# Symbolically Solving Partial Differential Equations using Deep Learning

Maysum Panju, Kourosh Parand, Ali Ghodsi

###### Abstract

We describe a neural-based method for generating exact or approximate
solutions to differential equations in the form of mathematical expressions.
Unlike other neural methods, our system returns symbolic expressions that can
be interpreted directly. Our method uses a neural architecture for learning
mathematical expressions to optimize a customizable objective, and is
scalable, compact, and easily adaptable for a variety of tasks and
configurations. The system has been shown to effectively find exact or
approximate symbolic solutions to various differential equations with
applications in natural sciences. In this work, we highlight how our method
applies to partial differential equations over multiple variables and more
complex boundary and initial value conditions.

##  1 Introduction

Much of the physics governing the natural world can be written in the language
of differential equations. Despite their simple appearances, these equations
can be as challenging to solve as they are common. With recent advancements in
machine learning, and deep learning in particular, a new tool became available
for finding solutions to problems that previously had seemed impenetrable.

The typical approach that neural networks take for solving differential
equations is to model the solution using a neural network function itself.
This network is trained to fit the differential equation and produces highly
accurate approximations to how the solution is supposed to behave. Although
this is practically useful, it does not have the advantage of clarity and
conciseness that symbolic mathematical expressions provide. Unfortunately, the
application of deep learning to produce symbolic solutions is relatively
underdeveloped.

In this work, we present a method for generating symbolic solutions to
differential equations that takes advantage of the flexibility and strength of
deep learning. Our method leverages the power of gradient-based back-
propagation training to provide symbolic solutions for differential equations
that can be easily and directly used and interpreted. Building on the single-
variable model shown in [11], we here focus on partial differential equations
over multiple variables, as these problems are more common and more complex
than differential equations over a single variable. Indeed, ordinary
differential equations can be solved as a special case using our proposed
method, along with other symbolic mathematical tasks such as integration,
function inversion, and symbolic regression. As an added benefit, when our
method fails to obtain an exact solution, including when no elementary
solution exists, it will return a symbolic function that approximates the true
solution, rather than leave you empty-handed.

In the sections that follow, we outline the core of our framework and the
architecture of the multivariate symbolic function learner, or MSFL. We then
show several experiments that demonstrate the power of our method on practical
examples of PDEs.

##  2 Related work

There are several papers that apply neural networks for solving both ordinary
and partial differential equations, including those by [10, 5, 1, 9]. These
approaches typically do not provide symbolic solutions to differential
equations, but highly accurate approximate functions. There are also
interesting works for discovering PDEs, such as by [8], which is a different
problem from what we are addressing here.

One excellent work that directly links deep learning with symbolic mathematics
is given by [7]. This paper attempts to train a transformer model to learn the
language of symbolic mathematics and “translate” between inputs and outputs of
mathematical tasks, such as integration and solving differential equations.
Although the results are remarkably impressive, they depend upon an extremely
costly training procedure that cannot scale well to new configurations, for
example, involving a different set of operations than the ones used in the
training set. On a deeper level, the model has faced criticism for issues
ranging from the artificiality of its testing to the fact that it “has no
understanding of the significance of an integral or a derivative or even a
function or a number” [2].

The method and results presented in this work are an extension and development
of the results in [11], which introduces a Symbolic Function Learner that can
be used to solve or find approximate symbolic solutions to ordinary
differential equations. In this work, we extend the method to apply to
multivariate partial differential equations.

##  3 Method

There are two components to our symbolic PDE solution system. The multivariate
symbolic function learning algorithm is a submodule that produces symbolic
mathematical expressions over many variables in order to minimize a given cost
function. The equation-solving wrapper is a system that uses the submodule to
find solutions to partial differential equations. We will start by describing
the wrapper system and how it relates to PDEs, and then outline a possible
framework for the symbolic function learner that it uses.

###  3.1 Equation Solver System

A general form for representing any partial differential equation (PDE) is

$g\left(x_{1},x_{2},\ldots,x_{d},y,\frac{\partial y}{\partial
x_{1}},\frac{\partial y}{\partial x_{2}},\ldots,\frac{\partial y}{\partial
x_{d}},\frac{\partial^{2}y}{\partial x_{1}^{2}},\frac{\partial^{2}y}{\partial
x_{1}\partial x_{2}},\ldots\right)=0.$ (1)

Here, $g$ is a real-valued mathematical expression in terms of the variables
$x_{1},\dots,x_{d}$, a function over these variables $y$, and any partial
derivative of any order of $y$ over any set of the variables.

It should be noted that this format encompasses far more than just PDEs; in
fact, integration, function inversion, and many functional equation problems
can be expressed in the form of Equation 1.

The solution to Equation 1 is the symbolic function $f(x_{1},\ldots,x_{d})$
that minimizes the expected error

$L_{1}(f)=\mathbb{E}\left[\left|g\left(x_{1},\ldots,x_{d},f(x_{1},\ldots,x_{d}),\frac{\partial
f}{\partial x_{1}},\ldots,\frac{\partial f}{\partial
x_{d}},\frac{\partial^{2}f}{\partial x_{1}^{2}},\frac{\partial^{2}f}{\partial
x_{1}\partial x_{2}},\ldots\right)\right|^{2}\right]$ (2)

where $x_{1},\ldots,x_{d}$ are distributed over the desired domain of $f$.
Note that in most cases, it is sufficient to use discrete approximations for
partial derivatives, such as

$\frac{\partial f}{\partial
x_{i}}\approx\frac{f(\textbf{{x}}+\textbf{{e}}^{(i)}\varepsilon/2)-f(\textbf{{x}}-\textbf{{e}}^{(i)}\varepsilon/2)}{\varepsilon}$
$\frac{\partial^{2}f}{\partial
x_{i}^{2}}\approx\frac{f(\textbf{{x}}+\textbf{{e}}^{(i)}\varepsilon)-2f(\textbf{{x}})+f(\textbf{{x}}-\textbf{{e}}^{(i)}\varepsilon)}{\varepsilon^{2}}$

where $\textbf{{e}}^{(i)}$ is the unit vector in the $i$th direction, and
$\varepsilon$ is some small constant.

PDEs are frequently accompanied by boundary conditions or initial value
constraints which solutions must satisfy. Often, these constraints come in the
form of an interval, such as $f(x,0)=c(x)$ for some specified $c$. We will
approximate these constraints by taking $N$ uniformly spaced points
$\left\\{(\textbf{{x}}_{i},y_{i})\right\\}$ along the interval and inserting
them into the secondary error function

$L_{2}(f)=\sum_{i}\left\|f(\textbf{{x}}_{i})-y_{i}\right\|^{2}.$ (3)

Combining these two error functions gives the total error

$L_{total}=L_{1}(f)+\lambda L_{2}(f).$ (4)

The best solution to the PDE is the function $f$ that minimizes $L_{total}.$
To find such an $f$, we must perform an optimization search using $L_{total}$
as an objective function. While neural networks do this naturally using back-
propagation, it is not immediately clear how to perform this optimization over
all symbolic functions. In the next section, we outline one way in which this
can be done.

###  3.2 Multivariate Symbolic Function Learner (MSFL)

Figure 1: The architecture of the multivariate symbolic function learner
(MSFL). Top: The neural representation of a single operator (interior) node in
the symbolic parse tree. The operator function p takes in two child node as
input and applies all available operators on their values. The discretized
softmax function $\textbf{s}^{\prime}$ is a gate that allows exactly one of
these operators to pass through, determined by learnable weight $\omega$. This
output is then scaled by learnable weight $w$. (Note: bias scalars $b$ are
omitted from the diagram to save space.) Bottom: After an initial layer of
leaf nodes which combine in a fully connected fashion, a series of operator
nodes form the template of a balanced binary parse tree. The weight parameters
determine how to interpret this tree as a single, well-formed symbolic
mathematical expression $f$ over multiple variables.

The multivariate symbolic function learner (or MSFL for short) is an algorithm
that generates symbolic mathematical expressions that minimize a given cost
function. Any algorithm meeting this requirement will be a suitable fit for
the proposed symbolic PDE solving system. In particular, many symbolic
regression algorithms would fit the role with minor adjustments, including
those by [14], [13], and [16]. In this section, we describe one model designed
for this task.

Recall that every mathematical expression can be represented as a syntactic
parse tree, where nodes in the interior represent operators and leaf nodes
represent variables or constants. To evaluate the represented expression, each
operator takes in the values of its children nodes as input. By introducing an
identity operator and defining unary operators to apply on a pre-specified
combination of two child nodes, these parse trees can be standardized to be
balanced and perfectly binary.

If the structure of a parse tree is taken as a template, then it is possible
to produce a mathematical function by identifying what operations and input
quantities to use in each node of the tree. This is the essence of the MSFL
algorithm. The goal is to learn what each node represents in the expression
that minimizes the given cost function. We will show how this can be done in a
way that is fully differentiable, and hence caters to deep learning
techniques, such as back-propagation.

Let $U$ be a list of allowable unary operators $[u_{1},\ldots,u_{r}]$ that map
$\mathbb{R}$ to $\mathbb{R}$, and let $V$ be a list of binary operators
$[v_{r+1},\ldots,v_{k}]$ that map $\mathbb{R}^{2}$ to $\mathbb{R}$, for a
total of $k$ allowable operators. We define the “operate” function
$\textbf{p}:\mathbb{R}^{2}\rightarrow\mathbb{R}^{k}$ by

$\displaystyle\textbf{p}(x_{1},x_{2})$ $\displaystyle=$
$\displaystyle[u_{1}(x_{1}+x_{2}),\ldots,u_{r}(x_{1}+x_{2}),v_{r+1}(x_{1},x_{2}),\ldots,v_{k}(x_{1},x_{2})].$

For example, if $U=[id,\sin,\exp{}]$ and $V=[\times]$, then

$\textbf{p}(x_{1},x_{2})=[x_{1}+x_{2},\sin(x_{1}+x_{2}),e^{x_{1}+x_{2}},x_{1}x_{2}].$

Note that we use $id$ to refer to the unary identity function that returns its
input value unchanged. In fact, it is equivalent to the addition operator
because of how unary operators are defined to combine two child node inputs
into one by using their sum.

We compute the p function in each interior node of the parse tree. In this
way, we are applying all possible operators to the node’s two children as
input. In order to interpret the parse tree as a mathematical expression, one
of these operators should be selected to pass on its output value, to the
exclusion of all others. This can be done using a trainable gating operation.

The gate can be set up as follows. Let $\omega$ be a learnable weight vector
in $\mathbb{R}^{k}$. Denote by s the softmax function, i.e.

$\textbf{s}(\omega)=\left[\frac{e^{\omega_{i}}}{\sum_{j=1}^{k}e^{\omega_{j}}}\right]_{i=1}^{k}.$

Now $\textbf{s}(\omega)$ is a nonnegative vector in $\mathbb{R}^{k}$ with
entries summing to 1. The dot product
$\textbf{p}(h_{1},h_{2})\cdot\textbf{s}(\omega)$ is then a convex linear
combination of the outputs over all operators allowed by p, skewed to give
most weight to the operator at the index corresponding to the largest entry of
$\omega$. The choice of operator that is represented by a given node can thus
be learned by updating the learnable weight vector $\omega$ during training.

