# Binaural Audio Rendering in the Spherical Harmonic Domain: A Summary of the
Mathematics and Its Pitfalls

Jens Ahrens

(Technical note111Find more technical notes at
http://www.ta.chalmers.se/education/texts-on-acoustics/ v. 2, Feb. 2022
Chalmers University of Technology
<EMAIL_ADDRESS>
###### Abstract

The present document reviews the mathematics behind binaural rendering of
sound fields that are available as spherical harmonic expansion coefficients.
This process is also known as binaural ambisonic decoding. We highlight that
the details entail some amount peculiarity so that one has to be well aware of
the precise definitions that are chosen for some of the involved quantities to
obtain a consistent formulation. We also discuss what sets of definitions
produce ambisonic signals that are compatible with the most common software
tools that are available.

##  1 Introduction

Any sound presentation method that uses loudspeakers can also be used for
binaural rendering simply by using head-related transfer functions (HRTFs) as
virtual loudspeakers. This approach has also been used in the context of
ambisonics in order to achieve what is often referred to as _binaural
decoding_ in (Jot et al., 1998; Wiggins et al., 2001; Noisternig et al., 2003)
and in a more elaborate way in (Duraiswami et al., 2005). The ambisonic
signals that are being decoded originate usually from a microphone array. Very
typical are spherical microphone arrays with a rigid baffle (Zotter and Frank,
2019).

More recent approaches to binaural rendering of such ambisonic sound scenes –
or spherical harmonic (SH) representations of sound scenes – use a different
formulation that does not employ discrete virtual loudspeakers but a
continuous distribution thereof (Menzies and Al-Akaidi, 2007; Rafaely and
Avni, 2010; Bernschütz, 2016). One also speaks of binaural decoding or
binaural rendering _in the SH domain_. Conceptually, these methods may be
summarized as follows: The SH representation of the captured sound field
provides information on what the sound field is; the HRTFs represent how the
human physiology influences a sound field on its way to the ear canal
entrances. The methods then establish a mathematical formulation that
’applies’ the HRTFs on the captured sound field. The result is the signals
that would arise at the ears of a listener when he/she were exposed to the
sound field that is represented by the SH signals. The listener is virtually
placed into the sound field.

We review this more recent rendering method in this document in the variant
proposed in (Rafaely and Avni, 2010). The purpose of this document is
summarizing the mathematical formulation in one single place with consistent
nomenclature and definitions of the building blocks. We highlight what
alternative definitions of some of the building blocks are commonly used, and
how the presented formulation needs to be adapted if those are being employed.
Mastering this will allow for arbitrarily combining the work from different
authors while maintaining a consistent formulation.

##  2 Definition of the Fourier Transform Over Time

In the following, we assume that the forward and inverse Fourier transforms
over time are defined as

$S(\omega)=\int\displaylimits_{-\infty}^{\infty}s(t)\
\mathrm{e}^{-\mathrm{i}\omega t}\mathrm{d}t$ (1)

and

$s(t)=\frac{1}{2\pi}\int\displaylimits_{-\infty}^{\infty}S(\omega)\
\mathrm{e}^{\mathrm{i}\omega t}\mathrm{d}\omega\,,$ (2)

where $s(t)$ is the time-domain representation of our signal under
consideration and $S(\omega)$ its frequency-domain representation. Note that
the algebraic sign of the exponent of the complex exponential is the aspect of
importance. Above definitions are compatible with the definition of the
discrete Fourier transform in MATLAB222Note that (Williams, 1999, Eq. (1.6)),
for example, uses a different sign convention..

The above definitions of the forward and inverse Fourier transform over time
determine that the spherical Hankel function of second kind
$h_{n}^{(2)}(\cdot)$ denotes outward propagating waves. Refer to (Ahrens,
2022b) for an in-depth discussion of the matter of sign conventions with
waves.

##  3 Spherical Harmonic Decomposition by Means of Spherical Microphone Arrays

Any square-integrable function $S(\beta,\alpha)$ that is defined on the
surface of a sphere as a function of the spatial coordinates colatitude333The
colatitude is also referred to as _polar angle_ or _zenith angle_. Note that
it is not elevation! $\beta$ and azimuth $\alpha$ can be expressed as a series
of surface spherical harmonics $Y_{n,m}(\beta,\alpha)$ as (Gumerov and
Duraiswami, 2005)

$S(\beta,\alpha)=\sum_{n=0}^{\infty}\sum_{m=-n}^{n}\\!\mathring{S}_{n,m}\,Y_{n,m}(\beta,\alpha)\
.$ (3)

$\mathring{S}_{n,m}$ are termed the SH coefficients of $S(\beta,\alpha)$.

We assume that $Y_{n,m}(\beta,\alpha)$ is defined as (Gumerov and Duraiswami,
2005, Eq. (2.1.59))

$Y_{n,m}(\beta,\alpha)=(-1)^{m}\,\sqrt{\frac{2n{+}1}{4\pi}\frac{(n{-}|m|)!}{(n{+}|m|)!}}\,P_{n}^{|{m}|}(\cos\beta)\,\,\mathrm{e}^{\mathrm{i}m\alpha},$
(4)

in which $P_{n}^{|{m}|}(\cdot)$ are the associated Legendre
functions444Associated Legendre functions are used with different
normalizations. We assume that $P_{n}^{m}(\mu)$ is defined via the following
Rodriguez formula (Gumerov and Duraiswami, 2005, Eq. (2.1.20)-(2.1.21)):
$P_{n}^{m}(\mu)=(-1)^{m}(1-\mu^{2})^{m/2}\frac{\mathrm{d}^{m}}{\mathrm{d}\mu^{m}}P_{n}(\mu),\
\ \forall n\geq 0,\ m\geq m\ ,$ (5) with
$P_{n}(\mu)=\frac{1}{2^{n}n!}\frac{\mathrm{d}^{n}}{\mathrm{d}\mu^{n}}(\mu^{2}-1),\
\ \forall n\geq 0\ .$ (6) This also the definition that MATLAB and SciPy use.
, $\mathrm{i}$ is the imaginary unit. Note that there are several alternative
viable definitions of $Y_{n,m}(\cdot)$ including purely real ones. We
highlight at this point that the formulation of some of the results derived
below depends to some extent on the definition of $Y_{n,m}(\cdot)$. We refer
the reader to Sec. 5.3 for a discussion of this.

The coefficients $\mathring{S}_{n,m}$ are obtained from

$\mathring{S}_{n,m}=\oint\displaylimits_{\Omega\in
S^{2}}\\!S(\beta,\alpha)\,\,Y_{n,m}(\beta,\alpha)^{*}\,\mathrm{d}\Omega\ .$
(7)

The asterisk ∗ denotes complex conjugation. $\Omega$ is an infinitesimal
surface element of the unit sphere $S^{2}$ (Sorry for using $S$ here again.
It’s the standard symbol for the unit sphere.). The integration over the
surface of the unit sphere in (7) is explicitly performed as

$\mathring{S}_{n,m}=\int\displaylimits_{0}^{2\pi}\\!\int\displaylimits_{0}^{\pi}S(\beta,\alpha)\,\,Y_{n,m}(\beta,\alpha)^{*}\cos\beta\,\mathrm{d}\beta\mathrm{d}\alpha\
.$ (8)

In the remainder, we assume that the quantity $S(\cdot)$ is frequency
dependent, i.e., $S(\beta,\alpha)=S(\beta,\alpha,\omega)$ and therefore
$\mathring{S}_{n,m}=\mathring{S}_{n,m}(\omega)$ where $\omega{\,=\,}2\pi f$ is
the radian frequency in $\mathrm{r}\mathrm{a}\mathrm{d}\mathrm{/}\mathrm{s}$,
$f$ is the frequency in $\mathrm{H}\mathrm{z}$. If $S(\cdot)$ is an interior
sound pressure field $S(\beta,\alpha,r,\omega)$, whereby $r$ is the radial
distance from the coordinate origin, it can be represented as (Gumerov and
Duraiswami, 2005)

$S(\vec{x},\omega)=S(\beta,\alpha,r,\omega)=\sum_{n=0}^{\infty}\sum_{m=-n}^{n}\\!\mathring{S}_{n,m}(r,\omega)\,\,Y_{n,m}(\beta,\alpha)\
,$ (9)

with

$\mathring{S}_{n,m}(r,\omega)=\breve{S}_{n,m}(\omega)\,\,j_{n}(r,\omega)\ .$
(10)

$j_{n}(\cdot)$ are the spherical Bessel functions.

The goal of the following endeavor is measuring the coefficients
$\breve{S}_{n,m}(\omega)$ of a given sound field. Pressure microphones that
are distributed over the surface of a rigid spherical scattering object were
shown to be a suitable apparatus for this (Meyer and Elko, 2002; Abhayapala
and Ward, 2002; Rafaely, 2005). In the following, we assume that
$\mathrm{e}^{-\mathrm{i}\vec{\mathbf{k}}_{\text{pw}}^{T}\vec{\mathbf{x}}}$
represents a plane wave that propagates in the direction into which
$\vec{\mathbf{k}}_{\text{pw}}$ points so that spherical Bessel functions of
second kind $h_{n}^{(2)}\\!\left(\cdot\right)$ denote outward propagating
waves (cf. also Sec. 5.1).

