}}^{2}$, then describe the
> stabilizer of its orbit, i.e. the set of plane automorphisms which leave
> invariant ${\rm O}_{H}(p)=\bigl{\\{}h(p):h\in H\bigr{\\}}$.

In particular, we show that when $\Delta$ is not closed in
$\overline{{\mathbb{k}}}^{2}$ an interesting phenomenon occurs: in many cases
the group of automorphisms stabilizing $\Delta$ is infinite countable and
therefore not algebraic.

In Section 2 we present the basic definitions and preliminary results
concerning the automorphisms group of a subset of the affine plane over
$\overline{{\mathbb{k}}}$.

In Section 3 we present our main result, namely, we describe the stabilizer of
${\rm O}_{H}(p)$ when it has irreducible closure in
$\overline{{\mathbb{k}}}^{2}$, see Theorem 3.4. As an application, in Section
4 we consider the case where $H$ is a cyclic group and the closure of ${\rm
O}_{H}(p)$ is a (non-necessarily irreducible) curve, see Theorem 4.8.

##  2\. Preliminaries

In this section we introduce the definition of the automorphisms group of a
subset of the affine plane ${\mathbb{A}}^{2}$ and recall the description of
the automorphisms group of a curve in ${\mathbb{A}}^{2}$.

######  Notation 2.1.

The following notations will be kept through the paper.

  1. (1)

In what follows $\Bbbk$ is an arbitrary perfect field and
$\overline{{\mathbb{k}}}$ is an algebraic closure of ${\mathbb{k}}$.

  2. (2)

We let ${\mathbb{A}}^{2}=\operatorname{Spec}\bigl{(}{\mathbb{k}}[x,y]\bigr{)}$
and denote the set of ${\mathbb{k}}$-points and
$\overline{{\mathbb{k}}}$-points in ${\mathbb{A}}^{2}$ by
${\mathbb{A}}^{2}({\mathbb{k}})$ and
${\mathbb{A}}^{2}(\overline{{\mathbb{k}}})$ respectively. Recall that
${\mathbb{A}}^{2}(\overline{{\mathbb{k}}})$ identifies with the
$\overline{{\mathbb{k}}}$-points of
${\mathbb{A}}^{2}_{\overline{{\mathbb{k}}}}=\operatorname{Spec}(\overline{{\mathbb{k}}})\times_{\mathbb{k}}{\mathbb{A}}^{2}$.

  3. (3)

If $\Delta\subset{\mathbb{A}}^{2}(\overline{{\mathbb{k}}})$, we denote the
_vanishing ideals_ of $\Delta$ by
$\mathcal{I}(\Delta)\subset{\mathbb{k}}[x,y]$ and
$\mathcal{I}_{\overline{{\mathbb{k}}}}(\Delta)\subset\overline{{\mathbb{k}}}[x,y]$.
If $J\subset{{\Bbbk}[x,y]}$ is an ideal, we denote
$\mathcal{V}(J)=\operatorname{Spec}\bigl{(}{\mathbb{k}}[x,y]/J\bigr{)}$ viewed
as a closed subscheme of ${\mathbb{A}}^{2}$, and
$\mathcal{V}_{\overline{{\mathbb{k}}}}(J)=\mathcal{V}(J)(\overline{{\mathbb{k}}})\subset{\mathbb{A}}^{2}_{\overline{{\mathbb{k}}}}(\overline{{\mathbb{k}}})\cong{\mathbb{A}}^{2}(\overline{{\mathbb{k}}})$
the _zero set of $J$_.

  4. (4)

If $\Delta\subset{\mathbb{A}}^{2}(\overline{{\mathbb{k}}})$, we set
$\widehat{\Delta}=\mathcal{V}_{\overline{{\mathbb{k}}}}\bigl{(}I(\Delta)\bigr{)}\subset{\mathbb{A}}^{2}(\overline{{\mathbb{k}}})$;
the Zariski closure of $\Delta$ in ${\mathbb{A}}^{2}(\overline{{\mathbb{k}}})$
is denoted by $\overline{\Delta}$ — hence,
$\overline{\Delta}\subset\widehat{\Delta}\subset{\mathbb{A}}^{2}(\overline{{\mathbb{k}}})$.

  5. (5)

The _automorphisms group_ of ${\mathbb{A}}^{2}$ is denoted by ${\rm
Aut}({\mathbb{A}}^{2})$ — recall that ${\rm Aut}({\mathbb{A}}^{2})$ is the
(abstract) group of isomorphisms of ${\mathbb{k}}$-schemes
$f:{\mathbb{A}}^{2}\to{\mathbb{A}}^{2}$.

The following well known result, that gives further insight on the
relationship between $\overline{\Delta}$ and $\widehat{\Delta}$, follows from
Galois descent (see for example [Mi2017, Appendix A.j]); we include its proof
for the sake of completeness.

######  Lemma 2.2.

Consider the canonical action of $G={\rm
Gal}(\overline{{\mathbb{k}}}/{\mathbb{k}})$ on
${\mathbb{A}}^{2}(\overline{{\mathbb{k}}})$. If
$\Delta\subset{\mathbb{A}}^{2}(\overline{{\mathbb{k}}})$, then
$\widehat{\Delta}=G\cdot\overline{\Delta}$. In particular, every irreducible
component of $\widehat{\Delta}$ is the image of an irreducible component of
$\overline{\Delta}$ under an element in $G$.

###### Proof.

It is clear that
$\widehat{\Delta}\subset{\mathbb{A}}^{2}(\overline{{\mathbb{k}}})$ is a
$G$–stable closet subset containing $\overline{\Delta}$. Therefore, by Galois
descent, there exists an unique closed subscheme
$X\subset\operatorname{Spec}\bigl{(}{\mathbb{k}}[x,y]\bigr{)}$ such that
$\widehat{\Delta}$ identifies with the set of $\overline{{\mathbb{k}}}$-points
of
$\operatorname{Spec}(\overline{{\mathbb{k}}})\times_{\operatorname{Spec}({\mathbb{k}})}X$.
Under the identification
${\mathbb{A}}^{2}({\mathbb{k}})\subset{\mathbb{A}}^{2}(\overline{{\mathbb{k}}})$,
we have that $X({\mathbb{k}})$ is the set of $\overline{{\mathbb{k}}}$-points
of $\widehat{\Delta}$ fixed by the Galois action and
$\widehat{\Delta}=G\cdot\bigr{(}X\cap{\mathbb{A}}^{2}({\mathbb{k}})\bigr{)}$.
It follows that
$X({\mathbb{k}})=\overline{\Delta}\cap{\mathbb{A}}^{2}({\mathbb{k}})$ and
$G\cdot\overline{\Delta}=\widehat{\Delta}$. The last assertion is an easy
consequence of Galois descent. ∎

We introduce now the main object of study of this work, namely the group of
automorphisms that stabilize an arbitrary subset of
$\overline{{\mathbb{k}}}$-points of the affine plane.

######  Definition 2.3.

Let $a:{\rm
Aut}({\mathbb{A}}^{2})\times{\mathbb{A}}^{2}_{\overline{{\mathbb{k}}}}(\overline{{\mathbb{k}}})\to{\mathbb{A}}^{2}_{\overline{{\mathbb{k}}}}(\overline{{\mathbb{k}}})$,
$a(f,p)=f(\overline{{\mathbb{k}}})(p)$, be the canonical action of ${\rm
Aut}({\mathbb{A}}^{2})$ on ${\mathbb{A}}^{2}(\overline{{\mathbb{k}}})$. If
$\Delta\subset{\mathbb{A}}^{2}(\overline{{\mathbb{k}}})$, we denote the
_stabilizer_ of $\Delta$ under $a$ by ${\rm Aut}({\mathbb{A}}^{2},\Delta)$.

######  Remark 2.4.

  1. (1)

Recall that an automorphism $\varphi\in{\rm Aut}({\mathbb{A}}^{2})$ is given
by a pair $(f,g)\in{\mathbb{k}}[x,y]\times{\mathbb{k}}[x,y]$, such that the
corresponding endomorphism of ${\mathbb{A}}^{2}_{\overline{{\mathbb{k}}}}$ is
an automorphism, with inverse
$(f,g)^{-1}=(h,j)\in{\mathbb{k}}[x,y]\times{\mathbb{k}}[x,y]$. Under this
identification, $a\bigl{(}(f,g),p\bigr{)}=\bigl{(}f(p),g(p)\bigr{)}$.

  2. (2)

Let $\Delta\subset{\mathbb{A}}^{2}(\overline{{\mathbb{k}}})$ and
$\varphi\in{\rm Aut}({\mathbb{A}}^{2},\Delta)$. If
$\varphi^{*}:\overline{{\mathbb{k}}}[x,y]\to\overline{{\mathbb{k}}}[x,y]$
denotes the induced automorphism (which is defined over ${\mathbb{k}}$), then
$\varphi^{*}$ stabilizes the ideals $\mathcal{I}(\Delta)$ and
$\mathcal{I}_{\overline{{\mathbb{k}}}}(\Delta)$. If follows that $\varphi$
stabilizes $\overline{\Delta}$ and $\widehat{\Delta}$; therefore $\varphi^{*}$
also stabilizes the ideals $\mathcal{I}(\widehat{\Delta})$ and and
$\mathcal{I}_{\overline{{\mathbb{k}}}}(\widehat{\Delta})$.

In other words:

${\rm Aut}({\mathbb{A}}^{2},\Delta)\subset{\rm
Aut}\bigl{(}{\mathbb{A}}^{2},\overline{\Delta}\bigr{)}\subset{\rm
Aut}\bigl{(}{\mathbb{A}}^{2},\widehat{\Delta}\bigr{)}.$ (1)

######  Notation 2.5.

If $H\subset{\rm Aut}({\mathbb{A}}^{2})$ is a subgroup and
$p\in{\mathbb{A}}^{2}(\overline{{\mathbb{k}}})$, we denote the _$H$ -orbit_ of
$p$ as ${\rm O}_{H}(p)$.

######  Example 2.6.

It is well known that the inclusions (1) may be strict. For example if
${\mathbb{k}}={\mathbb{Q}}$ and $\Delta=\bigl{\\{}(\sqrt{2},0)\bigr{\\}}$,
then $\widehat{\Delta}=\bigl{\\{}(\sqrt{2},0),(-\sqrt{2},0)\bigr{\\}}$ and
$\varphi=(-x,y+x^{2}-2)\in{\rm
Aut}({\mathbb{A}}^{2},\widehat{\Delta})\setminus{\rm
Aut}({\mathbb{A}}^{2},\Delta)$.

The fact that the first inclusion may be strict is a key point in our study of
the stabilizers of orbits.