Although the softmax gate places most weight on a single entry of the vector
of outputs passing through, it still retains nonzero contributions from all
other operators in the output of p. This can be corrected by adjusting the
output of $\textbf{s}(\omega)$ to become $\textbf{s}^{\prime}(\omega)$, where
$\textbf{s}^{\prime}$ is a discretized form of softmax that returns a vector
with 1 at the entry corresponding to the largest value of $\omega$ and 0 at
all other entries. One way to compute the discretized softmax is

$\textbf{s}^{\prime}(\omega)=\left[H_{1}\left(\
\frac{\textbf{s}(\omega)_{i}}{\max\textbf{s}(\omega)}\right)\right]_{i=1}^{k}$

where $H_{1}$ is a narrow hump function centred at 1, such as
$H_{1}(x)=e^{-1000(x-1)^{2}}$. Since the division operator, maximum function,
and hump function are all differentiable almost everywhere, the discretized
softmax preserves the differentiability of our model that allows deep learning
using gradient-based training methods.

The above framework for a single operator node forms the foundational unit for
the larger parse tree structure. Let $m$ represent the number of layers in the
entire parse tree. The number of layers is a measure of how complex the
represented mathematical expression is allowed to be. Trees with more layers
form a richer space of mathematical functions, but provide a challenge by
expanding the search space significantly.

For $i=0,\ldots,2^{m}-1$, define

$h_{i}^{(0)}=\textbf{w}_{i}^{(0)}\cdot\textbf{x}+b_{i}^{(0)}$

where $\textbf{x}=[x_{1},\ldots,x_{d}]$ is a vector of the variable in the
mathematical expression being constructed, and
$\textbf{w}_{i}^{(0)}\in\mathbb{R}^{d}$ and $b_{i}^{(0)}\in\mathbb{R}$ are
learnable parameters. This represents the lowest layer of the tree, consisting
of leaf nodes that denote numerical quantities. Each of these quantities is in
the form of a learnable linear combination of all possible variables.

Working up the layers of the tree, as $n=1,\ldots,m$, the value of each node
in recursively defined as

$h_{i}^{(n)}=w_{i}^{(n)}\left(\textbf{s}^{\prime}\left(\omega_{i}^{(n)}\right)\cdot\textbf{p}_{\delta}\left(h_{2i}^{(n-1)},h_{2i+1}^{(n-1)})\right)\right)+b_{i}^{(n)}$

for each $i=0,\ldots,2^{m-n-1}$. Here, each
$\omega_{i}^{(n)}\in\mathbb{R}^{k}$ and $w_{i}^{(n)},b_{i}^{(n)}\in\mathbb{R}$
are learnable weights whose values will be learned during the training
process.

The value at the root node of the tree, $h_{0}^{(m)}$, is the value of the
mathematical expression represented by the tree when evaluated using the input
x. It is by using this value for $\hat{f}(\textbf{x})$ that the MSFL can be
trained using the cost function in Equation 4. Note that
$\hat{f}(\textbf{x})=h_{0}^{(m)}$ is obtained as a differentiable function
over x and all learnable weights in the MSFL.

At the end of the training procedure, the MSFL returns the symbolic expression
represented by the final state of the parse tree. That is, it interprets each
interior tree node as the operator determined by the weight in its $\omega$
vector, and applies an affine transformation to each node using the defined by
the corresponding $w$ and $b$ values. If the resulting function is not the
exact solution to the PDE, it will usually be a close approximation, as it is
the result of a training procedure designed to optimize for a low fitting
error.

##  4 Experiments

We test our system on a number of PDE problems and demonstrate the results
below.

In each problem, we run our system over a function $g$ in the form of Equation
1. The MSFL algorithm is run using the unary operators $U=[id,\sin,\exp{}]$
and the binary operator $V=[\times]$. In order to avoid illegal argument
errors, we automatically compute the absolute value before entering any value
as input to operations defined only on the positive half-plane, such as the
square root function. As mentioned in Section 3.2, the nature of the $id$
operator removes the need for an explicit addition operator.

For each task, we run our method 20 times, training a model on 5000 randomly
generated points within the domain of $f$ each time. Each run returns the
function represented by the parse tree after 6000 iterations. The standard
softmax function s is used for the first 1250 (= 25%) of iterations of each
run, and the discrete form $\textbf{s}^{\prime}$ is used for the remainder of
the training. This allows an introductory exploratory training phase before
the model converges to a single expression structure, spending the remaining
iterations fine-tuning the values of constants. After 20 runs are complete, we
select from the 20 returned functions the one with the lowest validation
error, as per Equation 4.

In all of the solution visualizations shown below, the left-most graphs are
plots of the learned symbolic functions. The central graphs show the residual
error from the learned functions, that is, the value of $g$ evaluated at x as
in $L_{1}(f)$ used in Equation 2. The right graphs demonstrate how the learned
functions fit the specified boundary conditions.

###  4.1 Wave Equation

The motion of wave travelling in one spatial dimension over time can be
modelled by a function $u(x,t)$ that satisfies the PDE

$\frac{\partial^{2}u}{\partial t^{2}}(x,t)=c^{2}\frac{\partial^{2}u}{\partial
x^{2}}(x,t)$ (5)

for $x\in[0,\pi]$, $t>0$. This type of motion can be found in particles
vibrating around a rest position along a single direction [15].

Consider the case where $c=1$ and we are given the boundary conditions

$\displaystyle u(0,t)=0$ $\displaystyle u(\pi,t)=0$

for $t>0$, and initial conditions

$\displaystyle u(x,0)$ $\displaystyle=$ $\displaystyle 0$
$\displaystyle\frac{\partial u}{\partial t}(x,0)$ $\displaystyle=$
$\displaystyle\sin x$

for $x\in[0,\pi]$.

This system has the exact solution $u(x,t)=\sin x\sin t.$ Our method produced
the symbolic solution

$\hat{u}(x,t)=1.0002\sin(1.000x)\sin(0.9998t).$

This result is represented visually in Figure 2.

Figure 2: The learned solution to the wave equation (5), with residual error
and boundary values.

###  4.2 Heat Equation

The diffusion of heat through a medium along a single spatial dimension over
time can be modelled by a function $u(x,t)$ that satisfies the PDE

$\frac{\partial u}{\partial t}=\frac{\partial^{2}u}{\partial x^{2}}$ (6)

for $x\in[0,\pi]$, $t>0$. For example, this equation might be used to model
how an uneven initial distribution of heat disperses and levels out along a
rod of uniform composition.

Consider the case given by the boundary conditions

$\displaystyle u(0,t)=0$ $\displaystyle u(\pi,t)=0$

for $t>0$, and initial conditions

$\displaystyle u(x,0)=\sin x$

for $x\in[0,\pi]$.

This system has the exact solution $u(x,t)=e^{-t}\sin{x}.$ Our method produced
the symbolic solution

$\hat{u}(x,t)=(1.005e^{-0.994t}-0.005)\sin(0.9996x+0.001t).$

This result is represented visually in Figure 3.

Figure 3: The symbolic solution to the heat equation (6), with residual error
and boundary values.

###  4.3 Fokker-Planck Equations

The Fokker-Planck equation is a general model describing the behaviour of
particles undergoing Brownian motion, with applications ranging from
astrophysics to economics [12]. The general format of a linear Fokker-Planck
(FP) equation, also called a forward Kolmogorov equation, is

$\frac{\partial}{\partial t}u(x,t)=-\frac{\partial}{\partial
x}A(x,t)u(x,t)+\frac{\partial^{2}}{\partial x^{2}}B(x,t)u(x,t)$ (7)

with initial condition $u(x,0)=f(x)$, where $A(x,t)$ and $B(x,t)$ (known as
drift and diffusion coefficients, respectively), along with $f(x)$, are
specified real-valued functions.

In order to fit the structure of Equation 1, we can reformulate Equation 7 as

$g\left(u,\frac{\partial u}{\partial t},\frac{\partial x}{\partial
t},\frac{\partial^{2}x}{\partial t^{2}}\right)=\frac{\partial u}{\partial
t}-\left(\left(\frac{\partial^{2}B}{\partial x^{2}}-\frac{\partial A}{\partial
x}\right)u+\left(2\frac{\partial B}{\partial x}-A\right)\frac{\partial
u}{\partial x}+B\frac{\partial^{2}u}{\partial x^{2}}\right)=0.$ (8)

An alternative version for an FP equation, known as the backward Kolmogorov
equation, is

$\frac{\partial}{\partial t}u(x,t)=-A(x,t)\frac{\partial}{\partial
x}u(x,t)+B(x,t)\frac{\partial^{2}}{\partial x^{2}}u(x,t)$ (9)

with initial condition $u(x,0)=f(x)$, where $A(x,t),$ $B(x,t)$, and $f(x)$ are
specified real-valued functions. Note that this formulation is already close
to matching the format of Equation 1, and does not need to be significantly
rearranged.

We will look at three common examples of FP equations with known analytic
solutions. These examples have been investigated in several existing works on
FP equations, including those by [6], [3], and [4].

####  4.3.1 Example 1

Consider the case of Equation 7 over $x,t\in[0,1]$ where $A(x,t)=-1$ and
$B(x,t)=1$, subject to the initial condition $f(x)=x$.

This system has the exact solution $u(x,t)=x+t.$ Our method produced the
symbolic solution

$\hat{u}(x,t)=1.0001x+1.0001t.$

This result is represented visually in Figure 4.

Figure 4: The symbolic solution to FP Example 1, with residual error and
boundary values.

####  4.3.2 Example 2

Consider the case of Equation 7 over $x,t\in[0,1]$ where $A(x,t)=x$ and
$B(x,t)=x^{2}/2$, subject to the initial condition $f(x)=x$.

This system has the exact solution $u(x,t)=xe^{t}.$ Our method produced the
symbolic solution

$\hat{u}(x,t)=(0.972x-0.001t+0.002)e^{1.024t}+0.029x-0.002.$

This result is represented visually in Figure 5.

Figure 5: The symbolic solution to FP Example 2, with residual error and
boundary values.

####  4.3.3 Example 3

Consider the case of Equation 9 over $x,t\in[0,1]$ where $A(x,t)=-(x+1)$ and
$B(x,t)=x^{2}e^{t}$, subject to the initial condition $f(x)=x+1$.

This system has the exact solution $u(x,t)=(x+1)e^{t}.$ Our method produced
the symbolic solution

$\hat{u}(x,t)=(0.923x-0.006t+0.866)e^{0.011x+1.121t}+0.057x+0.136.$

This result is represented visually in Figure 6.

Figure 6: The symbolic solution to FP Example 3, with residual error and
boundary values.