A sound pressure field $S^{\text{\;\\!surf}}(\beta,\alpha,R,\omega)$ on the
surface of a rigid spherical scatterer of radius $R$ that is centered at the
coordinate origin is given by (Gumerov and Duraiswami, 2005, Eq. (3.1.1))

$S^{\text{\;\\!surf}}(\beta,\alpha,R,\omega)=\sum_{n=0}^{\infty}\sum_{m=-n}^{n}\\!\mathring{S}_{n,m}^{\text{\;\\!surf}}(R,\omega)\,\,Y_{n,m}(\beta,\alpha)\
,$ (11)

with

$\mathring{S}_{n,m}^{\text{\;\\!surf}}(R,\omega)=\breve{S}_{n,m}(\omega)\,\,b_{n}(R,\omega)\
.$ (12)

$b_{n}(\cdot)$ is given by (Gumerov and Duraiswami, 2005, Eq. (4.2.10) and
(4.2.13))

$\displaystyle b_{n}\\!\left(R,\omega\right)$ $\displaystyle=$ $\displaystyle
j_{n}\\!\left(\omega\frac{R}{c}\right)-\frac{j_{n}^{\;\\!\prime}\\!\left(\omega\frac{R}{c}\right)}{h_{n}^{\prime\;\\!(2)}\\!\left(\omega\frac{R}{c}\right)}\,h_{n}^{(2)}\\!\\!\left(\omega\frac{R}{c}\right)$
(13) $\displaystyle=$
$\displaystyle-\frac{\mathrm{i}}{\left(\omega\frac{R}{c}\right)^{2}}\,\frac{1}{h_{n}^{\prime\;\\!(2)}\\!\\!\left(\omega\frac{R}{c}\right)}\
.$ (14)

$c$ is the speed of sound in $\mathrm{m}\mathrm{/}\mathrm{s}$, and
$h_{n}^{\prime\;\\!(2)}\\!(\cdot)$ denotes the derivative of the $n$th order
spherical Hankel function of second kind with respect to the argument,
$j_{n}^{\;\\!\prime}\\!\left(\cdot\right)$ the derivative of the $n$th order
spherical Bessel function with respect to the argument.

$\mathring{S}_{n,m}^{\text{\;\\!surf}}(R,\omega)$ are the SH coefficients of
the sound pressure on the surface of the spherical scatterer with radius $R$.
$\breve{S}_{n,m}(\omega)$ are the SH coefficients – and thereby a complete
representation – of the incident sound field with the effect of the scatterer
removed. We are not aware that there is an established terminology that allows
for differentiating between the SH coefficients $\mathring{S}_{n,m}(\cdot)$
that we mark with a circle and the SH coefficients $\breve{S}_{n,m}(\cdot)$
that we mark with a breve. We have to use the mathematical notation for this.

Let us distribute pressure microphones on the surface of the spherical
scatterer and thereby establish a spherical microphone array (SMA). The
computation of $\breve{S}_{n,m}(\omega)$ from the microphone signals
$S^{\text{\;\\!surf}}(\beta,\alpha,R,\omega)$ can be performed via (Rafaely,
2005)

$\mathring{S}_{n,m}^{\text{\;\\!surf}}(R,\omega)=\oint\displaylimits_{\Omega\in
S^{2}}\\!S^{\text{\;\\!surf}}(\beta,\alpha,R,\omega)\,\,Y_{n,m}(\beta,\alpha)^{*}\,\mathrm{d}\Omega$
(15)

and

$\breve{S}_{n,m}(\omega)=\mathring{S}_{n,m}^{\text{\;\\!surf}}(R,\omega)\,\,b_{n}^{-1}(R,\omega)\
,$ (16)

or equivalently,

$\breve{S}_{n,m}(\omega)=b_{n}^{-1}(R,\omega)\,\oint\displaylimits_{\Omega\in
S^{2}}\\!S^{\text{\;\\!surf}}(\beta,\alpha,R,\omega)\,\,Y_{n,m}(\beta,\alpha)^{*}\,\mathrm{d}\Omega\
.$ (17)

$b_{n}^{-1}(R,\omega)$ is termed _radial filters_ in the SMA literature. These
filters exhibit impractically high gains at low frequencies at high orders
(because $b_{n}(R,\omega)$ tends to 0 there) so that they require
regularization. The effect of this is well documented in the SMA literature
(Moreau et al., 2006; Bernschütz, 2016).

In practical implementations, the integrals in (15) and (17) are discretized
because a continuous layer of pressure microphones cannot be implemented but a
finite set of discrete microphones has to be used. This bounds the maximum
order $n$ that can be extracted to $n\leq N$ because the orthogonality of
$Y_{n,m}$ is maintained only then. One speaks of an $N$th order decomposition.

##  4 Binaural Rendering

This section summarizes how a sound field whose SH coefficients
$\breve{S}_{n,m}(\omega)$ are known can be auralized – i.e. rendered/decoded –
binaurally. We define an HRTF $H(\phi,\theta,\omega)$ as the ear response to a
plane wave that carries a time-domain impulse and that propagates into the
direction $(\phi,\theta)$, whereby $\phi$ is the colatitude and $\theta$ is
the azimuth in a spherical coordinate system555HRTFs do indeed exhibit some
amount of distance dependence. But this is only relevant at distances of, say,
$1\text{\,}\mathrm{m}$ or shorter (cf., for example, (Wierstorf et al.,
2011)). HRTFs that where measured at farther distances can be interpreted as
plane-wave HRTFs..

Any interior sound field $S(\vec{x},\omega)$ can be represented by its
_Herglotz kernel_ or _signature function_ $\bar{S}(\phi,\theta,\omega)$ as
(Kress, 2001, Sec. 1.2.1.7, Eq. (72)), (Gumerov and Duraiswami, 2005, Eq.
(7.1.27))

$S(\vec{x},\omega)=\oint\displaylimits_{\Omega\in
S^{2}}\bar{S}(\phi,\theta,\omega)\,\mathrm{e}^{-\mathrm{i}\vec{k}(\phi,\theta,\omega)^{T}\vec{x}}\
\mathrm{d}\Omega\,,$ (18)

whereby $\Omega=\Omega(\phi,\theta)$ is an infinitesimal surface element of
the unit sphere $S^{2}$, $\vec{x}=[x,y,z]^{T}$ is a location in a Cartesian
coordinate system, and
$\vec{k}(\phi,\theta,\omega)=\frac{\omega}{c}[\cos\theta\sin\phi,\sin\theta\sin\phi,\cos\phi]^{T}$
is the wave vector. In other words, $S(\vec{x},\omega)$ is represented by a
continuum of propagating plane waves, and $\bar{S}(\phi,\theta,\omega)$ are
the coefficients of the plane waves666Despite using propagating plane waves as
basis functions, the representation (18) is a general representation that
applies to _any_ interior sound field (Kress, 2001, Sec. 1.2.1.7)..

It is presented in (Gumerov and Duraiswami, 2005, Eq. (7.1.26)-(7.1.28)) (and
more explicitly in (Ahrens, 2012, App. E.4.2)) that
$\bar{S}(\phi,\theta,\omega)$ is related to the SH coefficients
$\breve{S}_{n,m}(\omega)$ of $S(\vec{x},\omega)$ by

$\bar{S}(\phi,\theta,\omega)=\sum_{n=0}^{\infty}\sum_{m=-n}^{n}\frac{1}{4\pi\mathrm{i}^{-n}}\,\breve{S}_{n,m}(\omega)Y_{n,m}(\phi,\theta)\
.$ (19)

In order to compute the signal $B(\omega)$ that arises at the ears of a
listener due to exposure of the listener to $S(\vec{x},\omega)$, we can employ
the representation (18). We recall that an HRTF represents the ear response to
a plane wave. Instead of integrating over plane waves like in (18), we
integrate over the HRTFs $H(\phi,\theta,\omega)$ and weight each HRTF with the
strength $\bar{S}(\phi,\theta,\omega)$ of the corresponding plane wave
component as (Menzies and Al-Akaidi, 2007)

$\displaystyle B(\omega)$ $\displaystyle=$
$\displaystyle\oint\displaylimits_{\Omega\in
S^{2}}\bar{S}(\phi,\theta,\omega)\,H(\phi,\theta,\omega)\ \mathrm{d}\Omega$
(20) $\displaystyle=$ $\displaystyle\oint\displaylimits_{\Omega\in
S^{2}}\sum_{n}\sum_{m}\frac{1}{4\pi\mathrm{i}^{-n}}\breve{S}_{n,m}(\omega)Y_{n,m}(\phi,\theta)\times\,\sum_{n^{\prime}}\sum_{m^{\prime}}\mathring{H}_{n^{\prime},m^{\prime}}(\omega)Y_{n^{\prime},m^{\prime}}(\phi,\theta)\
\mathrm{d}\Omega$ $\displaystyle=$
$\displaystyle\sum_{n}\sum_{m}\frac{1}{4\pi\mathrm{i}^{-n}}\breve{S}_{n,m}(\omega)\sum_{n^{\prime}}\sum_{m^{\prime}}\mathring{H}_{n^{\prime},m^{\prime}}(\omega)\times\\!\\!\oint\displaylimits_{\Omega\in
S^{2}}\\!Y_{n,m}(\phi,\theta)\ Y_{n^{\prime},m^{\prime}}(\phi,\theta)\
\mathrm{d}\Omega\ .$

The basis functions $Y_{n,m}(\phi,\theta)$ are orthonormal, which means that

$\oint\displaylimits_{\Omega\in
S^{2}}Y_{n,m}(\phi,\theta)^{*}\,Y_{n^{\prime},m^{\prime}}(\phi,\theta)\
\mathrm{d}\Omega=\delta_{n,n^{\prime}}\delta_{m,m^{\prime}}\,,$ (21)

whereby $\delta$ denotes the Kornecker delta. Noting that
$Y_{n,-m}(\cdot)=Y_{n,m}(\cdot)^{*}$ for the present definition of
$Y_{n,m}(\cdot)$ (Eq. (4)), we reformulate (21) as

$\oint\displaylimits_{\Omega\in
S^{2}}Y_{n,-m}(\phi,\theta)\,Y_{n^{\prime},m^{\prime}}(\phi,\theta)\
\mathrm{d}\Omega=\delta_{n,n^{\prime}}\delta_{m,m^{\prime}}\,.$ (22)

In order to apply (22) in (20), we replace $m$ in (20) with $-m$ without loss
of generality. This leads finally to

$B(\omega)=\sum_{n=0}^{\infty}\sum_{m=-n}^{n}\frac{1}{4\pi\mathrm{i}^{-n}}\breve{S}_{n,-m}(\omega)\
\mathring{H}_{n,m}(\omega)\,.$ (23)

Eq. (23) may be termed the _binaural rendering equation_. A variant of it was
initially published in (Rafaely and Avni, 2010). It describes how the left and
right ear signals $B^{\text{L,R}}(\omega)$ are obtained from the SH
coefficients $\breve{S}_{n,-m}(\omega)$ of the sound field and the SH
coefficients $\mathring{H}_{n,m}^{\text{L,R}}(\omega)$ of the left-ear and
right-ear HRTFs. Recall (3) and (10)-(12) to appreciate the difference between
the coefficients marked with a breve ($\breve{X}_{n,m}$) and those marked with
a ring ($\mathring{X}_{n,m}$).