In [BlSt2015] the authors study the stabilizer of a closed subset of the
affine plane. In particular, they give an explicit description of the
geometrically irreducible curves $\mathcal{C}$ such that ${\rm
Aut}\bigl{(}{\mathbb{A}}^{2},\mathcal{C}(\overline{{\mathbb{k}}})\bigr{)}$ is
an algebraic group. We briefly recall now these results — according to Remark
2.4, we present an element of ${\rm Aut}({\mathbb{A}}^{2})$ as a pair of
polynomials $(f,g)\in{\mathbb{k}}[x,y]$.

######  Remark 2.7.

If $\Delta\subset{\mathbb{A}}^{2}({\overline{{\mathbb{k}}}})$ is finite, then
${\rm Aut}({\mathbb{A}}^{2},\Delta)$ is not an algebraic group (see [BlSt2015,
Proposition 3.11]) — notice that if
$\Delta=\bigl{\\{}(a_{i},b_{i})\in{\mathbb{A}}^{2}(\overline{{\mathbb{k}}});i=1,\ldots,\ell\bigr{\\}}$
and $m\gg 0$, then there exists a polynomial $P_{m}\in{\mathbb{k}}[x]$ of
degree $m$ such that $P(a_{i})=0$ for any $i=1,\ldots,\ell$. Therefore, ${\rm
Aut}_{\mathbb{k}}({\mathbb{A}}^{2},\Delta)$ contains the automorphisms
$\bigl{(}x,y+P_{m}(x)\bigr{)}\in{{\Bbbk}[x,y]}$ for all $m\gg 0$, and it
follows from [Ka1979, Theorem 1.3] that ${\rm
Aut}_{\mathbb{k}}({\mathbb{A}}^{2},\Delta)$ is not algebraic.

######  Theorem 2.8.

Let $\mathcal{C}\subset{\mathbb{A}}^{2}$ be a geometrically irreducible and
reduced curve. Then, up to applying a plane automorphism, $\mathcal{C}$ is one
of the curves of the following list:

  1. (1)

$\mathcal{C}=\mathcal{V}\bigl{(}x^{b}-\lambda y^{a}\bigr{)}$, where $a,b>1$
are coprime integers and $\lambda\in{\mathbb{k}}^{*}$. If this is the case,
then

${\rm
Aut}\bigl{(}{\mathbb{A}}^{2},\mathcal{C}(\overline{{\mathbb{k}}})\bigr{)}=\bigl{\\{}(t^{a}x,t^{b}y)\mathrel{:}t\in{\mathbb{k}}^{*}\bigr{\\}}\cong{\mathbb{k}}^{*}.$

  2. (2)

$\mathcal{C}=\mathcal{V}\bigl{(}x^{b}y^{a}-\lambda\bigr{)}$, where $a,b\geq 1$
are coprime integers, with $ab\neq 1$, and $\lambda\in{\mathbb{k}}^{*}$. If
this is the case, then

${\rm
Aut}\bigl{(}{\mathbb{A}}^{2},\mathcal{C}(\overline{{\mathbb{k}}})\bigr{)}=\bigl{\\{}(t^{a}x,t^{-b}y)\mathrel{:}t\in{\mathbb{k}}^{*}\bigr{\\}}\cong{\mathbb{k}}^{*}.$

  3. (3)

$\mathcal{C}=\mathcal{V}\bigl{(}xy-\lambda\bigr{)}$, where
$\lambda\in{\mathbb{k}}^{*}$. If this is the case, then

${\rm
Aut}\bigl{(}{\mathbb{A}}^{2},\mathcal{C}(\overline{{\mathbb{k}}})\bigr{)}=\bigl{\\{}(tx,t^{-1}y)\mathrel{:}t\in{\mathbb{k}}^{*}\bigr{\\}}\rtimes\\{\operatorname{id},\sigma\\}\cong{\mathbb{k}}^{*}\rtimes{\mathbb{Z}}_{2},$

where $\sigma(x,y)=(y,x)$ is the permutation of coordinates.

  4. (4)

$\mathcal{C}=\mathcal{V}\bigl{(}\lambda x^{2}+\nu y^{2}-1\bigr{)}$, where
${\rm char}({\mathbb{k}})\neq 2$, and $\lambda,\nu\in{\mathbb{k}}^{*}$ are
such that $-\lambda\nu$ is not a square in ${\mathbb{k}}$. If this is the
case, then ${\rm
Aut}\bigl{(}{\mathbb{A}}^{2},\mathcal{C}(\overline{{\mathbb{k}}})\bigr{)}$ is
the subgroup of $\operatorname{GL}_{2}({\mathbb{k}})$ preserving the form
$\lambda x^{2}+\nu y^{2}$ :

${\rm
Aut}\bigl{(}{\mathbb{A}}^{2},\mathcal{C}(\overline{{\mathbb{k}}})\bigr{)}=\bigl{\\{}\left(\begin{smallmatrix}a&-\nu
b\\\ \lambda
b&a\end{smallmatrix}\right)\mathrel{:}(a,b)\in{\mathbb{k}}\times{\mathbb{k}}\,,\
a^{2}+\lambda\nu
b^{2}=1\bigr{\\}}\rtimes\left\\{\operatorname{id},\tau\right\\}=T_{\lambda,\nu}\rtimes{\mathbb{Z}}_{2},$

where $\tau(x,y)=(x,-y)$. Notice that $T_{\lambda,\nu}$ is a
non-${\mathbb{k}}$-split torus.

  5. (5)

$\mathcal{C}=\mathcal{V}\bigl{(}x^{2}+\mu xy+y^{2}-1\bigr{)}$, where ${\rm
char}({\mathbb{k}})=2$ and $\mu\in\Bbbk^{*}$ is such that $x^{2}+\mu x+1$ has
no root in ${\mathbb{k}}$. If this is the case, then ${\rm
Aut}\bigl{(}{\mathbb{A}}^{2},\mathcal{C}(\overline{{\mathbb{k}}})\bigr{)}$ is
the subgroup of $\operatorname{GL}_{2}({\mathbb{k}})$ preserving the form
$x^{2}+\mu xy+y^{2}$:

${\rm
Aut}\bigl{(}{\mathbb{A}}^{2},\mathcal{C}(\overline{{\mathbb{k}}})\bigr{)}=\left\\{\left(\begin{smallmatrix}a&b\\\
b&a+\mu b\end{smallmatrix}\right)\mathrel{:}a,b\in{\mathbb{k}}\,,a^{2}+\mu
ab+b^{2}=1\right\\}\rtimes\left\\{\operatorname{id},\sigma_{\mu}\right\\}=T_{\mu}\rtimes{\mathbb{Z}}_{2},$

where $\sigma_{\mu}(x,y)=(x+\mu y,y)$. Notice that $T_{\mu}$ is a
non-${\mathbb{k}}$-split torus.

  6. (6)

$\mathcal{C}$ is the line of equation $x=0$, in which case

${\rm
Aut}\bigl{(}{\mathbb{A}}^{2},\mathcal{C}(\overline{{\mathbb{k}}})\bigr{)}=\Bigl{\\{}\bigl{(}ax,by+P(x)\bigr{)}\mathrel{:}a,b\in{\mathbb{k}}^{*}\,,\
P\in{\mathbb{k}}[x]\Bigr{\\}}.$

In particular, if $R:{\rm
Aut}\bigl{(}{\mathbb{A}}^{2},\mathcal{C}(\overline{{\mathbb{k}}})\bigr{)}\to{\rm
Aut}({\mathbb{A}}^{1})=\bigl{\\{}y\mapsto
by+c\mathrel{:}b\in{\mathbb{k}}^{*}\,,c\in{\mathbb{k}}\bigr{\\}}$ denotes the
restriction map, then
$\operatorname{Ker}(R)=\Bigl{\\{}(x,y)\mapsto\bigl{(}ax,y+P(x)\bigr{)}\mathrel{:}a\in{\mathbb{k}}^{*}\,,\
P\in{\mathbb{k}}[x]\,,\ P(0)=0\Bigr{\\}}$ and

${\rm
Aut}\bigl{(}{\mathbb{A}}^{2},\mathcal{C}(\overline{{\mathbb{k}}})\bigr{)}=\operatorname{Ker}(R)\rtimes{\rm
Aut}({\mathbb{A}}^{1}).$

  7. (7)

In any other case, the curve $\mathcal{C}$ is such that ${\rm
Aut}\bigl{(}{\mathbb{A}}^{2},\mathcal{C}(\overline{{\mathbb{k}}})\bigr{)}$ is
finite.

###### Proof.

See [BlSt2015, Theorem 2]. ∎

######  Definition 2.9.

We will abuse notations and say that
$D\subset{\mathbb{A}}^{2}(\overline{{\mathbb{k}}})$ is a _curve of type_ (1) –
(7), if $D=\mathcal{C}(\overline{{\mathbb{k}}})$, where $\mathcal{C}$ verifies
the respective condition in Theorem 2.8. The _canonical form_ of a curve of
type (1) – (6) is the curve of equations given in the Theorem 2.8 for the
corresponding type.

Following [BlSt2015], we say that a curve $\mathcal{C}$ is of _fence type_ if
up to applying an automorphism of ${\rm Aut}({\mathbb{A}}^{2})$, $\mathcal{C}$
has equation $P(x)=0$, with $P\in{\mathbb{k}}[x]$.

######  Corollary 2.10.

Assume that $\Delta$ is a proper closed subset of
${\mathbb{A}}^{2}(\overline{{\mathbb{k}}})$. Then one of the following
assertions holds:

$(a)$ $\Delta$ is contained in a curve of fence type, in which case ${\rm
Aut}({\mathbb{A}}^{2},\Delta)$ is not an algebraic group.

$(b)$ The group ${\rm Aut}({\mathbb{A}}^{2},\Delta)$ is conjugate to an
algebraic subgroup of either ${\rm Aff}({\mathbb{A}}^{2})$ or either $J_{n}$
for some $n\geq 2$ (notations as in [BlSt2015]).