##  5 Conclusions and future work

In this work, we have shown a framework for producing symbolic solutions to
partial differential equations over many variables using deep learning
techniques. We have illustrated the utility of our system on a number of
examples of PDEs taken from classical physics. Our method has shown its
ability to generate solutions that are either exactly or approximately correct
in many cases. In particular, the linear Fokker-Planck equations have provided
good testing grounds for the strengths of our method.

Although the multivariate symbolic function learner has shown its capability,
there is still room for improvement. As with all algorithms that seek to
optimze over symbolic functions, the problem of an immensely vast search space
poses a great challenge. It would be rewarding to see an MSFL sucessfully
scale to spaces of increasingly complex functions over large sets of operators
and variables. The modularity of our system allows the MSFL to be easily
swapped with any other function learning algorithm, offering a good
opportunity for future experiments.

In the bigger picture, we look forward to seeing more applications of deep
learning in the realm of symbolic mathematics, and hope that this contribution
will be a step towards that direction.

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#  Asymmetric quantum codes on non-orientable surfaces

Waldir S. Soares Jr., Douglas F. Copatti, Giuliano G. La Guardia
and Eduardo B. Silva<EMAIL_ADDRESS>
###### Abstract

In this paper, we construct new families of asymmetric quantum surface codes
(AQSCs) over non-orientable surfaces of genus $g\geq 2$ by applying tools of
hyperbolic geometry. More precisely, we prove that if the genus $g$ of a non-
orientable surface is even $(g=2h)$, then the parameters of the corresponding
AQSC are equal to the parameters of a surface code obtained from an orientable
surface of genus $h$. Additionally, if $S$ is a non-orientable surface of
genus $g$, we show that the new surface code constructed on a $\\{p,q\\}$
tessellation over $S$ has the ratio $k/n$ better than the ratio of an AQSC
constructed on the same $\\{p,q\\}$ tessellation over an orientable surface of
the same genus $g$.

Keywords. Asymmetric quantum codes, topological quantum codes, non-orientable
surfaces.

##  1 Introduction

One of the most important class of codes, which includes the Calderbank-Shor-
Steane (CSS) codes [1], is the class of stabilizer codes, introduced by
Gottesman [2]. These codes play similar rules of the linear codes in classical
coding theory. In this stabilizer formalism, the construction of quantum codes
is performed by choosing abelian subgroups of the Pauli group that leave
invariant the code space.

Among all stabilizer codes, some topological quantum codes stand out.
Topological quantum codes encode quantum words in the nonlocal degrees of
freedom of topologically ordered physical systems. Because of this, these
quantum codewords are intrinsically resistant to effects of noise, as long as
it remains local.

The well-known toric codes form an important class of topological quantum
codes, which was introduced by Kitaev [3]. Since then, such codes have been
widely explored, extended e generalized in many different ways, [4], [5], [6],
[7].

Surface codes are a powerful and promising method for protecting quantum
information from errors. By encoding quantum information into a two-
dimensional tessellation of qubits and using syndrome measurement as well as
logical operations to detect and correct errors, surface codes offer a way to
build large-scale, fault-tolerant quantum computers that can perform complex
computations and simulations. Surface codes are relatively robust to certain
types of errors, such as errors that occur in clusters or patterns.

Asymmetric quantum error-correcting codes (AQECCs) form a class of quantum
codes that are capable of correcting different types of errors, i.e., phase-
shift and qubit-flip errors. The combined amplitude damping and dephasing
channel [8] is an example of a quantum channel that satisfies $d_{z}>d_{x}$,
that is, the probability of occurrence of phase-shift errors is greater than
the probability of occurrence of qubit-flip errors. The idea of correcting
asymmetric errors with unequal probabilities was introduced by Steane [9].
After such a work, much research on this type of codes has already been
carried out: detailing construction methods, structures and performance, as
can be seen in [10], [11], [12], [13], [14], [15].

In an AQECC, the encoding operation is optimized for correcting errors that
occur during transmission, while the decoding operation is optimized for
correcting errors that occur during storage. This approach is particularly
useful in scenarios where the probability of errors occur during transmission
is different from that during storage. For example, in a quantum communication
scenario, errors may be more likely to occur during transmission due to the
presence of noise in the communication channel.

Asymmetric quantum codes can be designed using a variety of techniques, such
as stabilizer and subsystem codes. In case of stabilizer codes, the encoding
and decoding operations are based on the stabilizer formalism, which involves
the utilization of stabilizer generators to detect and correct errors. In
subsystem codes, the encoding and decoding operations are based on the theory
of quantum subsystems, which involves the division of the system into smaller
subsystems in which the errors can be individually corrected.

The main contribution of this paper is to construct new families of asymmetric
quantum surface codes (AQSCs) over non-orientable surfaces of genus $g\geq 2$,
utilizing tools of hyperbolic geometry. We show that if the genus $g$ of a
non-orientable surface is even $(g=2h)$, then the parameters of the
corresponding AQSC are equal to the parameters of an AQSCs over an orientable
surface of genus $h$ (see Theorem 4.1). Furthermore, we also show that if $S$
is a non-orientable surface of genus $g$, then the new surface code
constructed on a $\\{p,q\\}$ tessellation over $S$ has ratio $k/n$ better than
the ratio of an AQSC derived from the same tessellation over an orientable
surface of the same genus $g$ (see Theorem 4.2). Asymmetric surfaces codes
over orientable surfaces was presented in [16].

This paper is organized as follows. Section 2 presents a brief review on
hyperbolic geometry and the connection between hyperbolic polygons and two-
dimensional surfaces (orientable or non-orientable). In Section 3, we review
the concept of surface codes which, together with hyperbolic geometry tools,
provides a technique to generate new families of asymmetric surface codes. In
Section 4, we present the main contribution of this paper: the construction of
new families of AQSCs over non-orientable surfaces. Finally, in Section 5, the
final remarks are drawn.

##  2 Hyperbolic geometry

In this section we present definitions and results of hyperbolic geometry that
will be necessary for the development of the article. For more details on
hyperbolic geometry, the reader can consult [17, 18, 19].

As a model of the hyperbolic plane, we consider the upper-half plane
$\mathbb{H}^{2}=\\{z\in\mathbb{C}:Im(z)>0\\}$. A convex closed set formed by
$p^{\prime}$ geodesic segments is a _hyperbolic polygon_ (or a
$p^{\prime}$-gon) $\varPi^{\prime}$ with edges $p^{\prime}$. A
$p^{\prime}$-gon is regular if its edges have the same length and the interior
angles are equal. We denote by $\mathbb{S}^{2}$ the sphere in $\mathbb{R}^{3}$
and by $\mathbb{E}^{2}$ the Euclidean plane.

The _unimodular group_ $SL(2,\mathbb{R})$ (or $SL_{2}(\mathbb{R})$) is the
group of $2\times 2$ real matrices with determinant one:

$SL_{2}(\mathbb{R})=\left\\{\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\biggm{|}a,b,c,d\in\mathbb{R},ad-bc=1\\}\right\\};$

and the _linear special projective group_
${\operatorname{PSL}}_{2}(\mathbb{R})$ is the quotient

$PSL_{2}(\mathbb{R})=SL_{2}(\mathbb{R})/\\{\pm I_{2}\\}.$

The elements of $PSL_{2}(\mathbb{R})$ act on the complex plane by _Möbius
transformations_
$f:\mathbb{C}\cup\\{\infty\\}\rightarrow\mathbb{C}\cup\\{\infty\\}$, where
$f(z)=\frac{az+b}{cz+d}$, $a,b,c,d\in\mathbb{R}$ and $ad-bc=1$. This is
precisely the set of Möbius transformations that preserve the upper half-plane
and they are its isometries. The group $PSL_{2}(\mathbb{R})$ is the group of
conformal automorphisms of the upper half-plane (see [19] for more details).
From the Riemann mapping theorem, it is also isomorphic to the group of
conformal automorphisms of the unit disc. Thus, the hyperbolic area of a
hyperbolic polygon is also invariant under images of the members of
$PSL_{2}(\mathbb{R})$. A _Fuchsian group_ is a discrete subgroup of
$PSL_{2}(\mathbb{R})$.

It is well-known that every compact and connected $2$-manifold is homeomorphic
either to $\mathbb{S}^{2}$, or to a connect sum of one or more copies of the
torus, or a connect sum of one or more copies of the projective plane. It is
also known that these admit polygonal presentations (see [17], for instance).

The area of a hyperbolic triangle can be given only by its angles. Moreover,
the area of a hyperbolic triangle and the area of any regular hyperbolic
polygon can be computed from the well-known Gauss-Bonnet Theorem.

######  Theorem 2.1 (Gauss-Bonnet).

If $T$ is a hyperbolic triangle with internal angles $\alpha,\beta$ and
$\gamma$, then the area of $T$ is given by

$\displaystyle area(T)=\pi-\alpha-\beta-\gamma\,.$ (1)

A _regular tessellation_ of Euclidean or hyperbolic plane is a covering of the
plane by copies of a regular polygon, where the degree of each vertex is the
same, such that the intersection of two faces is either empty, or along entire
edges or at vertices, and the interior of two distinct faces are disjoint. The
degree of a vertex is the number of edges meeting at the vertex. A regular
tessellation is denoted by $\\{p,q\\}$ if $q$ regular $p$-gons meet at each
vertex. We write $\varPi^{\prime}$ to indicate a regular hyperbolic polygon
associated with a fundamental region of the $\\{p^{\prime},q^{\prime}\\}$
tessellation and, similarly, $\varPi$ for the $\\{p,q\\}$ tessellation.

It can be proved that any compact topological surface $M$ can be realized
geometrically from a polygon $\varPi^{\prime}$ by _edge-pairing
identification_ , see [17]. An edge-pairing identification is an element
$\zeta(\neq Id)$ of an isometry group $\Gamma$ that preserves the orientation,
such that $\zeta$ identifies an edge $e$ of $\varPi^{\prime}$ to another edge
$e^{\prime}$ of $\varPi^{\prime}$ (i.e., $\zeta(e)=e^{\prime}$). Moreover,
$\zeta^{-1}\in\Gamma\setminus Id$ and $\zeta^{-1}(e^{\prime})=e$. In general,
if $\zeta_{1}(e)=e^{\prime}$ and $\zeta_{2}(e^{\prime})=e^{\prime\prime}$ for
some $\zeta_{1},\zeta_{2}\in\Gamma$, then $\zeta(e)=e^{\prime\prime}$, where
$\zeta=\zeta_{2}\circ\zeta_{1}$. Hence, if $\Gamma$ is the Fuchsian group
generated by the edge-pairing identifications $\zeta$ of $\varPi^{\prime}$, it
gives rise to an identification space $S_{\varPi^{\prime}}$. This is a
hyperbolic surface since $\Gamma$ is a Fuchsian group and $\varPi^{\prime}$ is
a fundamental region of the $\\{p^{\prime},q^{\prime}\\}$ tessellation. Thus,
$S_{\varPi}^{\prime}=\varPi^{\prime}/\Gamma$ and $M=S_{\varPi}^{\prime}$.
Recall that given a space and a group acting on it, the images of a single
point under the group action form an orbit of the action. A fundamental region
[19] is a subset of the space that contains exactly one point from each of
these orbits. It serves as a geometric realization for the abstract set of
representatives of the orbits. In the context above, an identification may
also exist with vertices. We indicate a maximal set
$\\{v_{1},v_{2},\dots,v_{k}\\}$ of identified vertices as vertex cycle.