##  5 Peculiarities

###  5.1 Sign Conventions

As indicated in Sec. 2, the choice of the signs in the exponents of the
complex exponential in the time and the spatial Fourier transforms determine
what the propagation direction of a wave is that is represented by a given
mathematical expression. The chosen sign convention does indeed affect the
mathematical expressions that are used in the present document. We refer the
reader to (Ahrens, 2022b) for more information.

###  5.2 Definition of the HRTFs

HRTFs are usually defined as the ear response to sound impinging from a given
source position or direction. This is also how the data are stored. Yet, as
per the definitions above, $\mathring{H}_{n,m}(\omega)$ are the SH
coefficients under the interpretation that an HRTF is the ear response to a
plane wave that propagates into a given direction $(\phi,\theta)$. Otherwise,
Eq. (20) would not be applicable.

This means that we have to convert the coordinates of the sound incidence
directions that are usually part of the HRTF data into propagation directions
before being able to use (7) for expanding
$H(\phi_{\textnormal{prop.}},\theta_{\textnormal{prop.}},\omega)$ into
$\mathring{H}_{n,m}(\omega)$. This is conversion is straightforward:

$\displaystyle\phi_{\textnormal{prop.}}$ $\displaystyle=$
$\displaystyle\pi-\phi_{\textnormal{incid.}}$
$\displaystyle\theta_{\textnormal{prop.}}$ $\displaystyle=$
$\displaystyle\pi+\theta_{\textnormal{incid.}}\ .$ (24)

Recall that $\phi$ denotes colatitude, and $\theta$ denotes azimuth.

Let us quickly look at this matter from a mathematical point of view.
Replacing $\alpha$ with $\pi+\alpha$ and $\beta$ with $\pi-\beta$ in the SH
transformation in (15) means for the basis function that

$Y_{n,m}(\pi-\beta,\pi+\alpha)^{*}=(-1)^{n}(-1)^{m}(-1)^{-m}Y_{n,m}(\beta,\alpha)^{*}$
(25)

We exploited (Gumerov and Duraiswami, 2005, Eq. (2.1.46)) in (25). The terms
$(-1)^{m}$ and $(-1)^{-m}$ cancel out. This means that instead of obtaining
the coefficients $\mathring{H}_{n,m}(\omega)$, we obtain
$(-1)^{n}\mathring{H}_{n,m}(\omega)$. This in turn means that we can
compensate for such a confusion of the incidence and propagation directions by
applying a factor $(-1)^{n}$ to the ambisonic signals. As a side note,
applying this to (23) leads to

$B^{\prime}(\omega)=\sum_{n=0}^{\infty}\sum_{m=-n}^{n}\frac{1}{4\pi\mathrm{i}^{n}}\breve{S}_{n,-m}(\omega)\
\mathring{H}_{n,m}(\omega)\,.$ (26)

Replacing $\mathrm{i}^{-n}$ with $\mathrm{i}^{n}$ (and vice versa) therefore
also compensates for this confusion.

###  5.3 Definition of the Spherical Harmonics $Y_{n,m}(\phi,\theta)$

Sec. 3 stays entirely viable with any conceivable definition of
$Y_{n,m}(\phi,\theta)$. Using a real valued definition simply makes the
complex conjugation that is performed in some equations redundant.

Let us reconsider Sec. 4 on the example of the definition used in (Williams,
1999, Eq. (6.20)), which reads

$Y_{n,m}(\beta,\alpha)=\sqrt{\frac{2n{+}1}{4\pi}\frac{(n{-}m)!}{(n{+}m)!}}\,P_{n}^{{m}}(\cos\beta)\,\,\mathrm{e}^{\mathrm{i}m\alpha}\,.$
(27)

Compare (27) of (4) and appreciate the differences. Here’s the crux: While the
previous definition (4) leads to $Y_{n,-m}(\cdot)=Y_{n,m}(\cdot)^{*}$, the
present definition (27) leads to (Williams, 1999, Eq. (6.44))

$Y_{n,-m}(\cdot)=(-1)^{m}\,Y_{n,m}(\cdot)^{*}\ .$ (28)

As a consequence, we need to add a factor of $(-1)^{m}$ to (22) and (23) if
(27) is employed (Andersson, 2017).

Missing out on this factor of $(-1)^{m}$ does not make the binaural signals
unusable. It rather rotates the scene by
$180\text{\,}\mathrm{\SIUnitSymbolDegree}$ along the azimuth, because
$(-1)^{m}=\mathrm{e}^{\mathrm{i}m\pi}$, which one can detect easily and
correct for accordingly.

When using a real definition of $Y_{n,m}(\beta,\alpha)$777Such as (Whittaker
and Watson, 1963, Sec. 18.4)
$Y_{n,m}(\beta,\alpha)=(-1)^{m}\,\sqrt{\frac{2n{+}1}{4\pi}\frac{(n{-}|m|)!}{(n{+}|m|)!}}\,P_{n}^{|{m}|}(\cos\beta)\begin{cases}\sqrt{2}\sin{|m|\alpha},\
\forall m<0\\\ \ \ \ \ \ \ \ 1\ \ \ \ \ \ \,,\ \forall m=0\\\
\sqrt{2}\cos{|m|\alpha},\ \forall m>0\end{cases}\ ,$ (29) which corresponds to
the definition that is used in the N3D ambisonic format (Nachbar et al.,
2011). It is also identical to the definition that is used in Politis’s
getSH.m MATLAB function when called with the argument ’real’ (Politis, 2015).,
the orthonormality relation (21) reads

$\oint\displaylimits_{\Omega\in
S^{2}}Y_{n,m}(\phi,\theta)\,Y_{n^{\prime},m^{\prime}}(\phi,\theta)\
\mathrm{d}\Omega=\delta_{n,n^{\prime}}\delta_{m,m^{\prime}}\,.$ (30)

Exploiting this in (20) turns the rendering equation (23) into

$B(\omega)=\sum_{n=0}^{\infty}\sum_{m=-n}^{n}\frac{1}{4\pi\mathrm{i}^{-n}}\breve{S}_{n,m}(\omega)\
\mathring{H}_{n,m}(\omega)\,.$ (31)

###  5.4 Propagation Direction of a Plane Wave Vs. Incidence Direction

Some authors prefer to define plane waves not by their propagation direction
but by their direction of incidence (see, for example, (Rafaely and Avni,
2010)). This requires a lot of modifications of the present formulation as,
for example, (18) changes and also the whole spiel around the HRTFs that is
discussed in Sec. 5.2 is exactly the other way round. Not least does this
relate to the general topic of sign conventions (of the exponent of the
complex exponential that described the spatial part of a sound field), which
is in itself rather convoluted (we refer the reader to (Ahrens, 2022b) for
more information on that.). The matter of propagation direction vs. incidence
direction is discussed further in Sec. 5.5.

###  5.5 Ambisonics

The formulation presented in Sec. 3 is heavily inspired by (Gumerov and
Duraiswami, 2005). It can be considered more of a mathematical point of view.
When using the solutions in practice, it is worth adopting also the physical
perspective. One way doing this is defining the coefficients
$\breve{S}_{n,m}(\omega)$ via

$S(\vec{x},\omega)=S(\beta,\alpha,r,\omega)=\sum_{n=0}^{\infty}\sum_{m=-n}^{n}\\!4\pi\mathrm{i}^{-n}\breve{S}_{n,m}(\omega)\,\,j_{n}(r,\omega)\,\,Y_{n,m}(\beta,\alpha)$
(32)

rather than via (9) and (10). The main motivation for the definition (32) is
that it yields time-domain signals $\breve{s}_{n,m}(t)$ that are real (for
real sound pressures $s(\vec{x},t)$) when using a real-valued definition of
$Y_{n,m}(\cdot)$ so that they can be stored as multichannel audio signals888A
secondary aspect is that this formulation allows for interpreting
$\breve{S}_{n,m}(\omega)$ as a plane wave representation (Rafaely, 2019, Ch.
2.4 and Eq. (2.62)).. This changes the definition of the radial filters
$b_{n}^{-1}(R,\omega)$ because the factor $4\pi\mathrm{i}^{-n}$ now needs to
be included into (13) and (14) so that it also gets applied in (17).

The formulation that has established in the ambisonics community (Daniel,
2001; Nachbar et al., 2011) employs yet another difference compared to Sec. 3.
That is, everything in the ambisonics context is formulated in terms of
incidence directions rather than propagation directions so that Sec. 5.4 kicks
in. Instead of reformulating the entire Sec. 3 and 4, we remind ourselves of
the discussion from Sec. 5.2. We concluded that we can account for a change of
the reference direction by replacing the factor $\mathrm{i}^{-n}$, for example
in (32), with $\mathrm{i}^{n}$. This changes the radial filters
$b_{n}^{-1}(R,\omega)$ from the inverse of (13) to the inverse of

$b^{\text{ambi}}_{n}\\!\left(R,\omega\right)=4\pi\mathrm{i}^{n}\left[j_{n}\\!\left(\omega\frac{R}{c}\right)-\frac{j_{n}^{\;\\!\prime}\\!\left(\omega\frac{R}{c}\right)}{h_{n}^{\prime\;\\!(2)}\\!\left(\omega\frac{R}{c}\right)}\,h_{n}^{(2)}\\!\\!\left(\omega\frac{R}{c}\right)\right]\
.$ (33)

and accordingly for the representation from (14). As a consequence, the
factors $4\pi\mathrm{i}^{-n}$ or $4\pi\mathrm{i}^{n}$, respectively, need to
be removed from the right-hand sides of (19), (23) and (26), and the
coefficients $\mathring{H}_{n,m}(\omega)$ of the HRTFs need to be computed
based on the incidence directions, too. This formulation is compatible with
the N3D normalization that is used in the standard ambisonic software tools
such as SPARTA999https://leomccormack.github.io/sparta-site/ and the IEM
Plugin Suite101010https://plugins.iem.at/. MATLAB implementations are provided
in (Ahrens, 2022a, c).