###### Proof.

With the added hypothesis of the closedness of $\Delta$, the proof of
[BlSt2015, Proposition 3.11] now applies without obstructions. ∎

##  3\. The orbit of a group of automorphisms of the plane

Let $H\subset{\rm Aut}({\mathbb{A}}^{2})$ be a subgroup and
$p\in{\mathbb{A}}^{2}(\overline{{\mathbb{k}}})$. We are interested now in the
calculation of ${\rm Aut}\bigl{(}{\mathbb{A}}^{2},{\rm O}_{H}(p)\bigr{)}$.
Notice that we have that

$H\subset{\rm Aut}\bigl{(}{\mathbb{A}}^{2},{\rm O}_{H}(p)\bigr{)}\subset{\rm
Aut}\bigl{(}{\mathbb{A}}^{2},\overline{{\rm O}_{H}(p)}\bigr{)}.$

######  Lemma 3.1.

Let $H\subset{\rm Aut}({\mathbb{A}}^{2})$ be a subgroup and
$p\in{\mathbb{A}}^{2}(\overline{{\mathbb{k}}})$. Then $\overline{{\rm
O}_{H}(p)}$ is either:

  1. (1)

a finite set;

  2. (2)

a curve $\mathcal{C}=\mathcal{C}_{1}\cup\dots\cup\mathcal{C}_{s}$, with
$\mathcal{C}_{i}\cong\mathcal{C}_{j}$ for all $i,j$, where $\mathcal{C}_{1}$
is a curve of type (1) – (6);

  3. (3)

${\mathbb{A}}^{2}(\overline{{\mathbb{k}}})$.

###### Proof.

Assume that $\overline{{\rm O}_{H}(p)}$ is neither finite nor all the plane;
then $\dim\overline{{\rm O}_{H}(p)}=1$ and therefore $H$ is necessarily
infinite. Let $\overline{{\rm O}_{H}(p)}=\bigcup_{i=1}^{s}\mathcal{C}_{i}$ be
the decomposition in irreducible components, with $\dim\mathcal{C}_{1}=1$. By
continuity and the irreducibility of $\mathcal{C}_{1}$, it follows that there
exists $h_{i}\in H$ such that $h_{i}\cdot\mathcal{C}_{1}=\mathcal{C}_{i}$.
Hence, it remains to prove that $\mathcal{C}_{1}$ is a curve of type (1) –
(6).

Let $H_{i}=\\{h\in H\mathrel{:}h\cdot\mathcal{C}_{1}=\mathcal{C}_{i}\\}\subset
H$, $i=1,\dots s$. Since $H$ is infinite, there exists $i\in\\{1,\dots,s\\}$
such that $\\#H_{i}=\infty$. Let $h_{i}\in H_{i}$ be a fixed element. Then
$h_{i}^{-1}H_{i}\subset H_{1}\subset{\rm
Aut}({\mathbb{A}}^{2},\mathcal{C}_{1})$, and it follows that $\\#{\rm
Aut}({\mathbb{A}}^{2},\mathcal{C}_{1})=\infty$. We deduce from Theorem 2.8
that $\mathcal{C}_{1}$ is a curve of type (1) – (6). ∎

We collect now some technical remarks that we need in order to calculate the
stabilizers of an orbit.

######  Lemma 3.2.

Let $\mathcal{C}$ be a curve of type (1) – (6) given by its canonical form and
$p=(x_{p},y_{p})\in\mathcal{C}(\overline{{\mathbb{k}}})$. If we keep the
notations of Theorem 2.8, then the isotropy group ${\rm
Aut}({\mathbb{A}}^{2},\mathcal{C})_{p}$ is trivial unless

  1. (i)

$\mathcal{C}=\mathcal{V}\bigl{(}xy-\lambda\bigr{)}$, in which case ${\rm
Aut}\bigl{(}{\mathbb{A}}^{2},\mathcal{C}(\overline{{\mathbb{k}}})\bigr{)}_{p}=\bigl{\\{}{\rm
Id},\frac{x_{p}}{y_{p}}\sigma\bigr{\\}}$, where $\sigma=(y,x)$ is the
permutation of coordinates.

  2. (ii)

$\mathcal{C}=\mathcal{V}\bigl{(}\lambda x^{2}+\nu y^{2}-1\bigr{)}$, in which
case

${\rm
Aut}\bigl{(}{\mathbb{A}}^{2},\mathcal{C}(\overline{{\mathbb{k}}})\bigr{)}_{p}=\bigl{\\{}{\rm
Id},t_{\lambda x_{p}^{2}-\nu
y_{p}^{2},2x_{p}y_{p}}\tau\bigr{\\}}=\bigl{\\{}{\rm Id},t_{2\lambda
x_{p}^{2}-1,2x_{p}y_{p}}\tau\bigr{\\}},$

where $t_{a,b}=\left(\begin{smallmatrix}a&-\nu b\\\ \lambda
b&a\end{smallmatrix}\right)\in T_{\lambda,\nu}$ and
$\tau=\left(\begin{smallmatrix}1&0\\\ 0&-1\end{smallmatrix}\right)$.

  3. (iii)

$\mathcal{C}=\mathcal{V}\bigl{(}x^{2}+\mu xy+y^{2}-1\bigr{)}$, in which case

${\rm
Aut}\bigl{(}{\mathbb{A}}^{2},\mathcal{C}(\overline{{\mathbb{k}}})\bigr{)}_{p}=\bigl{\\{}{\rm
Id},t_{\mu x_{p}y_{p}+1,\mu y_{p}^{2}}\sigma_{\mu}\bigr{\\}}=\bigl{\\{}{\rm
Id},t_{x_{p}^{2}+y_{p}^{2},\mu y_{p}^{2}}\sigma_{\mu}\bigr{\\}},$

where $t_{a,b}=\left(\begin{smallmatrix}a&b\\\ b&a+\mu
b\end{smallmatrix}\right)\in T_{\mu}$ and
$\sigma_{\mu}=\left(\begin{smallmatrix}1&\mu\\\ 0&-1\end{smallmatrix}\right)$.

  4. (iv)

$\mathcal{C}$ is the line of equation $x=0$, in which case

$\begin{split}{\rm
Aut}\bigl{(}{\mathbb{A}}^{2},\mathcal{C}(\overline{{\mathbb{k}}})\bigr{)}_{p}=&\
\Bigl{\\{}\bigl{(}ax,by+P(x)\bigr{)}\mathrel{:}a,b\in{\mathbb{k}}^{*}\,,\
P\in{\mathbb{k}}[x],P(0)=-by_{p}\Bigr{\\}}\cong\\\ &\
\operatorname{Ker}(R)\rtimes\bigl{\\{}(0,y)\mapsto(0,by+(1-b)y_{p})\mathrel{:}\
b\in{\mathbb{k}}^{*}\bigr{\\}}.\end{split}$

###### Proof.

Let $G={\rm
Aut}\bigl{(}{\mathbb{A}}^{2},\mathcal{C}(\overline{{\mathbb{k}}})\bigr{)}$. If
$\mathcal{C}$ is of type (1) or (2), it is clear that $G_{p}=\\{{\rm Id}\\}$
for any $p\in\mathcal{C}(\overline{{\mathbb{k}}})$.

  1. (i)

Easy calculations show that if $p=(x_{p},y_{p})$ is a
$\overline{{\mathbb{k}}}$-point of $\mathcal{V}\bigl{(}xy-\lambda\bigr{)}$,
then $G_{p}=\bigl{\\{}{\rm Id},\frac{x_{p}}{y_{p}}\sigma\bigr{\\}}$.

  2. (ii)

Consider now $\mathcal{C}=\mathcal{V}\bigl{(}\lambda x^{2}+\nu
y^{2}-1\bigr{)}$ and let
$p=(x_{p},y_{p})\in\mathcal{C}(\overline{{\mathbb{k}}})$. Then
$(T_{\lambda,\nu})_{p}=\\{{\rm Id}\\}$ and an involution
$t_{a,b}\tau=\left(\begin{smallmatrix}a&\nu b\\\ \lambda
b&-a\end{smallmatrix}\right)$ fixes $p$ if and only if there exists
$s\in{\mathbb{k}}^{*}$ such that

$\left\\{\begin{array}[]{rcl}-s\nu b&=&x_{p}\\\ s(a-1)&=&y_{p}\\\
\end{array}\right.$ (2)

Indeed, $(-\nu b,a-1)$ is a basis of the space of eigenvectors of eigenvalue
$1$ of $t_{a,b}\tau$.

In order to solve Equation (2), we first recall that we also have the
compatibility conditions

$\left\\{\begin{array}[]{rcl}a^{2}+\lambda\nu b^{2}&=&1\\\ \lambda
x_{p}^{2}+\nu y_{p}^{2}&=&1\\\ \end{array}\right.$ (3)

If $(a,b)\neq(1,0)$ then we can eliminate $s$ in Equation (2). If we
substitute $x_{p}$ in Equation (3), we obtain that $(a,b)=(\lambda
x_{p}^{2}-\nu y_{p}^{2},2x_{p}y_{p})=(2\lambda x_{p}^{2}-1,2x_{p}y_{p})$.
Notice that since $a\neq 1$, we deduce that
$p\neq\bigl{(}\pm\frac{1}{\sqrt{\lambda}},0\bigr{)}$. An easy calculations
show that $G_{{}_{(\pm\frac{1}{\sqrt{\lambda}},0)}}=\\{{\rm Id},\tau\\}$.

  3. (iii)

If $\mathcal{C}=\mathcal{V}\bigl{(}\lambda x^{2}+\nu y^{2}-1\bigr{)}$ and
$p=(x_{p},y_{p})\in\mathcal{C}(\overline{{\mathbb{k}}})$, we proceed as in the
previous case: we have that $(T_{\mu})_{p}=\\{{\rm Id}\\}$ and
$t_{a,b}\sigma_{\mu}=\left(\begin{smallmatrix}a&\mu a+b\\\
b&a\end{smallmatrix}\right)$ fixes $p$ if and only if there exists
$s\in{\mathbb{k}}^{*}$ such that

$\left\\{\begin{array}[]{rcl}s(a-1)&=&x_{p}\\\ sb&=&y_{p}\\\
\end{array}\right.$

with compatibility conditions

$\left\\{\begin{array}[]{rcl}a^{2}+\mu ab+b^{2}&=&1\\\ x_{p}^{2}+\mu
x_{p}y_{p}+y_{p}^{2}&=&1\\\ \end{array}\right.$

If $(a,b)\neq(1,0)$ then we can eliminate $s$ and obtain that $(a,b)=(\mu
x_{p}y_{p}+1,\mu y_{p}^{2})=(x_{p}^{2}+y_{p}^{2},\mu y_{p}^{2})$. Notice that
$t_{1,\mu}\sigma_{\mu}$ has not $1$ as eigenvalue, and that since $a\neq 1$,
we deduce that $p\neq\bigl{(}1,0\bigr{)}$. An easy calculation show that
$G_{(1,0)}=\\{{\rm Id},\sigma_{\mu}\\}$.

  4. (iv)

For the case $\mathcal{C}=\mathcal{V}(x)$, just recall that ${\rm
Aut}\bigl{(}{\mathbb{A}}^{2},\mathcal{C}(\overline{{\mathbb{k}}})\bigr{)}$ is
the semi-direct product of $\operatorname{Ker}(R)$ and ${\rm
Aut}({\mathbb{A}}^{1})$.