We next present some results from [17] which will be utilized in this work.
Propositions 2.2 and 2.3 establish the conditions for a polygon to be a
fundamental region.

######  Proposition 2.2 (Edge and Angle condition).

Let $\varPi^{\prime}$ be a compact polygon and let $\Gamma$ be an isometry
group of $\mathbb{S}^{2}$, $\mathbb{E}^{2}$ or $\mathbb{H}^{2}$ which
preserves orientation, such that $\varPi^{\prime}$ is a fundamental region for
$\Gamma$. Then one has:

  1. 1.

For each edge $s$ in $\varPi^{\prime}$ there exists exactly one edge
$s^{\prime}(\neq s)$ of $\varPi$ such that $s^{\prime}=\gamma(s)$,
$\gamma\in\Gamma$ (the elements of $\Gamma$ are called side pairing
transformations of $\varPi^{\prime}$).

  2. 2.

For each set of paired edges of $\varPi^{\prime}$, the sum of angles of the
set of identified vertices is $2\pi$. This set is a _vertex cycle_.

######  Proposition 2.3 (Poincaré).

If a compact polygon $\varPi^{\prime}$ satisfies the Edge and Angle
conditions, then $\varPi^{\prime}$ is a fundamental region for the group
$\Gamma$ generated by the edge-pairing transformations of $\varPi^{\prime}$.

We know that every non-orientable surface of genus $g$ can be presented by a
$2g$-gon $\varPi^{\prime}$. We label the edges of the polygon
$\varPi^{\prime}$ that must be identified by the same symbols, and the
identification should be made according to the orientation given in the edge.
We label edges with a symbol, say “$a$” for a particular orientation, and
“$a^{-1}$” for the opposite orientation. Therefore, we can get a surface of
genus $g$ from a $2g$-gon $\varPi^{\prime}$, by labelling the edges as
$a_{1},a_{1},a_{2},a_{2},\ldots,a_{g},a_{g}$. We write this polygon,
$\varPi^{\prime}$ as $a_{1}a_{1}a_{2}a_{2}\dots a_{g}a_{g}$. The
identification space $a_{1}a_{1}a_{2}a_{2}\dots a_{g}a_{g}$ of a $2g$-gon can
be also generated by diagonal edge pairing identification of the $2g$-gon by
applying cut and paste processes (see [20]). For instance, the cut and paste
process for a normalized edge pairing identification to get the diagonal edge
pairing identification, is illustrated in Figure 1 for a non-orientable
surface of genus $3$.

Figure 1: Cut and paste process where in $(i)$ is the given polygon. Cutting
$(i)$ diagonally along $x_{1}$ and pasting along $b$ gives $(ii)$, which is
isomorphic to $(iii)$. Cutting $(iii)$ diagonally along $x_{2}$ and pasting
along $a$ gives $(iv)$, which is isomorphic to $(v)$.

The method of constructing topological quantum codes utilizes a $2g$-gon
$\varPi^{\prime}$ of $\mathbb{H}^{2}$. Let $S=\\{s_{1},s_{2},\ldots,s_{2g}\\}$
be the set of edges of $\varPi^{\prime}$. The edge pairing transformation for
the polygon $\zeta:S\rightarrow S$ is given by $\zeta(s_{i})=s_{g+i}$, where
the sum of the subscripts of $s$ is performed modulo $2g$. In $S$, $s_{i}$ is
adjacent with $s_{i-1}$ and $s_{i+1}$ for all $i\in\\{2,3,\dots,2g-1\\}$,
$s_{1}$ is adjacent with $s_{2g}$ and $s_{2}$, and $s_{2g}$ is adjacent with
$s_{2g-1}$ and $s_{1}$. These identifications produce a code having maximal
hyperbolic distance between the identified edges of $\varPi^{\prime}$ with the
edge-pairing. For instance, if $S=\\{s_{1},s_{2},\dots,s_{10}\\}$ then
$\zeta:S\rightarrow S$ satisfy $\zeta(s_{1})=s_{6},\leavevmode\nobreak\
\zeta(s_{2})=s_{7},\leavevmode\nobreak\
\zeta(s_{3})=s_{8},\leavevmode\nobreak\
\zeta(s_{4})=s_{9},\leavevmode\nobreak\ \zeta(s_{5})=s_{10}$ (see Fig. 2).

Figure 2: Edge-pairing transformation $\gamma(s_{i})=s_{g+i}$

_Euler characteristic_ is an important topological invariant for surfaces.
Given a compact surface $M$ that can be tessellated by a finite number of
polygons, let $V$, $E$ and $F$ be the set of vertices, edges, and faces,
respectively, determined by a tessellation of the surface. The Euler
characteristic $\chi(M)$ of $M$ is defined by

$\displaystyle\chi(M)=\\#V-\\#E+\\#F\,,$

where $\\#V,\\#E,\\#F$ are the cardinalities of $V,E,F$, respectively.
$\chi(M)$ depends only on the topology of the surface and not on a particular
tessellation. For closed orientable surfaces of genus $h$, it follows that
(see [17])

$\displaystyle\chi(M)=2-2h\,;$

for closed non-orientable surfaces of genus $g$, one has

$\displaystyle\chi(M)=2-g\,.$

We can obtain a relation between the number of elements in a regular
tessellation in a surface. Let $\varPi^{\prime}$ be a regular hyperbolic
polygon associated with the fundamental region of the
$\\{p^{\prime},q^{\prime}\\}$ tessellation, that is, $\varPi^{\prime}$ is a
polygon with $p^{\prime}$ edges where $q^{\prime}$ polygons with $p^{\prime}$
edges meet in each vertex. If we count the $q$ edges in each of the $V$
vertices, we have counted each edge of the tessellation twice. Analogously, if
we count all the $p$ edges corresponding to the border of each of the $F$
faces of the tessellation, we have counted each edge of the tessellation
twice. Therefore, we have $qV=2E=pF$.

##  3 Surface codes

A _quantum error-correcting code_ (QECC) $Q$ is a $K$-dimensional subspace of
the complex Hilbert space ${\mathbb{C}}^{q^{n}}$. If $Q$ has minimum distance
$d$, then $Q$ is an $((n,K,d))_{q}$ code. If $K=q^{k}$, we write
$[[n,k,d]]_{q}$. The length $n$, the dimension $K$ and the minimum distance
$d$ are the parameters of $Q$.

In this paper we only deal with $q=2$.

In [3], Kitaev proposed a particular case of stabilizer code, the well-known
toric code. Utilizing a $l\times l$ square as the fundamental polygon $P$ for
a torus, Kitaev considered a $\\{4,4\\}$ tessellation of $P$. He associated a
qubit to each edge of the corresponding tessellation. At each vertex $v$ of
the tessellation is associated an operator $X_{v}$ acting as tensor products
of the Pauli matrix $X$ on each edge adjacent to this vertex and as identity
operator in all other edges:

$X_{f}=\bigotimes_{i\in E_{f}}X^{i}$

. Each face $f$ of the tessellation is associated with an operator $Z_{f}$
which acts as the tensor product of Pauli matrix $Z$ on each edge of the
boundary of $f$ and as identity in all other faces:

$Z_{v}=\bigotimes_{j\in E_{v}}Z^{j}$

.

Figure 3: Face (f) e vertice (v) operators of Kitaev’s Toric Code act on the
highlighted qubits

In this way, Kitaev’s toric code is then defined by the space stabilized by
$X_{v}$’s and $Z_{f}$’s:

$\mathcal{C}=\\{|\psi\rangle;X_{v}|\psi\rangle=|\psi\rangle;Z_{f}|\psi\rangle=|\psi\rangle;\forall
v,f\\}\,.$

The code length is the number of edges of the $\\{4,4\\}$ tessellation, i.e.,
$n=2l^{2}$. The encoded qubits $k$ is the number of representatives of the
equivalence classes of homological non-trivial cycles; hence $k=2$, since the
torus has genus $g=1$. The minimum distance $d$ of a toric code is the minimum
among the number of edges contained in the shortest homological non-trivial
cycle in the original tessellation ($d_{x}$) and the number of edges contained
in the shortest homological non-trivial cycle in its dual tessellation
($d_{z}$), that is, $d=\min(d_{x},d_{z})=l$ (in this particular case, since
the $\\{4,4\\}$ tessellation is autodual, one has $d_{x}=d_{z}$).

There exist many types of extensions and generalizations of the toric code
[4], [21]). In [4], the authors worked with codes over orientable surfaces
whereas in [21], the construction was performed over non-orientable surfaces.

Applying similar ideas of Kitaev’s toric code in order to generate a code, it
is necessary to tile the fundamental polygon $P^{\prime}$ utilizing a
$\\{p,q\\}$ tessellation. Let us consider a $g$-torus with fundamental polygon
$P^{\prime}=\\{4g,4g\\}$, tessellated by a polygon $P=\\{p,q\\}$. To make this
tessellation possible, the quotient between the area $P^{\prime}$ and $P$ must
be a positive integer. If $\mu(M)$ is the area of a polygon $M$, the number of
faces $n_{f}$ can be computed by

$n_{f}=\frac{\mu({P^{\prime}})}{\mu({P})}=\frac{4q(g-1)}{pq-2p-2q}.$ (2)

The code length is given by

$n=\frac{p}{2}n_{f}.$ (3)

As was said previously, the encoded qubits $k$ is the number of homological
non-trivial cycles belonging to distinct equivalence classes of homology,
which implies $k=2g$. In order to compute the minimum distance of the code,
one must find out the hyperbolic distance $d_{h}$ between two opposite sides
of the fundamental polygon $P^{\prime}$. Thus, a homologically non-trivial
cycle has, at least, a number of edges greater than $d_{h}$.

Figure 4: The hiperbolic distance between two opposite sides of a polygon
$P^{\prime}$ is $d_{h}=2a$

From hyperbolic trigonometry (see Fig. (4), it follows that

$d_{h}=2arccos\frac{\cos(\pi/4g)}{\sin(\pi/4g)},$ (4)

and the length $l(p,q)$ of an edge of a $\\{p,q\\}$ polygon is given by

$l(p,q)=arccos\Bigg{[}\frac{\cos^{2}(\pi/q)+\cos(2\pi/p)}{\sin^{2}(\pi/q)}\Bigg{]}.$
(5)

Hence,

$d_{x}=\frac{d_{h}}{l(p,q)}\hbox{ and }d_{z}=\frac{d_{h}}{l(q,p)},$ (6)

which implies

$d=\lceil{\min(d_{x},d_{z})}\rceil.$ (7)

##  4 Asymmetric surface codes

In this section, new families of AQSCs obtained from a non-orientable surface
are constructed. As it is usual, we assume that $d_{X}$ is utilized to correct
$X$ errors and $d_{z}$ to correct $Z$ errors.