Refer to Table 10 for a summary.

As a side note, the coordinate system that is used in ambisonics is a right-
hand coordinate system where the $x$-axis points ’forward’, the $y$-axis
points ’to the left’, and the $z$-axis points ’upwards’ relative to the
reference orientation of a notional listener.

##  6 Conclusions

We revisited the mathematics of binaural audio rendering in the spherical
harmonic domain, and we highlighted the pitfalls, which relate to some of the
definitions of the employed quantities. Our findings are summarized in Table
10 on the subsequent page. We recommend in critical situations, such as when
implementing a given method, that the reader uses the present document as
reference to verify the consistency of the definitions in the literature that
presents the method to be implemented and the definitions used by the
implementation.

We also have to highlight that some of the literature is inconsistent in
itself, which includes some of our own publications.

$\displaystyle{\int^{I}}$ Definition of | Definition of the |  Definition of | Radial filters | Binaural rendering
---|---|---|---|---
$\breve{S}_{n,m}(\omega)$ |  temporal Fourier transform — Reference direction | $Y_{n,m}(\beta,\alpha)$ | $b_{n}^{-1}\left(\omega\frac{R}{c},R\right)$ |
$\displaystyle S(\vec{x},\omega)=\sum_{n=0}^{\infty}\sum_{m=-n}^{n}\breve{S}_{n,m}(\omega)\,\,j_{n}\\!\left(\omega\frac{r}{c}\right)\,Y_{n,m}(\beta,\alpha)$ |  Eq. (1) and (2) — Propagation |  Eq. (4) | $\left(-\frac{\mathrm{i}}{\left(\omega\frac{R}{c}\right)^{2}}\frac{1}{h_{n}^{\prime\;\\!(2)}\\!\left(\omega\frac{R}{c}\right)}\right)^{-1}$ | $\displaystyle B^{\text{L,R}}(\omega)=\sum_{n=0}^{N}\sum_{m=-n}^{n}\\!\\!\frac{1}{4\pi\mathrm{i}^{-n}}\,\breve{S}_{n,-m}(\omega)\,\,\mathring{H}_{n,m}^{\text{L,R}}(\omega)$
$\displaystyle S(\vec{x},\omega)=\sum_{n=0}^{\infty}\sum_{m=-n}^{n}\\!\\!4\pi\mathrm{i}^{-n}\,\breve{S}_{n,m}(\omega)\,\,j_{n}\\!\left(\omega\frac{r}{c}\right)\,Y_{n,m}(\beta,\alpha)$ |  Eq. (1) and (2) — Propagation |  Eq. (4) | $\left(-4\pi\mathrm{i}^{-n}\,\frac{\mathrm{i}}{\left(\omega\frac{R}{c}\right)^{2}}\frac{1}{h_{n}^{\prime\;\\!(2)}\\!\left(\omega\frac{R}{c}\right)}\right)^{-1}$ | $\displaystyle B^{\text{L,R}}(\omega)=\sum_{n=0}^{N}\sum_{m=-n}^{n}\\!\breve{S}_{n,-m}(\omega)\,\,\mathring{H}_{n,m}^{\text{L,R}}(\omega)$
$\displaystyle S(\vec{x},\omega)=\sum_{n=0}^{\infty}\sum_{m=-n}^{n}\\!\\!4\pi\mathrm{i}^{-n}\,\breve{S}_{n,m}(\omega)\,\,j_{n}\\!\left(\omega\frac{r}{c}\right)\,Y_{n,m}(\beta,\alpha)$ |  Eq. (1) and (2) — Propagation |  Eq. (27) | $\left(-4\pi\mathrm{i}^{-n}\,\frac{\mathrm{i}}{\left(\omega\frac{R}{c}\right)^{2}}\frac{1}{h_{n}^{\prime\;\\!(2)}\\!\left(\omega\frac{R}{c}\right)}\right)^{-1}$ | $\displaystyle B^{\text{L,R}}(\omega)=\sum_{n=0}^{N}\sum_{m=-n}^{n}\\!(-1)^{m}\,\breve{S}_{n,-m}(\omega)\,\,\mathring{H}_{n,m}^{\text{L,R}}(\omega)$
$\displaystyle S(\vec{x},\omega)=\sum_{n=0}^{\infty}\sum_{m=-n}^{n}\\!\\!4\pi\mathrm{i}^{n}\,\breve{S}_{n,m}(\omega)\,\,j_{n}\\!\left(\omega\frac{r}{c}\right)\,Y_{n,m}(\beta,\alpha)$ |  Eq. (1) and (2) — Incidence |  Eq. (29) | $\left(-4\pi\mathrm{i}^{n}\,\frac{\mathrm{i}}{\left(\omega\frac{R}{c}\right)^{2}}\frac{1}{h_{n}^{\prime\;\\!(2)}\\!\left(\omega\frac{R}{c}\right)}\right)^{-1}$ | $\displaystyle B^{\text{L,R}}(\omega)=\sum_{n=0}^{N}\sum_{m=-n}^{n}\\!\breve{S}_{n,m}(\omega)\,\,\mathring{H}_{n,m}^{\text{L,R}}(\omega)$
$\displaystyle S(\vec{x},\omega)=\sum_{n=0}^{\infty}\sum_{m=-n}^{n}\\!\\!4\pi\mathrm{i}^{-n}\,\breve{S}_{n,m}(\omega)\,\,j_{n}\\!\left(\omega\frac{r}{c}\right)\,Y_{n,m}(\beta,\alpha)$ |  Eq. (1) and (2) but with the sign of the exponent swapped — Incidence |  Eq. (29) | $\left(4\pi\mathrm{i}^{-n}\,\frac{\mathrm{i}}{\left(\omega\frac{R}{c}\right)^{2}}\frac{1}{h_{n}^{\prime\;\\!(1)}\\!\left(\omega\frac{R}{c}\right)}\right)^{-1}$ | $\displaystyle B^{\text{L,R}}(\omega)=\sum_{n=0}^{N}\sum_{m=-n}^{n}\\!\breve{S}_{n,m}(\omega)\,\,\mathring{H}_{n,m}^{\text{L,R}}(\omega)$
Table 1: Five typical combinations of definitions. The right two columns are a
consequence of the definitions in the left three columns. The fourth and fifth
rows both yield ambisonic signals $\breve{S}_{n,m}(\omega)$ that correspond to
the N3D ambisonic standard101010See
https://github.com/AppliedAcousticsChalmers/ambisonic-encoding.

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Note that $Z\in\partial^{u}D$ implies that the sequence
$(F^{-n}(Z))_{n=0}^{\infty}$ converges to the fixed point in $\partial^{u}D$
(that is $X$ for $b>0$, and $Y$ for $b<0$). Therefore, for every endpoint
$x\in E_{X}$, the sequence $(f^{-n}(x))_{n=0}^{\infty}$ converges to the fixed
point of $f$ that is related to the fixed point in $\partial^{u}D$.

In the following lemma, we show that the pre-image of any branch point from
$B_{X}$ always contains another branch point from $B_{X}$.

######  Lemma 7.6.

For any branch point $x\in B_{X}$ we have $f^{-1}(x)\cap B_{X}\neq\emptyset$.

###### Proof.

Let $x=[(s_{n})_{n=1}^{\infty}]\in B_{X}$. Then, by definition of the set
$B_{X}$, $\bigcap_{n=1}^{\infty}s_{n}=\alpha$, and $\alpha$ contains the
terminal point, say $\mathfrak{b}$, of a component denoted by
$K_{\mathfrak{b}}$ of $F^{-n}({\mathcal{K}})\cap D$, for some
$n\in{\mathbb{N}}_{0}$. Let us consider $F^{-1}(\alpha)$ and
$F^{-1}(K_{\mathfrak{b}})$.

If $F^{-1}(K_{\mathfrak{b}})\subset D$ then $F^{-1}(\alpha)\subset D$ and
hence $f^{-1}(x)$ contains only one point, denote it $x^{-1}$, and the
geometric realization of $x^{-1}=f^{-1}(x)$ is $F^{-1}(\alpha)$. Also
$F^{-1}(\mathfrak{b})\in F^{-1}(\alpha)$ and $F^{-1}(\mathfrak{b})$ is the
terminal point of $F^{-1}(K_{\mathfrak{b}})$, so $f^{-1}(x)$ is a branch
point.

If $F^{-1}(K_{\mathfrak{b}})\smallsetminus D\neq\emptyset$ then
$F^{-1}(\alpha)\cap D$ has at least two components and one of them, say
$\alpha^{\prime}$, contains the point $F^{-1}(\mathfrak{b})$ that is also the
terminal point of a component of $F^{-1}(K_{\mathfrak{b}})\cap D$. Therefore,
$\alpha^{\prime}$ is the geometric realization of one of the preimages of $x$.
Let us denote that preimage by $x^{-1}$. Then $x^{-1}$ is a branch point in
$B_{X}$. We can inductively define $x^{-n}$, for every $n\in{\mathbb{N}}$, and
the proof follows. ∎

Note that $\mathfrak{b}\in\partial^{u}D$ implies that the sequence
$(F^{-n}(\mathfrak{b}))_{n=0}^{\infty}$ converges to the fixed point in
$\partial^{u}D$. Therefore, for every branch point $x\in B_{X}$, the sequence
$(x^{-n})_{n=0}^{\infty}$ converges to the fixed point of $f$ that is related
to the fixed point in $\partial^{u}D$. .