∎

######  Remark 3.3.

(1) Let $\mathcal{C}$ be a curve of type (3) – (5) and consider $G={\rm
Aut}({\mathbb{A}}^{2},\mathcal{C})$. Let $p\in\mathcal{C}$ be such that
$\overline{{\rm O}_{G}(p)}=\mathcal{C}$ — it is clear that such a $p$ exists.
Then, it follows from Theorem 2.8 and Lemma 3.2 that

$G=T\rtimes G_{p}=T\rtimes\\{{\rm Id},\tau_{p}\\}=T\cup T\tau_{p},$

where $T$ is a possibly non-split torus and $\tau_{p}$ is the involution given
in Lemma 3.2.

Notice that $t\tau_{p}=t^{-1}\tau_{p}$ for all $t\in T$. In particular, any
element of $T\tau_{p}$ is also an involution and ${\rm
Aut}({\mathbb{A}}^{2},\mathcal{C})=T\rtimes\\{{\rm Id},t\tau_{p}\\}$ for all
$t\in T$.

(2) It follows that if $H\subset{\rm Aut}({\mathbb{A}}^{2},\mathcal{C})$ is a
subgroup, then either $H\subset T$ or

$H=H_{0}\cup H_{0}\gamma=H_{0}\rtimes\\{{\rm Id},\gamma\\},$

where $H_{0}=H\cap T$ and $\gamma\in H$ is a non trivial involution. Indeed,
if $\gamma,\gamma^{\prime}\in H\cap(T\gamma)$, then $\gamma^{\prime}=h\gamma$,
with $h\in H_{0}$.

(3) In particular, if a subgroup $H\subset{\rm Aut}({\mathbb{A}}^{2})$ and
$p\in{\mathbb{A}}^{2}(\overline{{\mathbb{k}}})$ are such that $\overline{{\rm
O}_{H}(p)}$ is a curve of type (3) – (5), it follows that either $H$ is a
subgroup of the corresponding torus $T\subset{\rm
Aut}\bigl{(}{\mathbb{A}}^{2},\overline{{\rm O}_{H}(0)}\bigr{)}$ or

$H=H_{0}\rtimes\\{{\rm Id},t_{0}\tau_{p}\\}=H_{0}\cup H_{0}t_{0}\tau_{p},$

where $H_{0}=T\cap H_{0}$ and $t_{0}\in T$.

Analogously, we have that $A={\rm Aut}\bigl{(}{\mathbb{A}}^{2},{\rm
O}_{H}(p)\bigr{)}$ is either a subgroup of $T$ or

$A=A_{0}\rtimes\\{{\rm Id},t_{1}\tau_{p}\\}=A_{0}\cup A_{0}t_{1}\tau_{p},$

where $A_{0}=T\cap A_{0}$ and $t_{1}\in T$.

######  Theorem 3.4.

Let $H\subset{\rm Aut}({\mathbb{A}}^{2})$ be a subgroup and let
$p\in{\mathbb{A}}^{2}(\overline{{\mathbb{k}}})$ such that
$\mathcal{C}=\overline{{\rm O}_{H}(p)}$ is an irreducible curve. In the
notations of Remark 3.3, we have that:

  1. (a)

If $\mathcal{C}$ is a curve of type (1) or (2) then $A=H$.

  2. (b)

If $\mathcal{C}$ is a curve of type (3), (4) or (5), then $G_{p}=\\{{\rm
Id},\tau_{p}\\}\cong{\mathbb{Z}}_{2}$ and:

    1. (i)

If $H=H_{0}\subset T$ then $A=H_{0}\cup H_{0}\tau_{p}=H_{0}\rtimes G_{p}$.

    2. (ii)

If $H=H_{0}\rtimes G_{p}=H_{0}\cup H_{0}\tau_{p}$, then $A=H$.

    3. (iii)

If $H=H_{0}\rtimes\\{{\rm Id},t_{0}\tau_{p}\\}=H_{0}\cup H_{0}t_{0}\tau_{p}$,
with $t_{0}\notin H_{0}$, then $A=H$ unless $t_{0}^{2}\in H_{0}$, in which
case

$A=\bigl{\langle}H_{0}\cup\\{t_{0}\\}\bigr{\rangle}\cup\bigr{\langle}H_{0}\cup\\{t_{0}\\}\bigr{\rangle}t_{0}\tau_{p}=\bigl{\langle}H_{0}\cup\\{t_{0}\\}\bigr{\rangle}\rtimes\\{{\rm
Id},t_{0}\tau_{p}\\};$

in particular, $A_{0}=\bigl{\langle}H_{0}\cup\\{t_{0}\\}\bigr{\rangle}$ and
$A_{0}/H_{0}\cong{\mathbb{Z}}_{2}$.

  3. (c)

If $\mathcal{C}$ is a curve of type (6), then

$A=H(G_{p}\cap A).$

In particular, $H\operatorname{Ker(R)}\subset A$.

###### Proof.

In what follows, we keep the notations of Theorem 2.8 and assume (without loss
of generality) that $\mathcal{C}$ is given by its canonical form. It is clear
that $H\subset A\subset G$ — the last inclusion follows by continuity of the
action; then $\mathcal{C}=\overline{{\rm O}_{H}(p)}=\overline{{\rm
O}_{A}(p)}=\overline{{\rm O}_{G}(p)}$.

  1. (a)

If $\mathcal{C}=\overline{{\rm O}_{H}(p)}$ is of type (1) or (2), it easily
follows that $G_{p}=\\{{\rm Id}\\}$. Thus, if $\varphi\in A$ then
$\varphi(p)\in{\rm O}_{H}(p)$, and therefore there exists $h\in H$ such that
$h\varphi\in G_{p}$. It follows that $A=H$.

  2. (b)

Let $G=T\rtimes\\{Id,\tau_{p}\\}$; since $H\subset A\subset G$, it follows
from Remark 3.3 that both $H$ and $A$ are determined by their intersections
with $T$ and the involutions $t_{0}\tau_{p}$, with $t_{0}\in T$, belonging to
the group. Hence, we may distinguish three cases:

    1. (i)

$H=H_{0}$. If $s\in A_{0}$ then $sp\in{\rm O}_{H}(p)$ and therefore $sp=hp$
for some $h\in H$, so $s=h\in H_{0}$. On the other hand
$\tau_{p}(hp)=h^{-1}\tau_{p}p=h^{-1}p\in{\rm O}_{H}(p)$ for any $h\in H$. It
follows that $A=H\rtimes\\{{\rm Id},\tau_{p}\\}$.

    2. (ii)

$H=H_{0}\cup H_{0}\tau_{p}=H_{0}\rtimes G_{p}$. If $s\in A_{0}$ then $sp=hp$
for some $h\in H$. If $h\in H_{0}$ we deduce that $s=h\in H_{0}$. If
$h=h_{0}\tau_{p}$, then $hp=h_{0}p$ and it follows that $s\in H_{0}$. Hence
$A=H$.

    3. (iii)

$H=H_{0}\cup H_{0}t_{0}\tau_{p}=H_{0}\rtimes\\{{\rm Id},t_{0}\tau_{p}\\}$,
where $t_{0}\notin H_{0}$.

If $s\in A_{0}$, then $sp=hp$ for some $h\in H$. If $h\in H_{0}$ we deduce
that $s=h\in H_{0}$. If $h=h_{0}t_{0}\tau_{p}$, then $hp=h_{0}t_{0}p$ and it
follows that $s=h_{0}t_{0}$. Hence, $s\in A_{0}$ if and only if $t_{0}\in
A_{0}$ (because $h_{0}\in H_{0}$), and it remains to determine under which
conditions we have that $t_{0}\in A_{0}$.

Consider $h_{1}\in H_{0}$; then
$t_{0}\cdot(h_{1}p)=h_{1}t_{0}p=h_{1}t_{0}\tau_{p}(p)\in{\rm O}_{H}(p)$ (since
$t_{0}\tau_{p}\in H$) and
$t_{0}\cdot\bigl{(}h_{1}t_{0}\tau_{p}(p)\bigr{)}=h_{1}t_{0}^{2}\tau_{p}(p)=h_{1}t_{0}^{2}p$.
Notice that $h_{1}t_{0}^{2}p\in{\rm O}_{H}(p)$ if and only if either
$h_{1}t_{0}^{2}p\in{\rm O}_{H_{0}}(p)$, or there exists $h_{2}\in H_{0}$ such
that $h_{1}t_{0}^{2}p=h_{2}t_{0}\tau_{p}(p)=h_{2}t_{0}p$. In the first case it
follows that $t_{0}^{2}\in H_{0}$, and in the last case it follows that
$h_{1}t_{0}^{2}=h_{2}t_{0}$, and therefore $t_{0}\in H_{0}$. Since
$t_{0}h_{1}t_{0}\tau_{p}(p)=h_{1}t_{0}^{2}p$, we conclude that $t_{0}\in
A_{0}$ if and only if $t_{0}^{2}\in H_{0}$.

  3. (c)

If $\mathcal{C}$ is the line of equation $x=0$ and
$\varphi=\bigl{(}ax,by+P(x)\bigr{)}\in A$, then
$\bigr{(}0,by_{p}+P(0)\bigr{)}=h\cdot p$ for some $h\in H$. It follows that
$h^{-1}\varphi\in G_{p}$ and therefore $\varphi\in HG_{p}$; the result
follows.

∎

If $\overline{{\rm O}_{H}(p)}$ is an irreducible curve of type (6), the
inclusion $H\operatorname{Ker}(R)\subset A$ may be strict, as the following
example shows.

######  Example 3.5.

Assume that $\operatorname{char}({\Bbbk})=0$ and let $\varphi=(x,y+c)$. If
$H={\langle\varphi\rangle}=\\{\varphi^{n}\mathrel{:}n\in{\mathbb{Z}}\\}$, then
$\psi=(ax,ny)$ belongs to ${\rm Aut}\bigl{(}{\mathbb{A}}^{2},{\rm
O}_{H}(0,0)\bigr{)}$ for all $n\in{\mathbb{Z}}$ – indeed
$\varphi^{n}(0,0)=(0,nc)$ for all $n\in{\mathbb{Z}}$.

######  Corollary 3.6.

Let $H\subset{\rm Aut}({\mathbb{A}}^{2})$ be an infinite countable subgroup of
plane automorphisms and $p\in{\mathbb{A}}^{2}(\overline{{\mathbb{k}}})$ a
point such that $\mathcal{C}=\overline{{\rm O}_{H}(p)}$ is an irreducible
curve. Then the stabilizer of ${\rm O}_{H}(p)$ is not an algebraic group.