######  Definition 1.

[22] An asymmetric quantum error correction code (AQECC) with parameters
$((n,K,d_{z}/d_{x}))$ is a $K$-dimensional subspace of the complex Hilbert
space ${\mathbb{C}}^{2^{n}}$. The code corrects all qubit-flip errors up to
$\frac{d_{x}-1}{2}$ and all phase-shift errors up to $\frac{d_{z}-1}{2}$. An
$((n,2^{k},d_{z}/d_{x}))$ code is denoted by $[[n,k,d_{z}/d_{x}]]$.

We here construct families of AQSCs derived from non-orientable surfaces by
applying similar approaches utilized in [16] and [21].

######  Theorem 4.1.

If $S_{g}$ is a non-orientable surface of even genus $g=2h$ then an AQSC
generated by a $\\{p,q\\}$ tessellation has the same parameters of an AQSC
generated by the same $\\{p,q\\}$ tessellation over an orientable surface
$S_{h}$ of genus $h$.

Proof.The fundamental region of the orientable surface $S_{h}$ is the polygon
$\\{4h,4h\\}$, and the fundamental region of a non-orientable surface $S_{g}$
is the polygon $\\{2g,2g\\}$. Therefore, $h=2g$; hence, the fundamental region
of both surfaces are the same $P^{\prime}=\\{2g,2g\\}$ polygon.

The code length $n$ is computed by $n=n_{f}\frac{p}{2}$. But
$n_{f}=\frac{\mu(P^{\prime})}{\mu(P)}$, where $P=\\{p,q\\}$. However, since
$p$ and $n_{f}$ are equal for these two cases, then $n$ is also equal for the
corresponding two codes.

In both cases – orientable and non-orientable surfaces–, $k$ can be computed
by $k=2-\mathcal{X}$, where $\mathcal{X}$ is the Euler characteristic of the
surface. For an orientable surface $S_{h}$, we have $\mathcal{X}(S_{h})=2-2h$,
and for a non-orientable surface $S_{g}$, one has $\mathcal{X}(S_{g})=2-g$.
Then $k=g=2h$ are equal for these two codes.

Finally, let us compute the corresponding distances. The computations of
$d_{x}$ and $d_{z}$, as shown in 7, depend only on three magnitudes: $d_{h}$
(hyperbolic distance between opposite edge-pairings of the fundamental
region), $l(p,q)$ (the edge length of the $\\{p,q\\}$ polygon) and $l(q,p)$
(the edge length of the $\\{q,p\\}$ polygon). Since
$d_{x}=\frac{d_{h}}{l(p,q)}$ and $d_{z}=\frac{d_{h}}{l(q,p)}$, and both –
fundamental polygon and tessellation polygon– are the same for these two
codes, it follows that the distances are also equal. This concludes the proof.
$\square$

In Theorem 4.1, we consider non-orientable surfaces of even genus. However,
when considering a surface of an arbitrary genus we have the following result.

######  Theorem 4.2.

Let $S$ be a non-orientable surface of genus $g$. Then an AQSC generated by a
$\\{p,q\\}$ tessellation over $S$ has ratio $k/n$ better than the ratio of an
AQSC generated by the same $\\{p,q\\}$ tessellation over an orientable surface
of the same genus $g$.

###### Proof.

Let $S_{1}$ be an orientable surface and $S_{2}$ be a non-orientable surface,
both with the same genus $g$. We want to prove that the ratio
$r_{1}=k_{1}/n_{1}$ of the AQSC $C_{1}$ derived from the $\\{p,q\\}$
tessellation over $S_{1}$ is lower than the ratio $r_{2}=k_{2}/n_{2}$ of the
AQSC $C_{2}$ derived from the $\\{p,q\\}$ tessellation over $S_{2}$.

For $C_{1}$ we have: $k_{1}=2g$ and $n_{1}=\frac{2pq(g-1)}{pq-2p-2q}$; thus,

$r_{1}=\frac{g(pq-2p-2q)}{pq(g-1)}.$ (8)

On the other hand, for $C_{2}$, it follows that $k_{2}=g$ and
$n_{2}=\frac{pq(g-2)}{pq-2p-2q}$, which implies

$r_{2}=\frac{g(pq-2p-2q)}{pq(g-2)}.$ (9)

Therefore,

$q=\frac{r_{1}}{r_{2}}=\frac{\frac{g(pq-2p-2q)}{pq(g-1)}}{\frac{g(pq-2p-2q)}{pq(g-2)}}=\frac{g-2}{g-1}<1,\forall
g\geq 2,$ (10)

and the result follows.

∎

In the sequence, we exhibit the parameters of some new AQSCs obtained from
non-orientable surfaces of odd genus.

Table 1: Parameters of AQSCs from non-orientable surface of genus $g=5$, $d_{h}\approx 3.5796$ $\\{p,q\\}$ | $n_{f}$ | $l(p,q)$ | $[[n,k,d_{z}/d_{x}]]$
---|---|---|---
$\\{p,q\\}$ | $n_{f}$ | $l(p,q)$ | $[[n,k,d_{z}/d_{x}]]$
$\\{3,7\\}$ | $42$ | $1.0905$ |
$\\{7,3\\}$ | $18$ | $0.5663$ | $[[63,5,7/4]]$
$\\{3,8\\}$ | $24$ | $1.5286$ |
$\\{8,3\\}$ | $9$ | $0.7270$ | $[[36,5,5/3]]$
$\\{3,9\\}$ | $18$ | $1.8551$ |
$\\{9,3\\}$ | $6$ | $0.8192$ | $[[27,5,5/2]]$
$\\{3,12\\}$ | $12$ | $2.5534$ |
$\\{12,3\\}$ | $3$ | $0.9516$ | $[[18,5,4/2]]$
$\\{3,15\\}$ | $10$ | $3.0486$ |
$\\{15,3\\}$ | $2$ | $1.0070$ | $[[15,5,4/2]]$
$\\{4,5\\}$ | $15$ | $1.2537$ |
$\\{5,4\\}$ | $12$ | $1.0613$ | $[[30,5,4/3]]$
$\\{4,7\\}$ | $7$ | $2.1408$ |
$\\{7,4\\}$ | $4$ | $1.4491$ | $[[14,5,3/2]]$
$\\{4,8\\}$ | $6$ | $2.4485$ |
$\\{8,4\\}$ | $3$ | $1.5286$ | $[[12,5,3/2]]$
$\\{4,10\\}$ | $5$ | $2.9387$ |
$\\{10,4\\}$ | $2$ | $1.6169$ | $[[10,5,3/2]]$