######  Lemma 7.7.

For every $i\in\mathbb{Z}$ there exists $k\in{\mathbb{N}}_{0}$ such that
$\pi(S_{i}^{-k-n})\in E_{X}\cup B_{X}$, for all $n\in\mathbb{N}_{0}$.
Conversely, for every $r\in E_{X}\cup B_{X}$ there exist $i\in\mathbb{Z}$ and
$k\in{\mathbb{N}}_{0}$ such that $\pi(S_{i}^{-k})=r$.

###### Proof.

Since for $b>0$ we have $S_{i}^{-1}\in W^{u}\cap{\mathcal{K}}$, and for $b<0$
we have $S_{i}^{-1}\in W^{u}_{Y}\cap{\mathcal{K}}$, there exists
$N_{i}\in{\mathbb{N}}_{0}$ such that $S_{i}^{-k}\in\partial^{u}D\cap
F^{-k}({\mathcal{K}})$ for every $k\geq N_{i}$ and by Lemma 7.1, Lemma 7.5 and
Lemma 7.6, $\pi(S_{i}^{-k})\in E_{X}\cup B_{X}$, for every $k\geq N_{i}$.

Let $r\in E_{X}\cup B_{X}$. By definitions of the sets $E_{X}$ and $B_{X}$,
there exists $k\in{\mathbb{N}}_{0}$ such that the geometric realization of $r$
is either the initial point of a component of $F^{-k}({\mathcal{K}})\cap D$,
or it contains the terminal point of a component of $F^{-k}({\mathcal{K}})\cap
D$. In any case that point, denote it by $P$, belongs to
$F^{-k}({\mathcal{K}})\cap\partial^{u}D$. Hence, for $b>0$ we have $P^{k}\in
F^{k}(F^{-k}({\mathcal{K}})\cap\partial^{u}D)\subset W^{u}\cap{\mathcal{K}}$
and $W^{u}$ intersects ${\mathcal{K}}$ at $P^{k}$, and for $b<0$ we have
$P^{k}\in F^{k}(F^{-k}({\mathcal{K}})\cap\partial^{u}D)\subset
W^{u}_{Y}\cap{\mathcal{K}}$ and $W^{u}_{Y}$ intersects ${\mathcal{K}}$ at
$P^{k}$. Thus $P^{k}=S_{i}^{-1}$ for some $i\in{\mathbb{Z}}$ and
$\pi(S^{-1-k})=r$. ∎

###  7.4. Classification

Let $F_{i}:=F_{a_{i},b_{i}}:D_{i}\to D_{i}$, $i=1,2$, be two Hénon maps within
the Wang-Young parameter set, and let $\Lambda_{i}$ denote the attractor of
$F_{i}$. Let $X_{i},Y_{i}$ be the fixed points of $F_{i}$, where
$X_{i}\in\Lambda_{i}$, and for $b<0$ we have $Y_{i}\in\partial^{u}D_{i}$. Let
$S_{i,j}$, $j\in{\mathbb{Z}}$ ($j\in{\mathbb{N}}_{0}$, respectively), be the
turning points od $F_{i}$ for $b>0$ (the qusi-turning points of $F_{i}$ for
$b<0$, respectively). We analogously denote all ’items’ related to the map
$F_{i}$ by an index $i$. Let also $f_{1}:{\mathbb{T}}_{1}\to{\mathbb{T}}_{1}$
and $f_{2}:{\mathbb{T}}_{2}\to{\mathbb{T}}_{2}$ be the corresponding tree
maps, whose natural extensions are conjugate to $F_{1}$ and $F_{2}$
respectively.

######  Lemma 7.8.

Suppose that $H:D_{1}\to D_{2}$ is a conjugacy between $F_{1}$ and $F_{2}$.
Let $\gamma_{X_{i}}\in\Gamma_{i}$ be such that $X_{i}\in\gamma_{X_{i}}$. Then
there exists $k\in{\mathbb{Z}}$ such that
$F_{2}^{k}(H(\partial\gamma_{X_{1}}))=\partial\gamma_{X_{2}}$.

###### Proof.

Since $H$ is a conjugacy, $H(X_{1})=X_{2}$, $H(W^{u}_{1})=W^{u}_{2}$ and
$H(W^{s}_{1})=W^{s}_{2}$. Since homoclinic points are invariant for a
conjugacy, there exists $k\in{\mathbb{Z}}$ such that for every $i\geq k$, we
have $F_{2}^{i}(H(\gamma_{X_{1}}))\subseteq\gamma_{X_{2}}$.

Case 1. Let $b>0$. Let
$\\{E^{\prime}_{1}\\}=\gamma_{X_{1}}\cap[S_{1,0}^{2},S_{1,1}]^{u}$,
$\\{E^{\prime}_{2}\\}=\gamma_{X_{2}}\cap[S_{2,0}^{2},S_{2,1}]^{u}$, and as
before $E_{i}\in\partial\gamma_{X_{i}}$, see Figue 13. It is easy to see that
there are only two homoclinic orbits of $F_{i}$,
$\\{E_{i}^{j}:j\in{\mathbb{Z}}\\}$ and $\\{E^{\prime
j}_{i}:j\in{\mathbb{Z}}\\}$, such that
$(X_{i},P_{i})^{u}\cap(X_{i},P_{i})^{s}=\emptyset$ for
$P_{i}\in\\{E^{j}_{i},E^{\prime j}_{i}:j\in{\mathbb{Z}}\\}$ (recall that
$Q=Q^{0}$ for every point $Q$).

$S_{0}$$z_{0}$$S^{1}_{0}$$z_{1}$$S_{-1}$$z_{2}$$S^{3}_{0}$$S^{4}_{0}$$z_{-5}$$z_{-3}$$z_{-4}$$S_{2}$$S^{1}_{-1}$$S_{1}$$S^{2}_{0}$$z_{-1}$$z_{-2}$$X$$W^{u}$$W^{s}$$E_{1}^{-1}$$E_{1}$
$S_{0}$$z_{0}$$S^{1}_{0}$$z_{1}$$S_{-1}$$z_{2}$$S^{3}_{0}$$S^{4}_{0}$$z_{-5}$$z_{-3}$$z_{-4}$$S_{2}$$S^{1}_{-1}$$S_{1}$$S^{2}_{0}$$z_{-1}$$z_{-2}$$X$$W^{u}$$W^{s}$$E_{1}^{-1}$$E^{\prime}_{1}$

Figure 13. Positions of points $E_{1}$ and $E^{\prime}_{1}$.

Note that for the endpoint $E_{i}$ of $\gamma_{X_{i}}$ there exists a sequence
of homoclinic points in $\gamma_{X_{i}}$ that converges to $E_{i}$. On the
other hand, for the endpoint $E^{\prime}_{i}$ of $[X_{i},E^{\prime}_{i}]^{s}$
there exists no sequence of homoclinic points in $[X_{i},E^{\prime}_{i}]^{s}$
converging to $E^{\prime}_{i}$. Therefore, $H(\gamma_{X_{1}})=[X_{2},Q]^{s}$
implies $Q\in\\{E_{2}^{j}:j\in{\mathbb{Z}}\\}$ and the claim follows.

Case 2. Let $b<0$. Similarly as in Case 1, let
$G_{1}=\gamma_{X_{1}}\cap[z_{1,0}^{3},z_{1,0}^{1}]^{u}$,
$G_{2}=\gamma_{X_{2}}\cap[z_{2,0}^{3},z_{2,0}^{1}]^{u}$,
$G^{\prime}_{1}=\gamma_{X_{1}}\cap[z_{1,0}^{3},z_{1,1}^{1}]^{u}$ and
$G^{\prime}_{2}=\gamma_{X_{2}}\cap[z_{2,0}^{3},z_{2,1}^{1}]^{u}$, see Figure
14. It is easy to see that there are only two homoclinic orbits of $F_{i}$,
$\\{G_{i}^{j}:j\in{\mathbb{Z}}\\}$ and $\\{G^{\prime
j}_{i}:j\in{\mathbb{Z}}\\}$, such that
$(P_{i},P^{1}_{i})^{u}\cap(P_{i},P^{1}_{i})^{s}=\\{X_{i}\\}$, for
$P_{i}\in\\{G^{j}_{i},G^{\prime j}_{i}:j\in{\mathbb{Z}}\\}$.