###### Proof.

It follows from Theorem 3.4 that if $\mathcal{C}$ is of type (1)–(5) then $A$
is infinite countable and therefore is not algebraic. If $\mathcal{C}$ is of
type (6) then $A$ contains $\operatorname{Ker}(R)$ and therefore $A$ contains
automorphisms of arbitrary degree; it follows from [Ka1979, Theorem 1.3] that
$A$ is not algebraic. ∎

##  4\. The orbit of an automorphism of the plane

Let $\varphi\in{\rm Aut}({\mathbb{A}}^{2})$ and
$p\in{\mathbb{A}}^{2}(\overline{{\mathbb{k}}})$. In this section, as an
application of Theorem 3.4, we study the geometry of the _orbit of $p$ by
$\varphi$_. More precisely, we consider $H=\langle\varphi\rangle$ and describe
$\mathcal{C}=\overline{{\rm O}_{H}(p)}$, ${\rm
Aut}({\mathbb{A}}^{2},\mathcal{C})$ and ${\rm
Aut}\bigr{(}{\mathbb{A}}^{2},{\rm O}_{H}(p)\bigr{)}$. Along this section, we
keep the previous notations and set ${\rm O}_{\varphi}(p)={\rm O}_{H}(p)$.

As Lemma 3.1 indicates, the closure of the orbit ${\rm O}_{\varphi}(p)$ is
either finite, a curve or the whole plane. We begin by showing that under mild
conditions, there exist pairs $p\in{\mathbb{A}}^{2}(\overline{{\mathbb{k}}})$
and $\varphi\in{\rm Aut}({\mathbb{A}}^{2})$ such that ${\rm O}_{\varphi}(p)$
is infinite and $\mathcal{C}=\overline{{\rm O}_{\varphi}(p)}$ is of type
(1)–(6) — in particular, ${\rm Aut}\bigr{(}{\mathbb{A}}^{2},{\rm
O}_{H}(p)\bigr{)}$ is not algebraic (see Corollary 3.6).

######  Example 4.1.

  1. (a)

Assume that ${\mathbb{k}}$ is such that it contains an element
$t_{0}\in{\mathbb{k}}$ that is not a root of unity. Then:

    1. (i)

If $p=(x_{p},y_{p})$, $x_{p}y_{p}\neq 0$ and
$\varphi(x,y)=(t_{0}^{a}x,t_{0}^{b}y)$, with $a,b>1$ coprimes, then
$\mathcal{C}=\mathcal{V}(x^{b}-\frac{x_{p}^{b}}{y_{p}^{a}}y^{a})$.

    2. (ii)

If $p=(x_{p},y_{p})$, $x_{p}y_{p}\neq 0$ and
$\varphi(x,y)=(t_{0}^{a}x,t_{0}^{-b}y)$, with $a,b\geq 1$ coprimes, then
$\mathcal{C}=\mathcal{V}(x^{b}y^{a}-x_{p}^{b}y_{p}^{a})$. Notice that if
$ab\neq 1$ then $\mathcal{C}$ is of type (2), whereas if $a=b=1$, then
$\mathcal{C}$ is of type (3).

    3. (iii)

If $p=(x_{p},y_{p})\neq(0,0)$, and
$\varphi(x,y)=t_{0}\cdot(x,y)=(t_{0}x,t_{0}y)$, then $\mathcal{C}$ is a line.

  2. (b)

Assume now that ${\Bbbk}$ is uncountable.

    1. (i)

If ${\rm char}({\mathbb{k}})\neq 2$ and there exist
$\lambda,\nu\in{\mathbb{k}}^{*}$ are such that $-\lambda\nu$ is not a square
in ${\mathbb{k}}$, consider
$T_{\lambda,\nu}=\bigl{\\{}\left(\begin{smallmatrix}a&-\nu b\\\ \lambda
b&a\end{smallmatrix}\right)\mathrel{:}(a,b)\in{\mathbb{k}}\times{\mathbb{k}}\,,\
a^{2}+\lambda\nu b^{2}=1\bigr{\\}}$. If
$\varphi=\left(\begin{smallmatrix}a&-\nu b\\\ \lambda
b&a\end{smallmatrix}\right)\in T_{\lambda,\nu}$, then $\varphi$ has
eigenvalues $a\pm\sqrt{-\lambda\mu}b$. Since ${\Bbbk}$ is uncountable, there
exists $(a,b)\in{\Bbbk}^{2}$ such that $a\pm\sqrt{-\lambda\mu}b$ is not a root
of unity, and therefore $(0,0)$ is the unique periodic point of $\varphi$. It
follows that if $p=(x_{p},y_{p})\in\mathcal{C}=\mathcal{V}(\lambda x^{2}+\nu
y^{2}-1)$, then ${\rm O}_{\varphi}(p)$ is infinite, with $\overline{{\rm
O}_{\varphi}(p)}=\mathcal{C}$; in particular, $\overline{{\rm
O}_{\varphi}(p)}$ is of type (4).

    2. (ii)

Analogously, if ${\rm char}({\mathbb{k}})=2$ and $\mu\in\Bbbk^{*}$ is such
that $x^{2}+\mu x+1$ has no root in ${\mathbb{k}}$, one can prove that there
exists $\varphi\in\left\\{\left(\begin{smallmatrix}a&b\\\ b&a+\mu
b\end{smallmatrix}\right)\mathrel{:}a,b\in{\mathbb{k}}\,,a^{2}+\mu
ab+b^{2}=1\right\\}$ such that $(0,0)$ is the unique periodic point of
$\varphi$, and therefore if $p\in\mathcal{C}=\mathcal{V}(x^{2}+\mu
xy+y^{2}-1)$ then ${\rm O}_{\varphi}(p)$ is infinite, with $\overline{{\rm
O}_{\varphi}(p)}=\mathcal{C}$; in particular, $\overline{{\rm
O}_{\varphi}(p)}$ is of type (5).

If $\varphi\in{\rm Aut}({\mathbb{A}}^{2})$, let $\mathcal{S}$ be the family of
$\varphi$-stable curves and $\mathcal{P}$ the set of periodic points of
$\varphi$. It is clear that there exists
$p\in{\mathbb{A}}^{2}(\overline{{\mathbb{k}}})$ such that $\overline{{\rm
O}_{\varphi}(p)}={\mathbb{A}}^{2}$ if and only if
$\mathcal{S}\cup\mathcal{P}\neq{\mathbb{A}}^{2}$. This simple remark allows us
to show that there exist automorphisms of the plane with orbits which closure
is all the plane as follows.

######  Example 4.2.

Let $\varphi\in{\rm Aut}({\mathbb{A}}^{2})$; for $n\geq 1$, we denote by
$F_{n}$ the (closed) subset of fixed points of $\varphi^{n}$. Clearly,
$\mathcal{P}=\bigcup F_{n}$.

Recall that the (first) _dynamical degree_ of $\varphi$ is defined as
$\lambda_{1}(\varphi)=\lim_{m\to\infty}\sqrt[m]{\deg\varphi^{m}}$ — notice
that $\lambda_{1}(\varphi)>1$ implies $\lambda_{1}(\varphi^{n})>1$ for any
$n\geq 1$; for more details see [DiFa2001], where the complex case is treated.
It follows from [Ca2010, Thm. B] and [Xi2015, Thm. 6.5] that if
$\lambda_{1}(\varphi)>1$, then $\mathcal{S}$ is a curve and $F_{n}$ is finite.
Therefore $\mathcal{S}\cup\mathcal{P}$ is contained in a countable union of
curves. Hence, if $\overline{{\mathbb{k}}}$ is uncountable and
$\lambda_{1}(\varphi)>1$, there exists
$p\in{\mathbb{A}}^{2}(\overline{{\mathbb{k}}})$ such that $\overline{{\rm
O}_{\varphi}(p)}={\mathbb{A}}^{2}$.

One particular example of this situation concerns the so-called Hénon maps,
that is morphisms of the form $\varphi=\bigl{(}y,-\delta x+P(y)\bigr{)}$,
where $\delta\in{\Bbbk}^{*}$ and $P\in{\Bbbk}[y]$ is a polynomial of degree
greater or equal that $2$, since in this case $\lambda_{1}(\varphi)=\deg P$.

_A priori_ , the closure of an orbit ${\rm O}_{\varphi}(p)$ is not necessarily
an irreducible curve. In what follows, we exploit Lemma 2.2 in order to
describe $\widehat{{\rm O}_{\varphi}(p)}$. If ${\rm O}_{\varphi}(p)$ is finite
then $\widehat{{\rm
O}_{\varphi}(p)}=\operatorname{Gal}(\overline{{\mathbb{k}}}/{\mathbb{k}})\cdot{\rm
O}_{\varphi}(p)$. On the other hand, if $\overline{{\rm
O}_{\varphi}(p)}={\mathbb{A}}^{2}(\overline{{\mathbb{k}}})$, then
$\widehat{{\rm O}_{\varphi}(p)}$ is the whole plane. Next proposition deal
with the case when $\overline{{\rm O}_{\varphi}(p)}$ is a curve.

######  Proposition 4.3.

Let $\varphi\in{\rm Aut}({\mathbb{A}}^{2})$ and
$p\in{\mathbb{A}}^{2}_{\overline{{\mathbb{k}}}}$ be such that $\overline{{\rm
O}_{\varphi}(p)}=\mathcal{C}$ is a curve, and let
$\mathcal{C}=\bigcup_{i=1}^{\ell}\mathcal{C}_{i}$ and $\widehat{{\rm
O}_{\varphi}(p)}=\bigcup_{i=1}^{s}\mathcal{C}_{i}$ be the decomposition in
irreducible components of $\mathcal{C}$ and $\widehat{{\rm O}_{\varphi}(p)}$
respectively (see Lemma 2.2). Then $s=k\ell$ and, up to reordering the
indexes, we have that:

  1. (1)

The morphism $\varphi$ restricts to isomorphisms
$\varphi|_{{}_{\mathcal{C}_{i+j\ell}}}:\mathcal{C}_{i+j\ell}\to\mathcal{C}_{i+j\ell+1}$
and
$\varphi|_{{}_{\mathcal{C}_{j\ell+\ell}}}:\mathcal{C}_{j\ell+\ell}\to\mathcal{C}_{j\ell+1}$,
for $i=1,\ldots,\ell-1$, $j=0,\dots,k-1$.

  2. (2)

$\mathcal{C}_{i+j\ell}=\overline{{\rm
O}_{\varphi^{\ell}}\bigl{(}\varphi^{i+j\ell-1}(p)\bigr{)}}$ for
$i=1,\ldots,\ell$, $j=0,\dots,k-1$.

  3. (3)

The curves $\mathcal{C}_{i}$ are isomorphic — in particular, the curves have
the same type.