Table 2: Parameters of AQSCs from non-orientable surfaces of genus $g=7$, $d_{h}\approx 4.3144$ $\\{p,q\\}$ | $n_{f}$ | $l(p,q)$ | $[[n,k,d_{z}/d_{x}]]$
---|---|---|---
$\\{p,q\\}$ | $n_{f}$ | $l(p,q)$ | $[[n,k,d_{z}/d_{x}]]$
$\\{3,7\\}$ | $70$ | $1.0905$ |
$\\{7,3\\}$ | $30$ | $0.5663$ | $[[105,7,8/4]]$
$\\{3,8\\}$ | $40$ | $1.5286$ |
$\\{8,3\\}$ | $15$ | $0.7270$ | $[[60,7,6/3]]$
$\\{3,9\\}$ | $30$ | $1.8551$ |
$\\{9,3\\}$ | $10$ | $0.8192$ | $[[45,7,6/3]]$
$\\{3,11\\}$ | $22$ | $2.3517$ |
$\\{11,3\\}$ | $6$ | $0.9210$ | $[[33,7,5/2]]$
$\\{3,12\\}$ | $20$ | $2.5534$ |
$\\{12,3\\}$ | $5$ | $0.9516$ | $[[30,7,5/2]]$
$\\{3,16\\}$ | $16$ | $3.1877$ |
$\\{16,3\\}$ | $3$ | $1.0186$ | $[[24,7,5/2]]$
$\\{3,21\\}$ | $14$ | $3.7611$ |
$\\{21,3\\}$ | $2$ | $1.0529$ | $[[21,7,4/2]]$
$\\{4,5\\}$ | $25$ | $1.2537$ |
$\\{5,4\\}$ | $20$ | $1.0613$ | $[[50,7,5/4]]$
$\\{4,6\\}$ | $15$ | $1.7627$ |
$\\{6,4\\}$ | $10$ | $1.3170$ | $[[30,7,4/3]]$
$\\{4,8\\}$ | $10$ | $2.4485$ |
$\\{8,4\\}$ | $5$ | $1.5286$ | $[[20,7,3/2]]$
$\\{4,9\\}$ | $9$ | $2.7101$ |
$\\{9,4\\}$ | $4$ | $1.5807$ | $[[18,7,3/2]]$
$\\{4,14\\}$ | $7$ | $3.6472$ |
$\\{14,4\\}$ | $2$ | $1.6900$ | $[[14,7,3/2]]$
Table 3: Parameters of AQSCs from non-orientable surfaces with $g=9$, $d_{h}\approx 4.8414$ $\\{p,q\\}$ | $n_{f}$ | $l(p,q)$ | $[[n,k,d_{z}/d_{x}]]$
---|---|---|---
$\\{p,q\\}$ | $n_{f}$ | $l(p,q)$ | $[[n,k,d_{z}/d_{x}]]$
$\\{p,q\\}$ | $n_{f}$ | $l(p,q)$ | $[[n,k,d_{z}/d_{x}]]$
$\\{3,7\\}$ | $98$ | $1.0905$ |
$\\{7,3\\}$ | $42$ | $0.5663$ | $[[147,9,9/5]]$
$\\{3,8\\}$ | $56$ | $1.5286$ |
$\\{8,3\\}$ | $21$ | $0.7270$ | $[[84,9,7/4]]$
$\\{3,9\\}$ | $42$ | $1.8551$ |
$\\{9,3\\}$ | $14$ | $0.8192$ | $[[63,9,6/3]]$
$\\{3,12\\}$ | $28$ | $2.5534$ |
$\\{12,3\\}$ | $7$ | $0.9516$ | $[[42,9,6/2]]$
$\\{3,13\\}$ | $26$ | $2.7341$ |
$\\{13,3\\}$ | $6$ | $0.9748$ | $[[39,9,5/2]]$
$\\{3,20\\}$ | $20$ | $3.6594$ |
$\\{20,3\\}$ | $3$ | $1.0481$ | $[[30,9,5/2]]$
$\\{3,27\\}$ | $18$ | $4.2792$ |
$\\{27,3\\}$ | $2$ | $1.0712$ | $[27,9,5/2]]$
$\\{4,5\\}$ | $35$ | $1.2537$ |
$\\{5,4\\}$ | $28$ | $1.0613$ | $[[70,9,5/4]]$
$\\{4,6\\}$ | $21$ | $1.7627$ |
$\\{6,4\\}$ | $14$ | $1.3170$ | $[[42,9,4/3]]$
$\\{4,8\\}$ | $14$ | $2.4485$ |
$\\{8,4\\}$ | $7$ | $1.5286$ | $[[28,9,4/2]]$
$\\{4,11\\}$ | $11$ | $3.1422$ |
$\\{11,4\\}$ | $4$ | $1.6432$ | $[[22,9,3/2]]$
$\\{4,18\\}$ | $9$ | $4.1637$ |
$\\{18,4\\}$ | $2$ | $1.7191$ | $[[18,9,3/2]]$
$\\{5,8\\}$ | $8$ | $2.7609$ |
$\\{8,5\\}$ | $5$ | $2.0481$ | $[[20,9,3/2]]$
$\\{5,15\\}$ | $6$ | $4.0698$ |
$\\{15,5\\}$ | $2$ | $2.1934$ | $[[15,9,3/2]]$
Table 4: Parameters of AQSCs from non-orientable surfaces with $g=11$, $d_{h}\approx 5.2548$ $\\{p,q\\}$ | $n_{f}$ | $l(p,q)$ | $[[n,k,d_{z}/d_{x}]]$
---|---|---|---
$\\{p,q\\}$ | $n_{f}$ | $l(p,q)$ | $[[n,k,d_{z}/d_{x}]]$
$\\{p,q\\}$ | $n_{f}$ | $l(p,q)$ | $[[n,k,d_{z}/d_{x}]]$
$\\{p,q\\}$ | $n_{f}$ | $l(p,q)$ | $[[n,k,d_{z}/d_{x}]]$
$\\{3,7\\}$ | $126$ | $1.0905$ |
$\\{7,3\\}$ | $54$ | $0.5663$ | $[[189,11,10/5]]$
$\\{3,8\\}$ | $72$ | $1.5286$ |
$\\{8,3\\}$ | $27$ | $0.7270$ | $[[108,11,8/4]]$
$\\{3,9\\}$ | $54$ | $1.8551$ |
$\\{9,3\\}$ | $18$ | $0.8192$ | $[[81,11,7/3]]$
$\\{3,12\\}$ | $36$ | $2.5534$ |
$\\{12,3\\}$ | $9$ | $0.9516$ | $[[54,11,6/3]]$
$\\{3,15\\}$ | $30$ | $3.0486$ |
$\\{15,3\\}$ | $6$ | $1.0070$ | $[[45,11,6/2]]$
$\\{3,24\\}$ | $24$ | $4.0374$ |
$\\{24,3\\}$ | $3$ | $1.0638$ | $[[36,11,5/2]]$
$\\{3,33\\}$ | $22$ | $4.6883$ |
$\\{33,3\\}$ | $2$ | $1.0803$ | $[[33,11,5/2]]$
$\\{4,6\\}$ | $27$ | $1.7627$ |
$\\{6,4\\}$ | $18$ | $1.3170$ | $[[54,11,4/3]]$
$\\{4,7\\}$ | $21$ | $2.1408$ |
$\\{7,4\\}$ | $12$ | $1.4491$ | $[[42,11,4/3]]$
$\\{4,8\\}$ | $18$ | $2.4485$ |
$\\{8,4\\}$ | $9$ | $1.5286$ | $[[36,11,4/3]]$
$\\{4,10\\}$ | $15$ | $2.9387$ |
$\\{10,4\\}$ | $6$ | $1.6169$ | $[[30,11,4/2]]$
$\\{4,13\\}$ | $13$ | $3.4932$ |
$\\{13,4\\}$ | $4$ | $1.6780$ | $[[26,11,4/2]]$
$\\{4,16\\}$ | $12$ | $3.9225$ |
$\\{16,4\\}$ | $3$ | $1.7073$ | $[[24,11,4/2]]$
$\\{4,22\\}$ | $11$ | $4.5720$ |
$\\{22,4\\}$ | $2$ | $1.7337$ | $[[22,11,4/2]]$
$\\{6,12\\}$ | $6$ | $3.7556$ |
$\\{12,6\\}$ | $3$ | $2.5534$ | $[[18,11,3/2]]$
$\\{p,q\\}$ | $n_{f}$ | $[[n,k,d_{z}/d_{x}]]$
---|---|---
$\\{7,3\\}$ | $6(g-2)$ | $[[21(g-2),g,d_{z}/d_{x}]]$
$\\{8,3\\}$ | $3(g-2)$ | $[[12(g-2),g,d_{z}/d_{x}]]$
$\\{9,3\\}$ | $2(g-2)$ | $[[9(g-2),g,d_{z}/d_{x}]]$
$\\{12,3\\}$ | $(g-2)$ | $[[6(g-2),g,d_{z}/d_{x}]]$
$\\{5,4\\}$ | $4(g-2)$ | $[[10(g-2),g,d_{z}/d_{x}]]$
$\\{6,4\\}$ | $2(g-2)$ | $[[6(g-2),g,d_{z}/d_{x}]]$
$\\{8,4\\}$ | $(g-2)$ | $[[4(g-2),g,d_{z}/d_{x}]]$
$\\{7,3\\}$ | $6(g-2)$ | $[[21(g-2),g,d_{z}/d_{x}]]$
Table 5: Parameters of AQSCs from $2g$ polygon Figure 5: Comparison between
$k/n$ for $\\{p,q\\}$ tessellation on orientable surface ($\\{p,q\\}$O) and
over a non orientable surface ($\\{p,q\\}$NO) for each odd genus $g$. Figure
6: $d_{z}-d_{x}$ versus $g$ from $\\{7,3\\}$ tessellation

##  5 Final remarks

We have constructed asymmetric quantum surface codes over non-orientable
surfaces. The new AQSC obtained from a $\\{p,q\\}$ tessellation over a non-
orientable surface of even genus $g=2h$ has the same parameters as the AQSC
generated by a $\\{p,q\\}$ tessellation over an orientable surface $S_{h}$ of
genus $h$. Moreover, if $S$ is a non-orientable surface of genus $g$, we have
shown that the new AQSC constructed on a $\\{p,q\\}$ tessellation over $S$ has
ratio $k/n$ better than the ratio of an AQSC constructed on the same
tessellation over an orientable surface of the same genus $g$. In some
specific cases, the new asymmetric codes present great asymmetry between
$d_{z}$ and $d_{x}$. For instance, if we consider the $\\{7,3\\}$
tessellation, we can observe that when considering surfaces of odd genus, the
difference between $d_{z}$ and $d_{x}$ increases.

## References

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Waldir Silva Soares Jr.
Department of Mathematics, Campus de Pato Branco, UTFPR, Universidade Técnica
Federal do Parana
Via do Conhecimento, s/n-KM 01-Fraron, Pato Branco - PR, 85503-390, Brazil
Email<EMAIL_ADDRESS>

Douglas Fernando Copatti
Department of Mathematics, Instituto Federal do Paraná - Campus Pitanga
Rua José de Alencar, 1.080 ? Vila Planalto, Pitanga - PR, 85.200-000, Brazil
Email<EMAIL_ADDRESS>

Giuliano Gadioli La Guardia
Department of Mathematics and Statistics, State University of Ponta Grossa
(UEPG), Campus Uvaranas - Bloco L - Uvaranas Av. General Carlos Cavalcanti,
4748
84030-900, Ponta Grossa - PR, Brazil
E-mail<EMAIL_ADDRESS>

Eduardo Brandani da Silva
Department of Mathematics, Maringa State University - UEM
Av. Colombo 5790, Maringa - PR, 87020-900 - Brazil
Email<EMAIL_ADDRESS>


# Holomorphically conjugate polynomial automorphisms of $\mathbb{C}^{2}$ are
polynomially conjugate

Serge Cantat  Serge Cantat, IRMAR, Campus de Beaulieu, bâtiments 22-23 263
avenue du Général Leclerc, CS 74205 35042 RENNES Cédex, France
<EMAIL_ADDRESS>and  Romain Dujardin  Romain Dujardin, Sorbonne
Université, CNRS, Laboratoire de Probabilités, Statistique et Modélisation
(LPSM), F-75005 Paris, France<EMAIL_ADDRESS>
###### Abstract.

We confirm a conjecture of Friedland and Milnor: if two polynomial
automorphisms $f$ and $g\in\mathsf{Aut}(\mathbb{A}^{2}_{\mathbf{C}})$ with
dynamical degree $>1$ are conjugate by some holomorphic diffeomorphism
$\varphi\colon\mathbf{C}^{2}\to\mathbf{C}^{2}$, then $\varphi$ is a polynomial
automorphism; thus, $f$ and $g$ are conjugate inside
$\mathsf{Aut}(\mathbb{A}^{2}_{\mathbf{C}})$. We also discuss a number of
variations on this result.

The research activities of the authors are partially funded by the European
Research Council (ERC GOAT 101053021). The authors benefited from the support
of the French government ”Investissements d’Avenir” program integrated to
France 2030 (ANR-11-LABX-0020-01).

##  1\. Introduction

###  1.1. Conjugacy classes in $\mathsf{Aut}(\mathbb{A}^{2}_{\mathbf{K}})$

In their very influential paper [10], Friedland and Milnor study the dynamical
and group-theoretic properties of polynomial automorphisms of the affine plane
defined over the field ${\mathbf{K}}=\mathbf{R}$ or $\mathbf{C}$. In
particular, they provide a dichotomy for conjugacy classes in
$\mathsf{Aut}(\mathbb{A}^{2}_{\mathbf{C}})$, which we describe below.

For any field ${\mathbf{K}}$, Jung’s theorem says that the group
$\mathsf{Aut}(\mathbb{A}^{2}_{\mathbf{K}})$ is the amalgamated product of the
group of _affine automorphisms_
${\sf{GL}}_{2}({\mathbf{K}})\ltimes{\mathbf{K}}^{2}$ and the group of
_elementary automorphisms_

$(x,y)\mapsto(\alpha x+p(y),\beta y+\gamma),$

with amalgamation along their intersection. If $f$ is an element of
$\mathsf{Aut}(\mathbb{A}^{2}_{\mathbf{K}})$, we denote by $\deg(f)$ the degree
of the formulas defining $f$, and set

$\lambda_{1}(f)=\lim_{n\to+\infty}\deg(f^{n})^{1/n}.$

Friedland and Milnor prove the following:

  * –

either $\lambda_{1}(f)=1$, and then $f$ is conjugate to an elementary or an
affine automorphism; moreover, if ${\mathbf{K}}$ is algebraically closed,
every affine automorphism is conjugate to an elementary automorphism by a
linear change of variable;

  * –

or $\lambda_{1}(f)$ is an integer larger than $1$, $f$ is conjugate to a
_generalized Hénon map_ , that is a finite composition of Hénon mappings
$(x,y)\mapsto(ay+p_{i}(x),x)$ with $\deg(p_{i})\geq 2$. In that case,
$\lambda_{1}(f)$ is the product of the degrees of the $p_{i}$.