$z^{1}_{-1}$$z^{1}_{-2}$$z^{1}_{-3}$$z^{1}_{0}$$z^{1}_{1}$$z^{1}_{-4}$$z_{0}^{4}$$z_{0}^{3}$$z_{-1}^{3}$$z_{-1}^{2}$$z_{0}^{2}$$z_{1}^{2}$$z_{2}$$z_{3}$$z_{6}$$z_{-1}$$z_{-2}$$z_{7}$$z_{-3}$$z_{0}$$z_{5}$$z_{4}$$z_{1}$$z_{-4}$$G_{1}$$X_{1}$$G_{1}^{1}$
$z^{1}_{-1}$$z^{1}_{-2}$$z^{1}_{-3}$$z^{1}_{0}$$z^{1}_{1}$$z^{1}_{-4}$$z_{0}^{4}$$z_{0}^{3}$$z_{-1}^{3}$$z_{-1}^{2}$$z_{0}^{2}$$z_{1}^{2}$$z_{2}$$z_{3}$$z_{6}$$z_{-1}$$z_{-2}$$z_{7}$$z_{-3}$$z_{0}$$z_{5}$$z_{4}$$z_{1}$$z_{-4}$$G^{\prime}_{1}$$X_{1}$$G^{\prime
1}_{1}$
$z^{1}_{-1}$$z^{1}_{-2}$$z^{1}_{-3}$$z^{1}_{0}$$z^{1}_{1}$$z^{1}_{-4}$$z_{0}^{4}$$z_{0}^{3}$$z_{-1}^{3}$$z_{-1}^{2}$$z_{0}^{2}$$z_{1}^{2}$$z_{2}$$z_{3}$$z_{6}$$z_{-1}$$z_{-2}$$z_{7}$$z_{-3}$$z_{0}$$z_{5}$$z_{4}$$z_{1}$$z_{-4}$$G_{1}^{2}$$X_{1}$$G_{1}^{3}$
$z^{1}_{-1}$$z^{1}_{-2}$$z^{1}_{-3}$$z^{1}_{0}$$z^{1}_{1}$$z^{1}_{-4}$$z_{0}^{4}$$z_{0}^{3}$$z_{-1}^{3}$$z_{-1}^{2}$$z_{0}^{2}$$z_{1}^{2}$$z_{2}$$z_{3}$$z_{6}$$z_{-1}$$z_{-2}$$z_{7}$$z_{-3}$$z_{0}$$z_{5}$$z_{4}$$z_{1}$$z_{-4}$$G^{\prime\prime}_{1}$$X_{1}$$G^{\prime\prime
1}_{1}$

Figure 14. Top: Positions of points $G_{1}$ and $G^{\prime}_{1}$. Bottom left:
Position of a point, $G_{1}^{2}$, from the orbit of $G_{1}$, and hence it
satisfies the property
$(G^{2}_{1},G^{3}_{1})^{u}\cap(G^{2}_{1},G^{3}_{1})^{s}=\\{X_{1}\\}$. Bottom
right: Position of a point, $G^{\prime\prime}_{1}$, that is not from the orbit
of $G_{1}$, and therefore it does not satisfy the property
$(G^{\prime\prime}_{1},G^{\prime\prime
1}_{1})^{u}\cap(G^{\prime\prime}_{1},G^{\prime\prime 1}_{1})^{s}=\\{X_{1}\\}$.

Let us denote by $A^{j}_{i}$, the region bounded by
$[G^{j}_{i},G^{j+1}_{i}]^{u}\cup[G^{j}_{i},G^{j+1}_{i}]^{s}$, and by
$A^{\prime j}_{i}$, the region bounded by $[G^{\prime j}_{i},G^{\prime
j+1}_{i}]^{u}\cup[G^{\prime j}_{i},G^{\prime j+1}_{i}]^{s}$. Note that
$\gamma_{i}\cap\mathop{\mathrm{missing}}{Int}A_{i}=\emptyset$, but
$\gamma_{i}\cap\mathop{\mathrm{missing}}{Int}A^{\prime}_{i}\neq\emptyset$ and
in that intersection there exists a sequence of homoclinic points that
converges to $G^{\prime}_{i}$. Therefore,
$H([G_{1},G^{1}_{1}]^{s})=[Q,Q^{1}]^{s}$ implies
$Q\in\\{G_{2}^{j}:j\in{\mathbb{Z}}\\}$ and the claim follows. ∎

######  Proposition 7.9.

Suppose that $H:D_{1}\to D_{2}$ is a conjugacy between $F_{1}$ and $F_{2}$.
Then there exist $k\in{\mathbb{Z}}$, and a conjugacy
$h:{\mathbb{T}}_{1}\to{\mathbb{T}}_{2}$ between $f_{1}$ and $f_{2}$, such that
$h\circ\pi_{1}\circ F_{1}^{-n}(P)=\pi_{2}\circ F_{2}^{-n+k}\circ H(P)$, for
every $n\in{\mathbb{N}}$ and $P\in\Lambda_{1}$ for $b>0$, and
$P\in\Lambda_{1}\cup W^{u}_{1,Y}$ for $b<0$. Moreover, $h$ preserves stems and
their levels; i.e., a stem of level $n\in{\mathbb{N}}_{0}$ of
${\mathbb{T}}_{1}$ is mapped to a stem of level $n$ of ${\mathbb{T}}_{2}$, and
the natural order of stems is preserved.

###### Proof.

By Lemma 7.8, there exists $k\in{\mathbb{Z}}$ such that
$F_{2}^{k}(H(\partial\gamma_{X_{1}}))=\partial\gamma_{X_{2}}$. Since
$F_{2}^{k}\circ H$ is also a conjugacy between $F_{1}$ and $F_{2}$, without
loss of generality we assume that $H(\gamma_{X_{1}})=\gamma_{X_{2}}$, and
moreover, the homoclinic (heteroclinic) points of $\gamma_{X_{1}}$ are mapped
onto the homoclinic (heteroclinic) points of $\gamma_{X_{2}}$ and their order
is preserved. Note that for $b>0$,
$H([E_{1}^{-1},X_{1}]^{u})=[E_{2}^{-1},X_{2}]^{u}$, the homoclinic points of
$[E_{1}^{-1},X_{1}]^{u}$ are mapped onto the homoclinic points of
$[E_{2}^{-1},X_{2}]^{u}$ and their order is preserved, as well as for
$F_{1}^{n}([E_{1}^{-1},X_{1}]^{u})$ for every $n\in{\mathbb{N}}$. Also, for
$b<0$, $H([E_{1}^{-1},E_{1}]^{u}_{Y_{1}})=[E_{2}^{-1},E_{2}]^{u}_{Y_{2}}$, the
heteroclinic points of $[E_{1}^{-1},E_{1}]^{u}_{Y_{1}}$ are mapped onto the
heteroclinic points of $[E_{2}^{-1},E_{2}]^{u}_{Y_{2}}$ and their order is
preserved, as well as for $F_{1}^{n}([E_{1}^{-1},E_{1}]^{u}_{Y_{1}})$ for
every $n\in{\mathbb{N}}$.

Let $\gamma_{1}\in\Gamma_{1}$. Since $\gamma_{1}\subset W^{s}_{1}$ and
$\partial\gamma_{1}$ are homoclinic points, there exists
$n\in{\mathbb{N}}_{0}$ such that
$\beta_{1}:=F_{1}^{n}(\gamma_{1})\subseteq\gamma_{X_{1}}$. Let
$\beta_{2}:=H(\beta_{1})\subseteq\gamma_{X_{2}}$. Then
$H(\gamma_{1})=F_{2}^{-n}\circ H\circ
F_{1}^{n}(\gamma_{1})=F_{2}^{-n}(\beta_{2})=:\gamma_{2}\in\Gamma_{2}$, and
hence $H$ induces a homeomorphism $\widetilde{H}:\Gamma_{1}\to\Gamma_{2}$,
$\widetilde{H}(\gamma_{1})=\gamma_{2}$, where $\gamma_{2}$ is defined as
above.

Since $\Gamma_{i}$ is dense in $D_{i}$, $\widetilde{H}$ induces a semi-
conjugacy $h:{\mathbb{T}}_{1}\to{\mathbb{T}}_{2}$ between $f_{1}$ and $f_{2}$
such that $\psi h\psi^{-1}|_{\Gamma_{1}}=\widetilde{H}$. Recall, $\psi$
assigns to each equivalence class $[(s_{n})_{n=1}^{\infty}]$ in ${\mathbb{T}}$
a continuum $\bigcap_{n=1}^{\infty}s_{n}$, which is a geometric realization of
that class,

$\psi([(s_{n})_{n=1}^{\infty}])=\bigcap_{n=1}^{\infty}s_{n}.$

Let us prove that $h$ is an injection, and therefore a conjugacy. Let
$x=[(s_{n})_{n=1}^{\infty}],x^{\prime}=[(s^{\prime}_{n})_{n=1}^{\infty}]\in{\mathbb{T}}_{1}$
and $x\neq x^{\prime}$. Then there exists $N\in{\mathbb{N}}$ such that
$s_{k}\cap s^{\prime}_{m}=\emptyset$ for all $k,m\geq N$. Thus, $H(s_{k})\cap
H(s^{\prime}_{m})=\emptyset$, for all $k,m\geq N$, and
$h(x)=[(H(s_{n}))_{n=1}^{\infty}]\neq[(H(s^{\prime}_{n}))_{n=1}^{\infty}]=h(x^{\prime})$.
Therefore, $h$ is a conjugacy. Moreover, it follows from construction that $h$
preserves stems, that is a stem of level $n\in{\mathbb{N}}_{0}$ of
${\mathbb{T}}_{1}$ is mapped to a stem of level $n$ of ${\mathbb{T}}_{2}$, and
the natural order of stems is preserved. ∎

Note that the conjugacy
$h:({\mathbb{T}}_{1},f_{1})\to({\mathbb{T}}_{2},f_{2})$ induces a conjugacy

$\widetilde{h}:(\displaystyle\lim_{\longleftarrow}({\mathbb{T}}_{1},f_{1}),\sigma_{1})\to(\displaystyle\lim_{\longleftarrow}({\mathbb{T}}_{2},f_{2}),\sigma_{2}),$

given by
$\widetilde{h}(t_{1},\dots,t_{n},\dots)=(h(t_{1}),\dots,h(t_{n}),\dots)$, such
that $\widetilde{h}\circ\Pi_{1}=\Pi_{2}\circ H$.

######  Theorem 7.10.

Suppose that $H:D_{1}\to D_{2}$ is a conjugacy between two Hénon maps $F_{1}$
and $F_{2}$. Then there exists $k\in{\mathbb{Z}}$ such that
$H(S_{1,i}^{n})=S_{2,i}^{k+n}$, for every $i,n\in{\mathbb{Z}}$.

###### Proof.

Let $H$ be a conjugacy between the Hénon maps $F_{1}$ and $F_{2}$ such that
$H(\gamma_{X_{1}})=\gamma_{X_{2}}$, as in the proof of Proposition 7.9. Then
$h$ maps every endpoint (every branch point) of ${\mathbb{T}}_{1}$ to the
corresponding endpoint (branch point) of ${\mathbb{T}}_{2}$.