###### Proof.

Without loss of generality we assume that $p\in\mathcal{C}_{1}$. We reorder
$\mathcal{C}_{1},\dots,\mathcal{C}_{\ell}$ in such a way that
$\mathcal{C}_{i}=\varphi^{i-1}(\mathcal{C}_{1})$ for $i=1,\dots,\ell$ (see
Lemma 3.1); then $\mathcal{C}_{1}=\varphi(\mathcal{C}_{\ell})$. It follows
from Galois descent that

$\widehat{{\rm
O}_{\varphi}(p)}=\operatorname{Gal}(\overline{{\mathbb{k}}}/{\mathbb{k}})\cdot\mathcal{C}=\bigcup_{i=1}^{\ell}\operatorname{Gal}(\overline{{\mathbb{k}}}/{\mathbb{k}})\cdot\mathcal{C}_{i},$

where if $i\neq j$, then
$\operatorname{Gal}(\overline{{\mathbb{k}}}/{\mathbb{k}})\cdot\mathcal{C}_{i}\cap\operatorname{Gal}(\overline{{\mathbb{k}}}/{\mathbb{k}})\cdot\mathcal{C}_{j}$
does not contain a curve. Since $\varphi$ is fixed by the Galois action, it
follows from Galois descent that if $i=1,\dots,\ell-1$, then
$\varphi|_{\mathcal{C}_{i}}:\mathcal{C}_{i}\to\mathcal{C}_{i+1}$ induces an
${\mathbb{k}}$-isomorphism
$\operatorname{Gal}(\overline{{\mathbb{k}}}/{\mathbb{k}})\cdot\mathcal{C}_{i}\to\operatorname{Gal}(\overline{{\mathbb{k}}}/{\mathbb{k}})\cdot\mathcal{C}_{i+1}$.
Assertion (1) follows; assertion (2) is clear.

In order to prove assertion (3), we first observe that
$\mathcal{C}_{j\ell+1},\dots,\mathcal{C}_{j\ell+\ell}$ being isomorphic for
$j=0,\dots k-1$, they have the same type. Up to reordering, we can assume that
there exists
$g_{j}\in\operatorname{Gal}(\overline{{\mathbb{k}}}/{\mathbb{k}})$,
$j=1,\dots,k-1$, such that $\mathcal{C}_{i+j\ell}=g_{j}\cdot\mathcal{C}_{i}$.
Since $\psi\in{\rm Aut}({\mathbb{A}}^{2},g_{j}\mathcal{C}_{1})$ if only if
$\psi=g^{-1}\psi g\in{\rm Aut}({\mathbb{A}}^{2},\mathcal{C}_{1})$, the result
follows. ∎

######  Example 4.4.

Suppose that $\Delta\subset{\mathbb{A}}^{2}(\overline{{\mathbb{k}}})$ is such
that $\widehat{\Delta}$ is a fence of equation $F(x)=0$, with
$F\in{\mathbb{k}}[x]$ — then it follows from [BlSt2015, Theorem 1] that

${\rm
Aut}\bigl{(}{\mathbb{A}}^{2},\widehat{\Delta}\bigr{)}=\bigl{\\{}\bigl{(}\alpha
x+\beta,\gamma y+P(x)\bigr{)}\mathrel{:}\alpha,\beta,\gamma\in{\mathbb{k}}\,,\
F(\alpha x+\beta)/F(x)\in{\mathbb{k}}^{*}\,,\ P\in{\mathbb{k}}[x]\bigr{\\}}.$

  1. (a)

If $F=x-1$, then $\varphi=\bigl{(}\alpha x+\beta,\gamma y+P(x)\bigr{)}\in{\rm
Aut}\bigl{(}{\mathbb{A}}^{2},\widehat{\Delta})$ if and only if $\alpha=1$ and
$\beta=0$. Choose $\gamma\in{\mathbb{k}}^{*}$ which is not a root of the unity
and consider $p=(1,1)\in{\mathbb{A}}^{2}(\overline{{\mathbb{k}}})$. Let
$P\in{\mathbb{k}}[x]$ be such that $P(1)=1$. Then $\varphi^{n}(p)\neq p$ for
all $n$ and ${\rm
O}_{\varphi}(p)=\bigl{\\{}(1,\sum_{i=0}^{n}\gamma^{i});n\in{\mathbb{Z}}\bigr{\\}}$.
It follows that $\overline{{\rm O}_{\varphi}(p)}=\widehat{{\rm
O}_{\varphi}(p)}_{\overline{{\mathbb{k}}}}=\widehat{\Delta}$. Thus, in this
case $\widehat{{\rm O}_{\varphi}(p)}_{\overline{{\mathbb{k}}}}$ is an
irreducible curve, stable by $\varphi$ — in the notations of Proposition 4.3,
$\ell=s=1$. Recall that in this case the group ${\rm
Aut}\bigl{(}{\mathbb{A}}^{2},{\rm O}_{\varphi}(p)\bigr{)}$ is not algebraic.

  2. (b)

If $F=x^{2}-1$ , then $\varphi=\bigl{(}\alpha x+\beta,\gamma
y+P(x)\bigr{)}\in{\rm Aut}\bigl{(}{\mathbb{A}}^{2},\widehat{\Delta}\bigr{)}$
if and only if $\alpha=\pm 1$ and $\beta=0$. Choose
$\gamma\in{\mathbb{k}}^{*}$ which is not a root of the unity, $d\geq 1$, and
consider the automorphism $\varphi_{d}=\bigl{(}-x,\gamma
y+(x^{2}-1)^{d}\bigr{)}$. If $p=(1,y_{0})$, with $y_{0}\neq 0$, then ${\rm
O}_{\varphi_{d}}(p)=\bigl{\\{}\bigl{(}(-1)^{n},\gamma^{n}y_{0}\bigr{)};n\in{\mathbb{Z}}\bigr{\\}}$
is dense in the fence $F(x)=0$. It follows that $\widehat{{\rm
O}_{\varphi}(p)}_{\overline{{\mathbb{k}}}}=\overline{{\rm
O}_{\varphi}(p)}=\widehat{\Delta}$, and therefore it is a curve with two
irreducible components. Moreover, in the notations of Proposition 4.3,
$\ell=s=2$ and once again ${\rm Aut}\bigl{(}{\mathbb{A}}^{2},{\rm
O}_{\varphi}(p)\bigr{)}$ is not algebraic because the orbit does not depend on
$d=\deg\varphi_{d}$.

The following example, very similar to the ones given above shows that is not
sufficient for $\Delta$ to be an orbit in order to guarantee
$\overline{\Delta}$ and $\widehat{\Delta}$ to be equals.

######  Example 4.5.

Assume that ${\Bbbk}\neq\overline{{\mathbb{k}}}$ and let
$\xi\in\overline{{\mathbb{k}}}\setminus{\Bbbk}$. Denote by $P\in{\Bbbk}[x]$
the minimal polynomial of $\xi$ over ${\Bbbk}$ and consider
$\varphi=\bigl{(}x,\gamma y+P(x)\bigr{)}\in{\rm Aut}({\mathbb{A}}^{2})$, where
$\gamma\in{\mathbb{k}}^{*}$ is not a root of unity. If $p=(\xi,1)$ , then
$\Delta={\rm
O}_{\varphi}(p)=\bigl{\\{}(\xi,\gamma^{n});n\in{\mathbb{Z}}\bigr{\\}}$ is
infinite. Clearly
$\overline{\Delta}=\mathcal{V}_{\overline{{\mathbb{k}}}}(x-\xi)$ and
$\widehat{\Delta}=\mathcal{V}_{\overline{{\mathbb{k}}}}(P)$ so
$\Delta\subsetneq\overline{\Delta}\subsetneq\widehat{\Delta}$.

Notice that the automorphisms $(x,y)\mapsto\bigl{(}x,y+P(x)^{d}\bigr{)}$,
$d\geq 1$, fix every point in $\Delta$ and therefore they belong to ${\rm
Aut}({\mathbb{A}}^{2},\Delta)$.

######  Remark 4.6.

Let $\varphi\in{\rm Aut}({\mathbb{A}}^{2})$ and
$p=(x_{p},y_{p})\in{\mathbb{A}}^{2}\bigl{(}\overline{{\mathbb{k}}}\bigr{)}$ be
such that $\overline{{\rm
O}_{\varphi}(p)}=\mathcal{C}_{1}\cup\dots\cup\mathcal{C}_{\ell}$ is a curve.
Then, by Proposition 4.3, up to reordering we can assume that
$\mathcal{C}_{i}=\varphi^{i-1}\mathcal{C}_{1}$, for $i=0,\dots,\ell-1$, with
$\varphi^{\ell}(\mathcal{C}_{1})=\mathcal{C}_{1}$. Therefore

$\mathcal{C}_{1}=\overline{{\rm O}_{\varphi^{\ell}}(p)}.$

Moreover, each curve $\mathcal{C}_{i}$ is of the same type and, up to
conjugation by an automorphism of ${\mathbb{A}}^{2}$, we can assume that
$\mathcal{C}_{1}$ is one of the list given in Theorem 2.8.

Since any automorphism $\psi\in{\rm
Aut}\bigl{(}{\mathbb{A}}^{2},\overline{{\rm O}_{\varphi}(p)}\bigr{)}$ induces
a permutation of the irreducible curves $\mathcal{C}_{i}$, we have the
following exact sequence of abstract groups:

$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bigcap_{i=1}^{\ell}{\rm
Aut}({\mathbb{A}}^{2},\mathcal{C}_{i})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
Aut}\bigl{(}{\mathbb{A}}^{2},\overline{{\rm
O}_{\varphi}(p)}\bigr{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{S}_{\ell}}$

where $\mathfrak{S}_{\ell}$ is group of permutations of $\ell$ elements. Since
$\varphi^{n\ell}\in{\rm Aut}\bigl{(}{\mathbb{A}}^{2},\overline{{\rm
O}_{\varphi^{\ell}}(\varphi^{i}(p))}\bigr{)}$, if follows that $\bigcap{\rm
Aut}({\mathbb{A}}^{2},\mathcal{C}_{i})$ is an infinite subgroup of ${\rm
Aut}({\mathbb{A}}^{2},\mathcal{C}_{1})$.

Moreover, ${\rm Aut}({\mathbb{A}}^{2},\mathcal{C}_{i})=\varphi^{i-1}{\rm
Aut}({\mathbb{A}}^{2},\mathcal{C}_{1})\varphi^{1-i}$ and if $\psi\in{\rm
Aut}\bigl{(}{\mathbb{A}}^{2},\overline{{\rm O}_{\varphi}(p)}\bigr{)}$, then
there exists $i\in\\{0,\dots\ell-1\\}$ such that $\varphi^{i}\psi\in{\rm
Aut}({\mathbb{A}}^{2},\mathcal{C}_{1})$. It follows that
$\psi\in\varphi^{-i}{\rm Aut}({\mathbb{A}}^{2},\mathcal{C}_{1})$.