As explained in [10], it follows that the dynamics of $f$ is interesting only
when $f$ is conjugate to a generalized Hénon map; for instance, for
$f\in\mathsf{Aut}(\mathbb{A}^{2}_{\mathbf{C}})$, the topological entropy of
the map $f\colon\mathbf{C}^{2}\to\mathbf{C}^{2}$ is equal to
$\log(\lambda_{1}(f))$ (see [13]).

The Bass-Serre theory shows that $\mathsf{Aut}(\mathbb{A}^{2}_{\mathbf{K}})$
acts on a simplicial tree, with stabilizers of vertices corresponding to
conjugates of the affine group and the elementary group. Then, the
classification of Friedland and Milnor corresponds to the following dichotomy:
$\lambda_{1}(f)=1$ if and only if the induced action of $f$ on the Bass-Serre
tree fixes some vertex, while $\lambda(f)>1$ when $f$ acts as a non-trivial
translation on some invariant geodesic axis. In accordance with this
description, we will refer to these two cases as _elliptic_ and _loxodromic_ ,
respectively.

###  1.2. Conjugation under biholomorphisms and rigidity

Let us now suppose that ${\mathbf{K}}=\mathbf{C}$. Friedland and Milnor also
study the following question: _if two polynomial automorphisms are conjugate
by some holomorphic diffeomorphism of $\mathbf{C}^{2}$, are they also
conjugate by an element of $\mathsf{Aut}(\mathbb{A}^{2}_{\mathbf{C}})$_?
Automorphisms with distinct dynamical degrees cannot be conjugate by a
homeomorphism $\mathbf{C}^{2}\to\mathbf{C}^{2}$, because their topological
entropies are distinct. In particular, elliptic and loxodromic classes should
be treated separately. A complete answer is given in [10] for elementary
automorphisms: the answer is _yes_ if such an automorphism admits a periodic
point, and _no_ otherwise [10, Theorem 6.10 and Lemma 6.12]. For generalized
Hénon maps, Friedland and Milnor also prove that the answer is yes for maps of
degree 2 and 3, and conjecture that the same result holds in the general case
[10, Theorem 7.1]. In this note we confirm this conjecture.

######  Theorem A.

Let $f$ and $g$ be loxodromic polynomial automorphisms of the affine plane,
defined over some subfield ${\mathbf{K}}$ of $\mathbf{C}$. If
$\varphi\colon\mathbf{C}^{2}\to\mathbf{C}^{2}$ is a biholomorphism that
conjugates $f$ to $g$, i.e. $\varphi\circ f\circ\varphi{{}^{-1}}=g$, then
$\varphi$ is a polynomial automorphism, and it is defined over a finite
extension of ${\mathbf{K}}$. Moreover, one can find
$\psi\in\mathsf{Aut}(\mathbb{A}^{2}_{\overline{{\mathbf{K}}}})$ such that
$\psi\circ f=g\circ\psi$ and $\deg(\psi)\leq 2^{57}(\deg(f)\deg(g))^{29}$.

A natural approach to this problem is to use eigenvalues at periodic orbits
and to show that loxodromic automorphisms with the same eigenvalues are
conjugate in $\mathsf{Aut}(\mathbb{A}^{2}_{\mathbf{C}})$. This is precisely
what is done in [10] in degree $\leq 3$. However, for arbitrary degree, such a
multiplier rigidity result is not available, so we use a different method. In
a nutshell, the main ingredient will be the existence and rigidity of the
canonical invariant currents of $f$ and $g$; these currents $T^{\pm}_{f}$ and
$T^{\pm}_{g}$ must be respected by the conjugacy, i.e.
$\varphi_{\varstar}T^{\pm}_{f}=cT^{\pm}_{g}$ for some $c>0$, and their
rigidity properties will be the key to our proof.

When $f=g$, the corresponding result was proven in [6, Proposition 8.1] and
[5]; we shall use this particular case below. Related results were also
obtained by Bera, Pal and Verma in [4], with an approach similar to ours.

### Acknowledgement

We thank Christophe Dupont and Sébastien Gouëzel for interesting discussions
related to this paper

##  2\. Proof of Theorem A

###  2.1. Preliminaries on Hénon maps (see [10] and [12])

Let $f\in\mathsf{Aut}(\mathbb{A}^{2}_{\mathbf{C}})$ be a generalized Hénon map
of degree $d$ (so $d=\lambda_{1}(f)$). Its birational extension to the
projective plane $\mathbb{P}^{2}(\mathbf{C})$ has an indeterminacy point at
$I^{+}_{f}:=[0:1:0]$, and it contracts the line at infinity
$\left\\{z=0\right\\}$ onto the indeterminacy point $I^{-}_{f}:=[1:0:0]$ of
$f{{}^{-1}}$. Let $K^{+}_{f}$ (resp. $K^{-}_{f}$) be the subset of
$\mathbf{C}^{2}$ of points with a bounded forward (resp. backward) orbit; its
closure in $\mathbb{P}^{2}(\mathbf{C})$ intersects the line at infinity at
$I^{+}_{f}$ (resp. at $I^{-}_{f}$). Let ${\left\|\cdot\right\|}$ denote the
usual euclidean norm on $\mathbf{C}^{2}$. Then, the function
$G^{+}_{f}\colon\mathbf{C}^{2}\to\mathbf{R}$ defined by

$G^{+}_{f}(p)=\lim_{n\to+\infty}\frac{1}{d^{n}}\log^{+}{\left\|f^{n}(p)\right\|}$

is continuous, plurisubharmonic, and non-negative; it vanishes exactly along
$K^{+}_{f}$ and satisfies $G^{+}_{f}\circ f=dG^{+}_{f}$. The difference
$p\mapsto G^{+}_{f}(p)-\log^{+}{\left\|p\right\|}$ extends to a continuous
function on $\mathbb{P}^{2}(\mathbf{C})\setminus\big{\\{}I^{+}_{f}\big{\\}}$,
as follows for instance from [12, Thm 1.7.1]. The closed, positive current

$T^{+}_{f}=\frac{\sf{i}}{\pi}\partial\overline{\partial}G^{+}_{f}$

satisfies $f^{\varstar}T^{+}_{f}=dT^{+}_{f}$; it is supported on the boundary
of $K^{+}_{f}$ and up to a positive multiplicative constant, _it is the unique
closed positive current supported by $K^{+}_{f}$_ (see [9, Theorem 7.12]).
Changing $f$ into $f^{{}^{-1}}$, one constructs similar objects $K^{-}_{f}$,
$G^{-}_{f}$ and $T^{-}_{f}$.

The intersection $K_{f}=K^{+}_{f}\cap K^{-}_{f}$ is a compact $f$-invariant
subset of $\mathbf{C}^{2}$, it is maximal for this property, and the
topological entropy of $f$ on $K_{f}$ is equal to $\log(d)$. The automorphism
$f$ has infinitely many saddle periodic points, all of them contained in
$K_{f}$ (see [1]). If $q\in\mathbf{C}^{2}$ is such a saddle periodic point,
its stable manifold $W^{s}(q)$ is the image of a holomorphic, injective,
immersion $\mathbf{C}\to\mathbf{C}^{2}$, _the image of which is dense in
$\partial K^{+}_{f}$_; the current $T^{+}_{f}$ can be recovered asymptotically
as a current of integration on $W^{s}(p)$ (see [3]) The function
$G_{f}:=\max(G^{+}_{f},G^{-}_{f})$ is continuous and plurisubharmonic and
vanishes exactly on $K_{f}$. Moreover

$G^{+}_{f}(p)\leq\log^{+}{\left\|p\right\|}+O(1)\;\text{ and
}\;G_{f}(p)=\log^{+}{\left\|p\right\|}+O(1)$

as ${\left\|p\right\|}$ goes to $+\infty$; see Corollary 2.6 and Proposition
3.8 in [2]. The product $T^{+}_{f}\wedge T^{-}_{f}$ is an invariant
probability measure, supported in $K_{f}$; it coincides with
$(\frac{\sf{i}}{\pi}\partial{\overline{\partial}}G_{f})^{2}$.

###  2.2. The proof

Conjugating $f$ and $g$ by elements of
$\mathsf{Aut}(\mathbb{A}^{2}_{\mathbf{C}})$, we may assume that both of them
are generalized Hénon maps. Let $\varphi\colon\mathbf{C}^{2}\to\mathbf{C}^{2}$
be a biholomorphism such that $\varphi\circ f\circ\varphi^{-1}=g$. Then
$\varphi\circ f^{n}=g^{n}\circ\varphi$ for all $n\in\mathbf{Z}$,
$\varphi(K^{\pm}_{f})=K^{\pm}_{g}$, $\varphi(K_{f})=K_{g}$, and $f$ and $g$
have the same topological entropy, hence the same degree $d$.

######  Lemma.

There exists a positive real number $c$ such that
$\varphi^{\varstar}T_{g}^{+}=cT_{f}^{+}$ and
$G_{g}^{+}\circ\varphi=cG_{f}^{+}$.

###### Proof.

The first statement follows from the fact, recalled above, that the only
positive closed $(1,1)$-currents supported on $K_{g}^{+}$ are of the form
$cT_{g}^{+}$, $c>0$.

To prove the second fact, we observe that $H=cG_{f}^{+}-G_{g}^{+}\circ\varphi$
is a pluriharmonic function on $\mathbf{C}^{2}$ with
$K_{f}^{+}\subset\left\\{H=0\right\\}$. Assume that $H\not\equiv 0$ and fix a
holomorphic function $h$ on $\mathbf{C}^{2}$ such that $H$ is the real part
$\Re(h)$. Then $\left\\{H=0\right\\}$ admits a unique singular foliation by
Riemann surfaces: its leaves are the level sets $\left\\{h=i\alpha\right\\}$,
$\alpha\in\mathbf{R}$. On the other hand, for any saddle periodic point $q$ of
$f$, the stable manifold $W^{s}(q)\subset\mathbf{C}^{2}$ is a connected,
immersed, Riemann surface contained in $K_{f}^{+}$; thus $W^{s}(q)$ is an
irreducible component of $\left\\{h=i\alpha\right\\}$ for some $\alpha$. This
is a contradiction since $W^{s}(q)$ is dense in the boundary of $K^{+}_{f}$,
so it is not embedded at any point. ∎

Similarly, there exists a constant $c^{\prime}>0$ such that
$G^{-}_{g}\circ\varphi=c^{\prime}G_{f}^{-}$. In fact, from

$\int\varphi^{\varstar}(T_{g}^{+}\wedge T_{g}^{-})=\int T_{f}^{+}\wedge
T_{f}^{-}=1$

we deduce that $c^{\prime}=1/c$, but this will not be used in what follows.
From this we infer that

$0\leq
G_{g}\circ\varphi=\max(G_{g}^{+}\circ\varphi,G_{g}^{-}\circ\varphi)=\max\left(cG_{f}^{+},c^{\prime}G_{f}^{-}\right)\leq\max(c,c^{\prime})G_{f}.$

On the other hand, $G_{f}(p)=\log{\left\|p\right\|}+O(1)$, and
$G_{g}(p)=\log{\left\|p\right\|}+O(1)$ as ${\left\|p\right\|}\to\infty$. Since
$\varphi$ is proper, we deduce that

$\log{\left\|\varphi(p)\right\|}+O(1)=G_{g}(\varphi(p))\leq\max(c,c^{\prime})G_{f}(p)\leq\max(c,c^{\prime})\log{\left\|p\right\|}+O(1),$

that is ${\left\|\varphi(p)\right\|}\leq C{\left\|p\right\|}^{e}$ for some
constant $C>0$ and for $e=\max(c,c^{\prime})$. This implies that $\varphi$ is
defined by polynomial formulas. Similarly, $\varphi^{-1}$ is also defined by
polynomial formulas, and $\varphi$ is a polynomial automorphism.