Recall that for $b>0$ we have
$H|_{\Lambda_{1}}=\Pi_{2}^{-1}\circ\widetilde{h}\circ\Pi_{1}$, and for $b<0$
we have $H|_{\Lambda_{1}\cup
W^{u}_{1,Y}}=\Pi_{2}^{-1}\circ\widetilde{h}\circ\Pi_{1}$. Therefore, for every
$i\in{\mathbb{Z}}$,

$H(S_{1,i})=\Pi_{2}^{-1}(h\circ\pi_{1}(S_{1,i}),h\circ\pi_{1}\circ
F_{1}^{-1}(S_{1,i}),\dots,h\circ\pi_{1}\circ F_{1}^{-n}(S_{1,i}),\dots).$

By Lemma 7.7 there exists $n\in{\mathbb{N}}$ such that $r_{n}:=\pi_{1}\circ
F_{1}^{-n}(S_{1,i})\in E_{X_{1}}\cup B_{X_{1}}$. If $r_{n}\in E_{X_{1}}$, by
Lemma 7.5, $r_{n+j}:=\pi_{1}\circ F_{1}^{-n-j}(S_{1,i})=f_{1}^{-j}(r_{n})\in
E_{X_{1}}$ for every $j\in{\mathbb{N}}$. If $r_{n}\in B_{X_{1}}$, by Lemma
7.6, $r_{n+j}:=\pi_{1}\circ F_{1}^{-n-j}(S_{1,i})\in f_{1}^{-j}(r_{n})\cap
B_{X_{1}}$ for every $j\in{\mathbb{N}}$. By Proposition 7.9, $h$ preserves
stems and ordering of stems, and hence $h$ maps the endpoint of $E_{X_{1}}$ of
a stem of ${\mathbb{T}}_{1}$ to the endpoint of $E_{X_{2}}$ of the
corresponding stem of ${\mathbb{T}}_{2}$, and the branch points of $B_{X_{1}}$
of a stem of ${\mathbb{T}}_{1}$ to the corresponding branch points of
$B_{X_{2}}$ of a corresponding stem of ${\mathbb{T}}_{2}$. Thus, $\pi_{2}\circ
F_{2}^{-n-j}\circ H(S_{1,i})=h\circ\pi_{1}\circ F_{1}^{-n-j}(S_{1,i})\in
E_{X_{2}}\cup B_{X_{2}}$, for every $j\in{\mathbb{N}}$. By Lemma 7.7 and since
we assumed that $H(\gamma_{X_{1}})=\gamma_{X_{2}}$, it follows that
$F_{2}^{-n-j}\circ H(S_{1,i})=F_{2}^{-n-j}(S_{2,i})$ for every
$j\in{\mathbb{N}}$. Therefore $H(S_{1,i})=S_{2,i}$, for every
$i\in{\mathbb{Z}}$. In general (without the assumption that
$H(\gamma_{X_{1}})=\gamma_{X_{2}}$) there exists $k\in{\mathbb{Z}}$ such that
$H(S_{1,i})=S_{2,i}^{k}$ and consequently $H(S_{1,i}^{n})=S_{2,i}^{k+n}$, for
every $i,n\in{\mathbb{Z}}$. ∎

The above result guarantees that for $b>0$ the orbits of basic critical points
of $F_{1}$ are mapped onto the orbits of basic critical points of $F_{2}$. For
$b<0$ this is guaranteed by the following corollary.

######  Corollary 7.11.

Suppose that $H:D_{1}\to D_{2}$ is a conjugacy between two orientation-
preserving Hénon maps $F_{1}$ and $F_{2}$. Then there exists
$k\in{\mathbb{Z}}$ such that $H(z_{1,i})=z_{2,i}^{k}$, for every
$i\in{\mathbb{Z}}$.

###### Proof.

Without loss of generality we suppose that $H(\gamma_{X_{1}})=\gamma_{X_{2}}$,
and we want to prove that $H(z_{1,j})=z_{2,j}$ for every $j\in{\mathbb{Z}}$.
Recall that for $b<0$ we have that $S_{i,j}=F_{i}(z^{\prime}_{i,j})$ for
$i=1,2$ and $j\in{\mathbb{N}}_{0}$. Moreover, for every basic critical point
$z_{i,j}$, $i=1,2$, $j\in{\mathbb{Z}}$, there exists a sequence of quasi-
critical points $(z^{\prime}_{i,j_{n}})_{n\in{\mathbb{N}}}$,
$j_{n}\in{\mathbb{N}}_{0}$, that converges to $z_{i,j}$. Also, if a sequence
of quasi-critical points converges, then its limit point is a critical point.
Since by the proof of Theorem 7.10 we have that $H(S_{1,j_{n}})=S_{2,j_{n}}$,
for every $n\in{\mathbb{N}}$, and $H(W^{u}_{1})=W^{u}_{2}$, it follows that
for every $j\in{\mathbb{Z}}$ there exists $k\in{\mathbb{Z}}$ such that
$H(z_{1,j})=z_{2,k}$.

Let us denote by $\zeta_{i,j}$, $i=1,2$, $j\in{\mathbb{Z}}$, the (connected)
components of $W^{u}_{i}\cap\Omega_{i}$, that are indexed in the following
way: $X_{i}\in\zeta_{i,0}$, $\zeta_{i,j}\subset W^{u+}_{i}$,
$\zeta_{i,-j}\subset W^{u-}_{i}$, for every $j\in{\mathbb{N}}$, and
$\zeta_{i,j},\zeta_{i,k}$ are consecutive (there are no other $\zeta_{i,m}$
between them) if and only if $|k-j|=1$, $j,k\in{\mathbb{Z}}$. Note that
$z_{i,-j}\in\zeta_{i,j}$ for $j=-1,0,1$, but in general if
$z_{i,-j}\in\zeta_{i,k}$, then $|j|\leq|k|$, since there are $\zeta_{i,m}$
that do not contain any basic critical point. Since $H$ is a conjugacy with
$H(\gamma_{X_{1}})=\gamma_{X_{2}}$, it follows that
$H(\zeta_{1,j})=\zeta_{2,j}$ for every $j\in{\mathbb{Z}}$. Therefore,
$H(z_{1,0})=z_{2,0}$ and consequently $H(z_{1,j})=z_{2,j}$ for every
$j\in{\mathbb{Z}}$. ∎

######  Corollary 7.12.

Suppose that $H:D_{1}\to D_{2}$ is a conjugacy between two Hénon maps $F_{1}$
and $F_{2}$. Then the sets of kneading sequences of $F_{1}$ and $F_{2}$
coincide.

###### Proof.

Without loss of generality, we assume that $H$ is a conjugacy such that
$H(\gamma_{X_{1}})=\gamma_{X_{2}}$. Then, by Theorem 7.10 and Corollary 7.11,
it follows that $H(z_{1,i})=z_{2,i}$, $i\in{\mathbb{Z}}$. We want to prove
that $z_{1,i}^{n}\in D_{1}^{r}$ if and only if $H(z_{1,i}^{n})=z_{2,i}^{n}\in
D_{2}^{r}$, for every $n,i\in{\mathbb{Z}}$, $n\neq 0$.

Let us first assume by contradiction that there exist $i\in{\mathbb{Z}}$ and
$n\in{\mathbb{N}}$ such that $z_{1,i}^{n}\in D_{1}^{l}$, $z_{2,i}^{n}\in
D_{2}^{r}$, and let $n$ and $i$ be such that $z_{1,i}^{n}$ is the closest
point to $X_{1}$ with that property (that is, the arc
$[X_{1},z_{1,i}^{n})\subset W^{u}_{1}$ does not contain any point
$z_{1,j}^{k}$ with that property). Let $\ell_{1}=[X_{1},z_{1,i}^{n}]\subset
W^{u}_{1}$ and $\ell_{2}=[X_{2},z_{2,i}^{n}]\subset W^{u}_{2}$ be arcs. Then
$H(\ell_{1})=\ell_{2}$, and all the turning and post-turning points of
$\ell_{1}$ are mapped onto all the turning and post-turning points of
$\ell_{2}$ and their order is preserved. Let $P\in\ell_{1}$ be the closest
(post-)turning point to $z_{1,i}^{n}$, so $H(P)\in\ell_{2}$ is the closest
(post-)turning point to $z_{2,i}^{n}$, and $P$ and $H(P)$ lie in the same
component of $D_{1}$ and $D_{2}$, respectively. Then one of the arcs
$[P,z_{1,i}^{n}]\subset W^{u}_{1}$, $[H(P),z_{2,i}^{n}]\subset W^{u}_{2}$
intersects, and the other one does not intersect the critical locus. Without
loss of generality, we assume that $[P,z_{1,i}^{n}]$ intersects
${\mathcal{K}}_{1}$ (in the other case consider $H^{-1}$), and let denote by
$Q$ the intersection point. Then $H\circ F_{1}(\ell_{1})=F_{2}(\ell_{2})$, and
hence $H([F_{1}(P),z_{1,i}^{n+1}])=[F_{2}(H(P)),z_{2,i}^{n+1}]$. Moreover,
$[F_{1}(P),z_{1,i}^{n+1}]$ contains a turning point $F_{1}(Q)$, and
$[F_{2}(H(P)),z_{2,i}^{n+1}]$ does not contain any turning point, a
contradiction with $H(z_{1,i})=z_{2,i}$, $i\in{\mathbb{Z}}$. Therefore,
$z_{1,i}^{n}\in D_{1}^{r}$ if and only if $H(z_{1,i}^{n})=z_{2,i}^{n}\in
D_{2}^{r}$, for every $n\in{\mathbb{N}}$ and $i\in{\mathbb{Z}}$.