######  Notation 4.7.

If $G$ is a group and $A,B\subset G$ arbitrary subsets, we denote
$AB=\\{ab\mathrel{:}a\in A\,,\ b\in B\\}$.

######  Theorem 4.8.

Let $\varphi\in{\rm Aut}({\mathbb{A}}^{2})$ and
$p=(x_{p},y_{p})\in{\mathbb{A}}^{2}\bigl{(}\overline{{\mathbb{k}}}\bigr{)}$ be
such that $\overline{{\rm
O}_{\varphi}(p)}=\mathcal{C}_{1}\cup\dots\cup\mathcal{C}_{\ell}$ is a curve.
Then up to applying an automorphism of ${\mathbb{A}}^{2}$ one of the following
assertions holds:

  1. (a)

$\mathcal{C}_{1}$ is of type (1) or (2) and $p\neq(0,0)$. In this case, ${\rm
Aut}\bigl{(}{\mathbb{A}}^{2},{\rm
O}_{\varphi}(p)\bigr{)}=\langle\varphi\rangle$.

  2. (b)

$\mathcal{C}_{1}$ is of type (3), (4) or (5). If we denote ${\rm
Aut}({\mathbb{A}}^{2},\mathcal{C}_{1})_{p}=\\{{\rm Id},\tau_{p}\\}$ (see Lemma
3.2), we have two possibilities:

    1. (i)

If $\tau_{p}\varphi=\varphi^{i}\tau_{p}$ for some $i\in{\mathbb{Z}}$, then

${\rm Aut}\bigl{(}{\mathbb{A}}^{2},{\rm
O}_{\varphi}(p)\bigr{)}=\langle\varphi\rangle\rtimes\\{{\rm
Id},\tau_{p}\\}=\langle\varphi\rangle\cup\langle\varphi\rangle\tau_{p}$

    2. (ii)

In other case, ${\rm Aut}\bigl{(}{\mathbb{A}}^{2},{\rm
O}_{\varphi}(p)\bigr{)}=\langle\varphi\rangle$.

  3. (c)

$\mathcal{C}_{1}$ is of type (6). In this case

$\begin{split}{\rm Aut}\bigl{(}{\mathbb{A}}^{2},{\rm
O}_{\varphi}(p)\bigr{)}=&\
\bigcap_{i=0}^{\ell-1}\langle\varphi\rangle\Bigl{(}\varphi^{i}\Bigl{(}G_{p}\cap{\rm
Aut}\bigl{(}{\mathbb{A}}^{2},{\rm
O}_{\varphi^{\ell}}(p)\bigr{)}\Bigr{)}\varphi^{-i}\Bigr{)}\\\ =&\
\bigcap_{i=0}^{\ell-1}\langle\varphi\rangle\Bigl{(}\Bigl{(}G_{p}\cap{\rm
Aut}\bigl{(}{\mathbb{A}}^{2},{\rm
O}_{\varphi^{\ell}}(p)\bigr{)}\Bigr{)}\varphi^{-i}\Bigr{)}\\\ =&\
\bigcap_{i=0}^{\ell-1}\langle\varphi\rangle\Bigl{(}{\rm
Aut}\bigl{(}{\mathbb{A}}^{2},{\rm
O}_{\varphi^{\ell}}(p)\bigr{)}\varphi^{-i}\Bigr{)},\end{split}$

where $G={\rm
Aut}\bigl{(}{\mathbb{A}}^{2},\mathcal{C}_{1}(\overline{{\mathbb{k}}})\bigr{)}$.

Moreover, if ${\mathbb{k}}$ is uncountable, there exist infinitely many
automorphisms $\varphi\in G$ such that

${\rm Aut}\bigl{(}{\mathbb{A}}^{2},{\rm
O}_{\varphi}(p)\bigr{)}=\bigcap_{i=0}^{\ell-1}\langle\varphi\rangle\bigl{(}\varphi^{i}\operatorname{Ker}(R)\varphi^{-i}\bigr{)}=\bigcap_{i=0}^{\ell-1}\langle\varphi\rangle\bigl{(}\operatorname{Ker}(R)\varphi^{-i}\bigr{)},$

where $R:{\rm
Aut}\bigr{(}{\mathbb{A}}^{2},\mathcal{C}_{1}(\overline{{\mathbb{k}}})\bigr{)}\to{\rm
Aut}\bigl{(}\mathcal{C}_{1}(\overline{{\mathbb{k}}})\bigr{)}$ is the
restriction morphism (see 2.8).

###### Proof.

By Proposition 4.3, we have that $\\{\varphi^{n}\\}_{n\in{\mathbb{Z}}}$ is
infinite and $\mathcal{C}_{1}$ is of type (1) – (6). Hence, it remains to
describe ${\rm Aut}\bigl{(}{\mathbb{A}}^{2},{\rm O}_{\varphi}(p)\bigr{)}$ in
each case. We assume that $\mathcal{C}_{i}=\varphi^{i-1}(\mathcal{C}_{1})$,
and $\varphi^{\ell}\in{\rm Aut}({\mathbb{A}}^{2},\mathcal{C}_{1})$ (see Remark
4.6).

  1. (a)

If $\mathcal{C}_{1}=\mathcal{V}(x^{b}-y^{a})$ then, since
$\varphi^{\ell}\in{\rm Aut}({\mathbb{A}}^{2},\mathcal{C}_{1})$, we have that
$\varphi^{\ell}(0,0)=(0,0)$ and therefore $p\neq(0,0)$. If $\psi\in{\rm
Aut}\bigl{(}{\mathbb{A}}^{2},{\rm O}_{\varphi}(p)\bigr{)}$, then by Remark 4.6
$\psi=\varphi^{j}t$ for some $t\in{\rm Aut}({\mathbb{A}}^{2},\mathcal{C}_{1})$
and some $j\in{\mathbb{Z}}$. But if $n\in{\mathbb{Z}}$, then there exists
$s_{n}$ such that $\varphi^{j}t\varphi^{n\ell}(p)=\varphi^{s_{n}}(p)$. If
follows that $\varphi^{s_{n}-j}(p)=t\varphi^{n\ell}(p)\in\mathcal{C}_{1}$, and
therefore $s_{n}-j$ is a multiple of $\ell$ and $t\in{\rm
Aut}({\mathbb{A}}^{2},{\rm O}_{\varphi^{\ell}}(p))$. We deduce from Theorem
3.4 that $t\in\langle\varphi^{\ell}\rangle$, and therefore ${\rm
Aut}\bigl{(}{\mathbb{A}}^{2},{\rm
O}_{\varphi}(p)\bigr{)}=\langle\varphi\rangle$.

If $\mathcal{C}_{1}$ is of type (2) an analogous argument applies.

  2. (b)

Since $\overline{{\rm O}_{\varphi^{\ell}}(p)}=\mathcal{C}_{1}$ and
$\varphi^{\ell}\in{\rm Aut}({\mathbb{A}}^{2},\mathcal{C}_{1})$ then, in the
notations of Theorem 3.4, there exists
$t_{0}\in{\Bbbk}^{*}\subset{\Bbbk}^{*}\rtimes\\{{\rm Id},\tau_{p}\\}={\rm
Aut}({\mathbb{A}}^{2},\mathcal{C}_{1})$ such that $\varphi^{\ell}=t_{0}$. As
in the previous case, if $\psi\in{\rm Aut}\bigl{(}{\mathbb{A}}^{2},{\rm
O}_{\varphi}(p)\bigr{)}$ we deduce that there exist $i\in{\mathbb{Z}}$ and
$h_{0}\in{\rm Aut}\bigl{(}{\mathbb{A}}^{2},{\rm
O}_{\varphi^{\ell}}(p)\bigr{)}$ such that $\psi=\varphi^{i}h_{0}$. Applying
Theorem 3.4 to $H=\langle\varphi^{\ell}\rangle$, we deduce that
$h_{0}\in\langle\varphi^{\ell}\rangle\rtimes\\{{\rm Id},\tau_{p}\\}$.

If there exists $\psi\in{\rm Aut}\bigl{(}{\mathbb{A}}^{2},{\rm
O}_{\varphi}(p)\bigr{)}\setminus\langle\varphi\rangle$, then
$h_{0}\notin\langle\varphi^{\ell}\rangle$ and there exists $n\in{\mathbb{Z}}$
such that $h_{0}=\varphi^{n\ell}\tau_{p}\in{\rm
Aut}\bigl{(}{\mathbb{A}}^{2},{\rm O}_{\varphi}(p)\bigr{)}$ — because
$\varphi\in{\rm Aut}\bigl{(}{\mathbb{A}}^{2},{\rm O}_{\varphi}(p)\bigr{)}$ —,
and it follows that $\tau_{p}\in{\rm Aut}\bigl{(}{\mathbb{A}}^{2},{\rm
O}_{\varphi}(p)\bigr{)}$.

If $j\in{\mathbb{Z}}$, then
$\tau_{p}\bigl{(}\varphi^{j}(p)\bigr{)}=\varphi^{i_{j}}(p)$ if and only if
$\varphi^{i_{j}}\tau_{p}\varphi^{j}\in{\rm
Aut}({\mathbb{A}}^{2},\mathcal{C}_{1})_{p}$. Taking $i=-1$ we deduce that
there exists $i\in{\mathbb{Z}}$ such that
$\tau_{p}\varphi=\varphi^{i}\tau_{p}$ for some $i\in{\mathbb{Z}}$ and the
assertion is proved.

  3. (c)

Assume now that $\mathcal{C}_{1}=\mathcal{V}(x)$. Since
$\mathcal{C}_{1}\cap\mathcal{C}_{i}$ is finite and $\varphi^{\ell}$-stable for
all $i=2,\dots,\ell$, it follows that
$p\notin\mathcal{C}_{1}\cap\mathcal{C}_{i}$, because ${\rm
O}_{\varphi^{\ell}}(p)$ is infinite — more in general,
$\mathcal{C}_{i}\cap\mathcal{C}_{j}\cap{\rm O}_{\varphi}(p)=\emptyset$.
Moreover, we deduce from Theorem 2.8 that
$\varphi^{\ell}=\bigl{(}ax,by+P(x)\bigr{)}$, where $a,b\in{\Bbbk}^{*}$ and
$P\in{\Bbbk}[x]$ — notice that $b$ is not a root of unity different from $1$,
since $\varphi^{n\ell}(p)=\bigr{(}0,b^{n}y_{p}+(1+\dots+b^{n-1})P(0)\bigr{)}$,
and that if $\operatorname{char}({\Bbbk})\neq 0$, then also $b\neq 1$.