If $\varphi$ and $\psi$ are two automorphisms conjugating $f$ to $g$, then
$\varphi\circ\psi^{-1}$ commutes with $f$. In [11, 5], it is proven that the
centralizer of $f$ in $\mathsf{Aut}(\mathbb{A}^{2}_{\mathbf{C}})$ contains
$f^{\mathbf{Z}}$ as a finite index subgroup. Thus, given any integer $D\geq
1$, the set
$\left\\{\psi\in\mathsf{Aut}(\mathbb{A}^{2}_{\mathbf{C}})\,;\;\psi\circ
f\circ\psi^{-1}=g,\;\deg(\psi)\leq D\right\\}$ is finite, and it is non-empty
for $D\geq\deg(\varphi)$. Write $\psi=(\psi_{1},\psi_{2})$, with $\psi_{1}$
and $\psi_{2}$ in $\mathbf{C}[x,y]$ of degree $\leq D$. The constraints
$\psi\circ f\circ\psi^{-1}=g$ and
$\psi\in\mathsf{Aut}(\mathbb{A}^{2}_{\mathbf{C}})$ correspond to polynomial
equations in the coefficients of $\psi_{1}$ and $\psi_{2}$; the coefficients
of these equations are in the field of definition ${\mathbf{K}}$ of $f$ and
$g$. Thus, the solutions lie in a finite extension of ${\mathbf{K}}$.

It remains to show that one can find an automorphism $\psi$ that conjugates
$f$ to $g$ and has degree at most $2^{57}(\deg(f)\deg(g))^{29}$. From [5,
Theorem 4.10], one can find such a conjugacy in the group
${\mathsf{Bir}}(\mathbb{P}^{2}_{\overline{{\mathbf{K}}}})$. Thus, to conclude
one only needs to apply the following lemma of independent interest. ∎

######  Lemma.

Let $f$ and $g\in\mathsf{Aut}(\mathbb{A}^{2}_{\mathbf{C}})$ be two Hénon
automorphisms. If $\psi$ is a birational transformation of the affine plane
that $f$ to $g$, then $\psi$ is an automorphism.

###### Proof.

Let us consider $f$, $g$, and $\psi$ as birational transformations of the
projective plane $\mathbb{P}^{2}_{\mathbf{C}}$. Let $E$ be the union of the
irreducible curves $E_{i}\subset\mathbb{A}^{2}_{\mathbf{C}}$ which are
contracted by $\psi$ (there strict transform is a point of
$\mathbb{P}^{2}_{\mathbf{C}}$). From $\psi\circ f=g\circ\psi$ we deduce that
$E$ is $f$-invariant, and since a Hénon map does not preserve any algebraic
curve, we deduve that $E$ is empty. Similarly, $\psi^{-1}$ does not contract
any curve $F\subset\mathbb{A}^{2}_{\mathbf{C}}$.

It remains to show that $\psi$ does not have any indeterminacy point in
$\mathbb{A}^{2}(\mathbf{C})$. To see this, we resolve the indeterminacies of
$\psi$ by a finite sequence of blow-ups: this provides a smooth projective
surface $X$ together with two birational morphisms $\varepsilon,\,\eta\colon
X\to\mathbb{P}^{2}_{\mathbf{C}}$ such that $\psi=\eta\circ\varepsilon^{-1}$;
we assume that $X$ is minimal for this property. Suppose that
$q\in\mathbb{A}^{2}(\mathbf{C})$ is an indeterminacy point of $\psi$ and
denote by $D\subset X$ the tree of rational curves which is mapped to $q$ by
$\varepsilon$. The curve $D$ must intersect an irreducible curve $C\subset X$
which is contracted by $\eta$ and is not contained in $D$; indeed, otherwise,
$X$ would not be a minimal resolution of the indeterminacies. But then, $C$ is
the strict transform by $\varepsilon$ of an irreducible plane curve
$C^{\prime}$ that contains $q$. This curve $C^{\prime}$ is contracted by
$\psi$ and contains $q$, in contradiction with $E=\emptyset$. ∎

######  Example.

Fix an integer $m\geq 2$, and a positive integer $D$ which is not the $m$-th
power of an integer. One easily sees that the Hénon maps
$f(x,y)=(y,x+y^{m+1})$ and $g(x,y)=(y,x+Dy^{m+1})$ are conjugate over
$\mathbf{Q}(D^{1/m})$ but not over $\mathbf{Q}$. Indeed, $h(x,y)=(\alpha
x,\alpha y)$ conjugates $f$ to $g$ if $\alpha^{m}=1/D$, and from [11] we know
that the centralizer of $f$ is the semi-direct product of $f^{\mathbf{Z}}$ and
the cyclic group of order $m$ generated by
$(x,y)\mapsto(e^{2i\pi/m}x,e^{2i\pi/m}y)$.

##  3\. Discussion and complements

(a) – Theorem A is still true when $\varphi$ is a proper holomorphic
semiconjugacy:

######  Theorem B.

If $f$ and $g$ are loxodromic automorphisms of $\mathbf{C}^{2}$ and
$\varphi:\mathbf{C}^{2}\to\mathbf{C}^{2}$ is a proper holomorphic map such
that $\varphi\circ f=g\circ\varphi$, then $\varphi$ is a polynomial
automorphism.

###### Proof.

Note that we also have $\varphi\circ f^{-1}=g^{-1}\circ\varphi$. The
properness of $\varphi$ implies successively that

  1. (1)

$\varphi(\mathbf{C}^{2})=\mathbf{C}^{2}$ by Remmert’s proper mapping theorem;

  2. (2)

$\varphi^{-1}(K_{g}^{\pm})=K_{f}^{\pm}$;

  3. (3)

$\varphi^{\varstar}(T_{g}^{+})$ (resp. $\varphi^{\varstar}(T_{g}^{-})$) is a
positive closed current supported on $K_{f}^{+}$ (resp. $K_{f}^{-}$).

Indeed, $\varphi^{\varstar}(T_{g}^{+})$ (resp.
$\varphi^{\varstar}(T_{g}^{-})$) is non-zero by the first item. Hence
$\varphi^{\varstar}T_{g}^{\pm}=c^{\pm}T_{f}^{\pm}$, with $c^{\pm}>0$, and
exactly as in Theorem A we conclude that $\varphi$ is polynomial. Finally, the
Jacobian of $\varphi$ does not vanish, because otherwise, by the relation
$\varphi\circ f=g\circ\varphi$, $\left\\{\mathrm{Jac}(\varphi)=0\right\\}$
would be an $f$-invariant algebraic curve, and such a curve does not exist by
[2, Proposition 4.2]. Thus, since $\varphi$ is proper, it is a covering, hence
an automorphism, and we are done. ∎

(b) – Take ${\mathbf{K}}=\mathbf{R}$. If
$f\in\mathsf{Aut}(\mathbb{A}^{2}_{\mathbf{R}})$ has no fixed point in
$\mathbf{R}^{2}$ and $f$ preserves the orientation of $\mathbf{R}^{2}$, then
$f$ is conjugate to a translation by some real analytic diffeomorphism of
$\mathbf{R}^{2}$ (see [7]). Thus, _Theorem A fails for polynomial
automorphisms of $\mathbf{R}^{2}$._ On the other hand one may expect some
rigidity properties when $h_{\mathrm{top}}(f\arrowvert_{\mathbf{R}^{2}})>0$,
for instance when $K_{f}\subset\mathbf{R}^{2}$.

(c) – If $U$ is a small ball in the complement of $K^{+}_{f}\cup K^{-}_{f}$,
then its orbit under the action of $f$ is wandering. So, we can find a
$C^{\infty}$ diffeomorphism of $\mathbf{C}^{2}$ that commutes to $f$, is the
identity on the complement of the $f$-orbit of $U$, and is not the identity on
$U$. This shows that, as it is stated, _Theorem A fails in the $C^{\infty}$
setting._

On the other hand, the following lemma shows that the canonical currents are
preserved under $C^{1}$ conjugacy. Thus, the rigidity of positive closed
currents with support in $K^{+}$ is not really needed in the proof of Theorem
A, we could replace it by this lemma.

######  Lemma.

Let $f$ and $g$ be Hénon automorphisms of the plane
$\mathbb{A}^{2}_{\mathbf{C}}$. If $\varphi$ is a $C^{1}$ diffeomorphism of
$\mathbf{C}^{2}$ conjugating $f$ to $g$, then
$\varphi^{\varstar}T^{+}_{g}=cT^{+}_{f}$ for some real number $c\neq 0$.

###### Sketch of proof.

For both $f$ and $g$, $T^{+}$ is a laminar current whose laminar structure is
entirely defined in terms of the smooth dynamics [1]. More precisely, the
structure theory of strongly laminar currents [8, §5] shows that these
currents can be viewed in a precise way as foliation cycles on a weak
lamination, whose leaves are Pesin stable manifolds and whose transverse
measure is induced by the unstable conditionals of the unique measure of
maximal entropy. It follows that if $\varphi$ is a $C^{1}$ diffeomorphism
conjugating $f$ and $g$, $\varphi^{\varstar}T^{+}_{g}$ is a closed laminar
current subordinated to the same weak lamination as $T^{+}_{f}$; more
precisely, $\varphi^{\varstar}T^{+}_{g}$ is

  * –

a closed current;

  * –

a laminar current with respect a transverse signed measures (the sign takes
care of the orientation of the local leaves, since $\varphi$ may reverse this
orientation);

  * –

absolutely continuous with respect to $T_{f}^{+}$; in particular it is
supported by $K^{+}_{f}$.

On the other hand, the fact that $T_{f}^{+}$ is an extremal point in the cone
of positive closed currents (see e.g. [12, §2.2]) entails that the transverse
invariant measure of $T^{+}_{f}$ is ergodic (see [8, Cor. 5.8]) and this
implies that $\varphi^{\varstar}T^{+}_{g}$ is a constant multiple of
$T^{+}_{f}$. ∎

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