# Invariant measures of torus piecewise isometries

Michael Blank  Institute for Information Transmission Problems RAS (Kharkevich
Institute);National Research University “Higher School of Economics”; e-mail:
<EMAIL_ADDRESS>
(September 2, 2021)

###### Abstract

We study measure-theoretical aspects of torus piecewise isometries. Not much
is known about this type of dynamical systems, except for the special case of
one-dimensional interval exchange mappings. The last case is fundamentally
different from the general situation in the presence of an invariant measure
(Lebesgue measure), which helps a lot in the analysis. Due to the absence of
good methods of analysis of general systems with discontinuities, even the
existence of invariant measures of the torus piecewise isometries was an open
question. We establish sufficient conditions for the existence/absence of
invariant measures for this class of systems. Technically, our results are
based on the approximation of the maps under study by weakly periodic ones.

2010 Mathematics Subject Classification. Primary: 37A05; Secondary: 28D05,
28D15, 37B05.
Key words and phrases. Piecewise isometry, invariant measure, semigroup,
periodic point.

##  1 Introduction

By now, we have learned reasonably well how to study hyperbolic (locally
expanding/contracting or both) chaotic dynamical systems, thanks to a large
extent to the development of the so called operator approach. Contrary to this
not much is known about piecewise isometries, except for a special case of
one-dimensional interval exchange transformations (IET) and a few similar very
exceptional multidimensional systems. It is worth noting that IETs are
fundamentally different from the general situation in the clear presence of an
invariant measure (Lebesgue measure), which helps a lot in the analysis.

While the IET represent mainly pure theoretical constructions, the general
piecewise isometries appear naturally in various applications like
contemporary methods of machine learning, some piecewise smooth physical
models (in particular Fermi-Ulam models), etc. (see, e.g. [9] for applications
in engineering, and further references about other possibilities therein).
Indeed, local translations of an IET disrupt its structure completely, while
preserve the class of general piecewise isometries. Note also that the maps
from this class in general are non-invertible, which adds possibilities for
applications for “real life” modeling, but also additional problems for their
analysis.

In general from a measure-theoretical point of view one of the first steps in
the analysis of a dynamical system is the study of its invariant measures. In
some cases one can easily find a “good” invariant measure (e.g., for IETs), or
to prove its existence if the map, defining the dynamical system, is
continuous or satisfies some special properties (e.g., piecewise expanding).
For systems with singularities (e.g., discontinuities) even the question about
the existence of an invariant measure might become a difficult problem. The
class of piecewise isometries represents an example of the latter sort. The
point is that the methods (e.g., operator approach or symbolic dynamics) well
established for other types of chaotic dynamical systems do not work in this
setting and one needs to look for new approaches, one of which will be
proposed in the present paper.

Let us start with a simple example of an isometry in ${\mathbb{R}}^{2}$
defined by the map $\tilde{T}x:={\cal R}{x}+b$, where $x,b\in{\mathbb{R}}^{2}$
are vectors and ${\cal R}$ is a rotation matrix. Applying the same map to the
2-dimensional torus $X$ we get $Tx:=\tilde{T}x\,({\rm mod\ }1)\,$, which is no
longer a global isometry. Moreover, the map $T$ is not a bijection and has
both lines of discontinuities and regions of points having no preimages. On
the other hand, one can easily make a partition of $X$ into a finite number of
regions, such that the restriction of $T$ to each region is an isometry (see
Section 7 for details). This is a basic model for our study. Some nontrivial
invariant sets for such a map are demonstrated at Fig. 1. We are interested in
asymptotic properties of the dynamics of piecewise isometric torus maps like
the one shown in this figure.

$0$$1$$1$$x\to{\cal R}x+b$ mod1${\cal R}={\cal
R}(\varphi),~{}\varphi=\frac{\pi}{e},~{}b=0$ Figure 1: Invariant sets of the
torus rotation by the angle $\pi/e$

Recall that an isometry is a map which preserves distances between all points
in the phase space. Thus a piecewise isometry (PWI) is a map $T$ from a space
$X$ equipped with a special partition $\\{X_{i}\\}$ into itself such that the
restriction of $T$ to each element of the partition preserves distances (i.e.
is a true isometry). An example of a torus PWI is demonstrated at Fig. 2.

Naturally, the list of available isometries sensitively depends on the choice
of a metric. In particular, under the discrete metric each bijection turns out
to be an isometry. We will consider a natural family of translational
invariant metrics
$\tilde{\rho}_{p}(x,y):=(\sum_{i}|x_{i}-y_{i}|^{p})^{\frac{1}{p}},~{}p\geq 1$,
where $x_{i},y_{i}$ are the coordinates of the points $x,y\in
X\in{\mathbb{R}}^{d}$. The Euclidean metric corresponds to
$\tilde{\rho}_{2}(\cdot,\cdot)$, while the uniform metric to
$\tilde{\rho}_{\infty}(\cdot,\cdot)$. The corresponding family of torus
metrics is defined as
$\rho_{p}(x,y):=\min_{z\in{\mathbb{Z}}^{d}}\tilde{\rho}_{p}(x-z,y)$.

$0$$1$$1$$TX_{1}$$X_{1}$$X_{2}$$X_{3}$$X_{4}$$X_{5}$$X=\sqcup_{i=1}^{5}X_{i}$$X_{i}\to
TX_{i}$$T_{i}:=T_{|X_{i}}$ – isometry Figure 2: A special partition for a
torus isometry. One of the elements of the partition and its image are marked
in yellow and green colors correspondingly.