If $\psi\in{\rm Aut}\bigl{(}{\mathbb{A}}^{2},{\rm O}_{\varphi}(p)\bigr{)}$,
then $\psi(p)=\varphi^{i}(p)$ for some $i$, and therefore
$\varphi^{-i}\psi(p)=p\in\mathcal{C}_{1}$. Since
$\varphi^{-i}\psi(\mathcal{C})=\mathcal{C}$, it follows that
$\varphi^{-i}\psi(\mathcal{C}_{1})=\mathcal{C}_{1}$. Hence, in the notations
of Theorem 2.8 and Lemma 3.2,

$\begin{split}\varphi^{-i}\psi\in G_{p}=&\
\Bigl{\\{}\bigl{(}a^{\prime}x,cy+S(x)\bigr{)}\mathrel{:}a^{\prime},c\in{\mathbb{k}}^{*}\,,\
S\in{\mathbb{k}}[x],S(0)=-cy_{p}\Bigr{\\}}\cong\\\ &\
\operatorname{Ker}(R)\rtimes\bigl{\\{}(0,y)\mapsto(0,cy+(1-c)y_{p})\mathrel{:}c\in{\Bbbk}^{*}\bigr{\\}}\end{split}$

Moreover, since ${\rm O}_{\varphi^{\ell}}(p)={\rm
O}_{\varphi}(p)\cap\mathcal{C}_{1}$, we deduce that $\varphi^{-i}\psi\in{\rm
Aut}\bigl{(}{\mathbb{A}}^{2},{\rm O}_{\varphi^{\ell}}(p)\bigr{)}$ and
therefore

$\psi\in\langle\varphi\rangle\bigl{(}G_{p}\cap{\rm
Aut}\bigl{(}{\mathbb{A}}^{2},{\rm
O}_{\varphi^{\ell}}(p)\bigr{)}\bigr{)}=\bigl{\\{}\varphi^{j}\gamma\mathrel{:}j\in{\mathbb{Z}}\,,\
\gamma\in G_{p}\cap{\rm Aut}\bigl{(}{\mathbb{A}}^{2},{\rm
O}_{\varphi^{\ell}}(p)\bigr{)}\bigr{\\}}.$

Analogously, since $\psi\bigl{(}\varphi^{i}(p)\bigr{)}=\varphi^{j_{i}}(p)$ for
$i=1,\dots,\ell-1$, it follows that $\varphi^{i-j_{i}}\psi\in{\rm
Aut}({\mathbb{A}}^{2},\mathcal{C}_{i})_{\varphi^{i}(p)}$. Hence

$\begin{split}\psi\in&\langle\varphi\rangle\Bigl{(}{\rm
Aut}({\mathbb{A}}^{2},\mathcal{C}_{i+1})_{\varphi^{i}(p)}\cap{\rm
Aut}\bigl{(}{\mathbb{A}}^{2},{\rm
O}_{\varphi^{\ell}}(\varphi^{i}(p))\bigr{)}\Bigr{)}=\\\
&\langle\varphi\rangle\Bigl{(}\bigl{(}\varphi^{i}G\varphi^{-i}\bigl{)}_{\varphi^{i}(p)}\cap\bigl{(}\varphi^{i}{\rm
Aut}\bigl{(}{\mathbb{A}}^{2},{\rm
O}_{\varphi^{\ell}}(p)\bigr{)}\varphi^{-i}\bigr{)}\Bigr{)}=\\\
&\langle\varphi\rangle\Bigl{(}\varphi^{i}G_{p}\varphi^{-i}\cap\bigl{(}\varphi^{i}{\rm
Aut}\bigl{(}{\mathbb{A}}^{2},{\rm
O}_{\varphi^{\ell}}(p)\bigr{)}\varphi^{-i}\bigr{)}\Bigr{)}=\\\
&\langle\varphi\rangle\Bigl{(}\varphi^{i}\Bigl{(}G_{p}\cap{\rm
Aut}\bigl{(}{\mathbb{A}}^{2},{\rm
O}_{\varphi^{\ell}}(p)\bigr{)}\Bigr{)}\varphi^{-i}\Bigr{)}.\end{split}$

All in all, we have that

$\psi\in\bigcap_{i=0}^{\ell-1}\langle\varphi\rangle\Bigl{(}\varphi^{i}\Bigl{(}G_{p}\cap{\rm
Aut}\bigl{(}{\mathbb{A}}^{2},{\rm
O}_{\varphi^{\ell}}(p)\bigr{)}\Bigr{)}\varphi^{-i}\Bigr{)}=\bigcap_{i=0}^{\ell-1}\langle\varphi\rangle\Bigl{(}\Bigl{(}G_{p}\cap{\rm
Aut}\bigl{(}{\mathbb{A}}^{2},{\rm
O}_{\varphi^{\ell}}(p)\bigr{)}\Bigr{)}\varphi^{-i}\Bigr{)}$

Let
$\psi\in\bigcap_{i=0}^{\ell-1}\langle\varphi\rangle\bigl{(}\varphi^{i}\bigl{(}G_{p}\cap{\rm
Aut}\bigl{(}{\mathbb{A}}^{2},{\rm
O}_{\varphi^{\ell}}(p)\bigr{)}\bigr{)}\varphi^{-i}\bigr{)}$; in order to prove
the first assertion of (c) we need to prove that $\psi\in{\rm
Aut}\bigl{(}{\mathbb{A}}^{2},{\rm O}_{\varphi}(p)\bigr{)}$. If
$j\in{\mathbb{Z}}$, let $j=q\ell+r$ with $0\leq r<\ell$ and consider the
decomposition $\psi=\varphi^{r}\gamma\varphi^{-r}$, with $\gamma\in
G_{p}\cap{\rm Aut}\bigl{(}{\mathbb{A}}^{2},{\rm
O}_{\varphi^{\ell}}(p)\bigr{)}$. Then

$\psi(\varphi^{j}(p))=\varphi^{r}\gamma\varphi^{-r+j}(p)=\varphi^{r}\gamma\varphi^{q\ell}(p).$

Since $\gamma\in{\rm Aut}\bigl{(}{\mathbb{A}}^{2},{\rm
O}_{\varphi^{\ell}}(p)\bigr{)}$, the assertion follows.

In order to prove the last assertion, we first observe that, up to conjugate
with the translation of vector $(0,y_{p})$, we can assume that $p=(0,0)$. Let
$\psi=\bigl{(}a^{\prime}x,b^{\prime}y+Q(x)\bigr{)}\in\bigcap_{i=0}^{\ell-1}\varphi^{i}G_{p}\varphi^{-i}\setminus\operatorname{Ker}(R)$.
Then $Q(0)=0$ and, since $\psi\in{\rm Aut}({\mathbb{A}}^{2},{\rm
O}_{\varphi^{\ell}}(p)\bigr{)}$, there exists $j\in{\mathbb{Z}}$ such that

$\bigl{(}0,b^{\prime}P(0)\bigr{)}=\psi\bigl{(}0,P(0)\bigr{)}=\psi\bigl{(}\varphi^{\ell}(p)\bigr{)}=\varphi^{j\ell}(p).$

If $j=1$, then
$\psi=\bigl{(}a^{\prime}x,y+Q(x)\bigr{)}\in\operatorname{Ker}(R)$. Thus, up to
work with $\psi^{-1}$ if necessary, we can assume that $j>1$. We deduce that
$b^{\prime}=1+\dots+b^{j-1}$; in particular $\varphi$ is such that $1\neq
1+\dots+b^{j-1}$ — that is, $b\neq 0$ is not a $(j-1)$-root of unity.

Analogously, we have that
$\psi\bigl{(}0,(1+b)P(0)\bigr{)}=\psi\bigl{(}\varphi^{2\ell}(p)\bigr{)}=\varphi^{m\ell}(p)$,
with $m\neq 2$. Since
$\varphi^{-\ell}=\bigl{(}a^{-1}x,b^{-1}y-b^{-1}P(x)\bigr{)}$, we deduce that

$\bigr{(}0,(1+\dots+b^{j-1})(1+b)P(0)\bigr{)}=\psi\bigl{(}\varphi^{2\ell}(p)\bigr{)}=\begin{cases}\bigl{(}0,(1+\dots+b^{m-1})P(0)\bigr{)},\quad\mbox{if}\
m>2\\\ \bigl{(}0,P(0)\bigr{)},\quad\mbox{if}\ m=1\\\ (0,0),\quad\mbox{if}\
m=0\\\ (0,-b^{-m}(1+\dots+b^{m})P(0)),\quad\mbox{if}\ m<0\\\ \end{cases}$

For all four cases above we deduce a polynomial condition on $b$. Since there
are countably many such conditions, each of them with finite solutions, we
deduce that if ${\Bbbk}$ is uncountable there exist infinite values of $b$ for
which no such a $\psi$ can be found.

∎

######  Remark 4.9.

Notice that the proof of the last assertion of Theorem 4.8(c), we omitted
additional conditions on $b$ that must be satisfied in order to allow the
existence of
$\psi\in\bigcap_{i=0}^{\ell-1}\varphi^{i}G_{p}\varphi^{-1}\setminus\operatorname{Ker}(R)$;
namely, the conditions arising from the equations
$\psi\bigl{(}\varphi^{j\ell}(p)\bigr{)}\in{\rm O}_{\varphi^{\ell}}(p)$, for
$j=3,\dots,\ell-1$. However, Example 3.5 shows that there exists an
automorphism $\varphi\in{\rm Aut}\bigl{(}{\mathbb{A}}^{2},\\{x=0\\}\bigr{)}$
such that ${\rm Aut}\bigl{(}{\mathbb{A}}^{2},{\rm
O}_{\varphi}(p)\bigr{)}\neq\langle\varphi\rangle\bigl{(}\bigcap_{i=0}^{\ell-1}\varphi^{i}\operatorname{Ker}(R)\varphi^{-i}\bigr{)}$,
therefore, these conditions may be satisfied in particular examples.

######  Remark 4.10.

In examples 4.4 and 4.5 we exhibited cases where $\overline{{\rm
O}_{\varphi}(p)}$ is a fence and ${\rm Aut}\bigl{(}{\mathbb{A}}^{2},{\rm
O}_{\varphi}(p)\bigr{)}$ is not algebraic. It is an open question to prove
that this is always the case or to provide an example where $\overline{{\rm
O}_{\varphi}(p)}$ is an union of curves of type (6) but ${\rm
Aut}\bigl{(}{\mathbb{A}}^{2},{\rm O}_{\varphi}(p)\bigr{)}$ is algebraic.

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