Let us now consider $z_{1,i}^{-n}$ and $z_{2,i}^{-n}$ for every
$n\in{\mathbb{N}}$ and $i\in{\mathbb{Z}}$. For every $i\in{\mathbb{Z}}$,
$z_{1,i}$ and $z_{2,i}$ are basic critical points of $F_{1}$ and $F_{2}$
respectively, and lie in ${\mathcal{K}}_{1}$ and ${\mathcal{K}}_{2}$
respectively. Let $n\in{\mathbb{N}}$. Then $z_{1,i}^{-n}$ lies in the basic
arc $[z_{1,j}^{k},z_{1,m}^{l}]^{u}$ for some $j,m\in{\mathbb{Z}}$ and at least
one of $k,l$ is positive, say $k>0$. Since
$z_{2,i}^{-n}\in[z_{2,j}^{k},z_{2,m}^{l}]^{u}$ and, as we have already proved,
$z_{1,j}^{k}\in D_{1}^{r}$ if and only if $z_{2,j}^{k}\in D_{2}^{r}$, it
follows that $z_{1,i}^{-n}\in D_{1}^{r}$ if and only if $z_{2,i}^{-n}\in
D_{2}^{r}$. ∎

The proof of Theorem 1.4 follows now from Corollary 7.12 and Theorem 1.3.

##  8\. The Lozi maps

We note that all the results presented here for the Hénon attractors with the
Wang-Young parameters, hold also for the Lozi maps within the Misiurewicz
parameter set $\mathcal{M}=\\{(a,b)\in{\mathbb{R}}^{2}:b>0,\ a\sqrt{2}-b>2,\
2a+b<4\\}$. For the parameters in this set, Misiurewicz [29] (see also [31])
has shown the existence of strange attractors, with topological mixing. Also,
the kneading theory (for the Lozi maps) has been developed by Misiurewicz and
the second author in [30], where analogues of Theorems 1.1 and 1.2 are proven.
The analogue of Theorem 1.4 is obtained as follows. One uses the following
dictionary, which can be extracted from [11].

Dictionary

Hénon maps from [2] |  Lozi maps from [29]
---|---
$D$ |  $\Delta$
${\mathcal{K}}$ |  $y$-axis$\,\cap\Delta$
$\Omega$ |  $T_{0}$

To prove that two Lozi maps $F_{1}$ and $F_{2}$ with the parameters in
$\mathcal{M}$ are topologically conjugate if their sets of kneading sequences
coincide, $\mathfrak{K}_{1}=\mathfrak{K}_{2}$, small additional effort is
needed, since for the Lozi map $F$ from $\mathcal{M}$, we do not know whether
a point $P\in\Lambda_{F}\smallsetminus W^{u}$ has at most two itineraries.

First, using the entries from the dictionary, one can apply the proof of
Proposition 5.6 to prove the following proposition:

######  Proposition 8.1.

Let $F_{1}$ and $F_{2}$ be two Lozi maps such that
$\mathfrak{K}_{1}=\mathfrak{K}_{2}$. Let
$\overline{p},\overline{q}\in\Sigma_{F_{1}}$ be two different elements such
that $\pi_{1}(\overline{p})=\pi_{1}(\overline{q})$, and there exists only one
$j\in{\mathbb{Z}}$ with $p_{j}\neq q_{j}$ ($p_{i}=q_{i}$ for all integers
$i\neq j$). Then $\pi_{2}(\overline{p})=\pi_{2}(\overline{q})$.

Now we consider the possibility that a point $P$ has more than two
itineraries.

######  Proposition 8.2.

Let $F_{1}$ and $F_{2}$ be two Lozi maps such that
$\mathfrak{K}_{1}=\mathfrak{K}_{2}$. Let $n\in{\mathbb{N}}$. Suppose that
$\overline{p},\overline{q}\in\Sigma_{F_{1}}$ are such that
$\pi_{1}(\overline{p})=\pi_{1}(\overline{q})$, and there exist
$j_{1},\dots,j_{n}\in{\mathbb{Z}}$ such that $p_{j_{i}}\neq q_{j_{i}}$ and
$p_{k}=q_{k}$ for all integers $k\neq j_{i}$ and for every $i=1,\dots,n$. Then
$\pi_{2}(\overline{p})=\pi_{2}(\overline{q})$.

###### Proof.

Let $P=\pi_{1}(\overline{p})$. By the assumptions of the lemma,
$F_{1}^{j_{i}}(P)\in y$-axis for $i=1,\dots,n$. Therefore, the set
$\pi^{-1}_{1}(P)$ contains all itineraries of the form

$\overleftarrow{p}_{j_{1}-1}\pm p_{j_{1}+1}\dots p_{j_{2}-1}\pm
p_{j_{2}+1}\dots p_{j_{n}-1}\pm\overrightarrow{p}_{j_{n}+1},$

where $p_{k}$, $k\neq j_{i}$, are coordinates of the itinerary $\overline{p}$,
and for the $j_{i}$-th coordinates we can choose $+$ or $-$ in an arbitrary
way.

Now, the proof follows by induction and Lemma 8.1. ∎

######  Proposition 8.3.

Let $F_{1}$ and $F_{2}$ be two Lozi maps such that
$\mathfrak{K}_{1}=\mathfrak{K}_{2}$. Suppose that
$\overline{p},\overline{q}\in\Sigma_{F_{1}}$ are such that
$\pi_{1}(\overline{p})=\pi_{1}(\overline{q})$, and there is a sequence
$(j_{i})_{i\in{\mathbb{Z}}}\subset{\mathbb{Z}}$, such that $p_{j_{i}}\neq
q_{j_{i}}$ and $p_{k}=q_{k}$ for all integers
$k\notin(j_{i})_{i\in{\mathbb{Z}}}$, then
$\pi_{2}(\overline{p})=\pi_{2}(\overline{q})$.

###### Proof.

Let us assume that there exist $\overline{p},\overline{q}\in\Sigma_{F_{1}}$
that satisfy the assumptions of the lemma. Let $P=\pi_{1}(\overline{p})$. By
the assumptions, $F_{1}^{j_{i}}(P)\in y$-axis for $i\in{\mathbb{Z}}$.
Therefore, the set $\pi^{-1}_{1}(P)$ contains all itineraries of the form

$\dots p_{j_{-1}-1}\pm p_{j_{-1}+1}\dots p_{j_{0}-1}\pm p_{j_{0}+1}\dots
p_{j_{1}-1}\pm p_{j_{1}+1}\dots,$

where $p_{k}$, $k\neq j_{i}$, are coordinates of the itinerary $\overline{p}$,
and for the $j_{i}$-th coordinates we can choose $+$ or $-$ in an arbitrary
way.

Let $j_{i}$ be chosen such that the coordinate $j_{0}$ is the closest one to
the decimal point (if there are two such indices take any of them), and
$j_{-k-1}<j_{-k}<j_{0}<j_{k}<j_{k+1}$ for every $k\in{\mathbb{N}}$ (as it is
already shown in the sequence above). Let us consider a sequence
$(\overline{q}^{n})_{n\in{\mathbb{N}}_{0}}$,
$\overline{q}^{n}\in\pi_{1}^{-1}(P)$, such that $\overline{p}$ and
$\overline{q}^{0}$ disagree only at $j_{0}$ coordinate, and for every
$n\in{\mathbb{N}}_{0}$, $\overline{q}^{n}$ and $\overline{q}^{n+1}$ disagree
only at $j_{|n|}$ coordinates. By Proposition 8.2,
$\pi_{2}(\overline{p})=\pi_{2}(\overline{q}^{n})$ for every
$n\in{\mathbb{N}}_{0}$. Note that the sequence $(\overline{q}^{n})_{n}$
converges to $\overline{q}$. Since $\pi_{2}$ is continuous,
$\pi_{2}(\overline{p})=\pi_{2}(\overline{q})$, as claimed in the proposition.
∎

Now again, using the entries from the dictionary, one introduces the
equivalence relation on the space of symbolic sequences, to get the following
theorem.

######  Theorem 8.4.

Two Lozi maps $F_{1}$ and $F_{2}$ with the parameters in $\mathcal{M}$ are
topologically conjugate on their strange attractors if their sets of kneading
sequences coincide, $\mathfrak{K}_{1}=\mathfrak{K}_{2}$.

The results of Section 6 for Lozi maps are proven in [30]. Finally, for the
results of Section 7 one uses the dictionary again, and the results in [11],
and almost verbatim copies the proofs of all the results in that section to
get the following.

######  Theorem 8.5.

Suppose that $H:\Delta_{1}\to\Delta_{2}$ is a conjugacy between two Lozi maps
$F_{1}$ and $F_{2}$ with parameters in $\mathcal{M}$, and with turning points
$\\{S_{1,i}:i\in{\mathbb{Z}}\\}$ and $\\{S_{2,i}:i\in{\mathbb{Z}}\\}$
respectively. Then there exists $k\in{\mathbb{Z}}$ such that
$H(S_{1,i}^{n})=S_{2,i}^{k+n}$, for every $i,n\in{\mathbb{Z}}$.

That gives the following result for the Lozi maps.

######  Corollary 8.6.

Suppose that $H:\Delta_{1}\to\Delta_{2}$ is a conjugacy between two Lozi maps
$F_{1}$ and $F_{2}$ with parameters in ${\mathcal{M}}$. Then the sets of
kneading sequences of $F_{1}$ and $F_{2}$ coincide.

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Jan P. Boroński
Faculty of Mathematics and Computer Science
Jagiellonian University in Kraków
ul. Łojasiewicza 6, 30-348 Kraków, Poland
– and –
National Supercomputing Centre IT4Innovations
IRAFM, University of Ostrava
30\. dubna 22, 70103 Ostrava, Czech Republic

<EMAIL_ADDRESS>
https://matinf.uj.edu.pl/en_GB/pracownicy/wizytowka?person_id=Jan_Boronski

Sonja Štimac
Department of Mathematics
Faculty of Science, University of Zagreb
Bijenička 30, 10 000 Zagreb, Croatia

<EMAIL_ADDRESS>
https://web.math.pmf.unizg.hr/~sonja/