The paper is organized as follows. In Section 2 we give a brief historical
description of results related to piecewise isometries and motivations of the
present research. Section 3 is dedicated to basic definitions and the
formulation of the main result – Theorem 3.1. Examples demonstrating the
possibility of the absence of invariant measures for PWI and the focusing
phenomenon (due to which this happens) will be described in Section 4. In
Section 5 we introduce and study weakly periodic maps and semigroups. In
Section 6 we apply the approximation by weakly periodic maps to prove Theorem
3.1. Finally in Section 7 we study a seemingly simple example of a PWI: torus
restriction of a plane rotation. Note that this situation is excluded from
Theorem 3.1 by the extendability assumption.

Acknowledgments. The author is grateful to Gregory Olshansky and Leonid
Polterovich for useful discussions.

##  2 Historical remarks

We start this short review with the discussion of known results related to
“bijective” PWI $T:X\to X$, when the union of the images under $T$ of the
elements of the special partition $\\{X_{i}\\}$ covers the entire phase space
$X$. Since in the multidimensional setting ($d>1$) this situation is rather
exotic, we will discuss only the one-dimensional case, known in the literature
as interval exchange transformations (IET). Quotes marks in the word
“bijective” emphasize a problem with the boundary points of the elements of
the partition $\Gamma:=\cup_{i}\partial{X_{i}}$. In fact, the authors either
assume that either $TX_{i}\cap TX_{j}\in T(\Gamma)$ for $i\neq j$ (may
intersect only on boundary points), or instead of the partition consider the
covering $\\{{\rm Clos}{X_{i}}\\}$. The former case leads to an
endomorphism,111Except for orientation preserving PWI with a very special
structure of the partition, consisting of semi-closed intervals of type
$X_{i}:=[a_{i},a_{i+1})$. while the latter to a multi-valued map, because the
map is not uniquely defined at $\Gamma$. This is especially unpleasant because
under the action of a multi-valued map a total measure of the phase space may
grow, which cannot happen for a conventional single-valued map. Fortunately,
in the case of IET for any version of the action on boundary points the
Lebesgue measure is preserved under dynamics. Therefore, since $\Gamma$ is the
set of zero Lebesgue measure, one does not care much about these
peculiarities.

Analysis of IETs is a very popular field of research and the list of
publications related to them is very long and still actively growing.
Therefore we mention only some basic publications in a historical order and
refer the reader to look for further reference therein. The first results and
important constructions related to orientation preserving IET were described
in V.I. Oseledets (1966) [19], M. Keane (1975) [13], A. Katok (1980) [12], W.
Veech (1982) [23], H. Masur (1982) [16], M. Viana (2006) [24], A.I. Bufetov(
2006) [5]. Starting already from the work of W. Veech the main object of
interest here was shifted from a pure dynamical point of view to connections
with geometry, in particular with Teichmuller flows. Since the latter goes out
of the scope of the present paper, we will not discuss this.

In the present paper we are interested mainly in measure-theoretic points of
view to dynamics, which is relatively simple and well understood in the case
of IETs.

The situation is becoming rather different if a PWI is not “bijective” even in
the above mentioned weak sense. This means that the images of the elements of
the special partition might have arbitrary intersections. The main body of
results in this direction are again about the one-dimensional orientation
preserving case, known in the literature as interval translation maps (ITM).
The most interesting among them in a historical order belongs to M.
Boshernitzan & I. Kornfeld (1995) [3], J. Buzzi & P. Hubert (2004) [6], H.
Bruin (2007,2012) [4], S. Marmi, P. Moussa, J-C. Yoccoz (2012) [17], D. Volk
(2014) [25]. A further generalization when a general cover is used instead of
a special partition (which inevitably leads to a multi-valued map) was
considered in A. Skripchenko & S. Troubetzkoy (2015) [21]. In any case the
problem with the boundary points already mentioned in the case of IET is
becoming a serious obstacle here. Nevertheless from a topological point of
view or in the discussion of attractors one can go around it in order to
reduce the analysis of ITM to a better undrestood IET. A popular approach
introduced in [3] for the case of the circle orientation preserving PWI is to
classify them into finite and infinite types with respect to the number of
iterations under which the images of the phase space are stabilized. Namely,
the finite type is characterized by the property $T^{n+1}X=T^{n}X$ for some
$n<\infty$, and otherwise we refer to an infinite type. Indeed, for the finite
type case in the one-dimensional orientation preserving case one obtains an
ITM in a finite number of iterations.

A number of attempts to follow this idea in the multidimensional setting have
been tried (see, e.g. [25, 22] and further references therein). Unfortunately,
in distinction to the one-dimensional case, the multidimensional dynamics is
much more complicated and cannot be reduced to some version of ITM, which has
been demonstrated by A. Goetz in [11]. In his example $X$ is an isosceles
triangle divided by the special partition into two smaller isosceles triangles
$X_{1}$ and $X_{2}$. Under the action of the piecewise isometry $T$ the
smaller triangles are rotated one clockwise and another anticlockwise in such
a way that their images intersect only along boundaries and their closures
cover $X$ completely. Similarly to the discussion above, there is a problem
how to define the action of $T$ on the boundary of the special partition, but
on the level of numerics this is not so important. The aim of the author was
to study numerically the structure of limit sets under dynamics, which turns
out to be extremely complicated. Moreover the simulation indicates that this
structure is self similar up to a very fine scales. This conjecture is still
not proven on the mathematical level.

From the most interesting for us measure-theoretic point of view, there are no
results at all for $d>1$ and only a few in the one-dimensional setting, which
we will discuss now. The main problem here is that there is no obvious
candidate for an invariant measure, while the images of a good enough
reference measure in general might concentrate at boundary points, which
complicates their analysis. In [6] it has been shown that in the absence of
periodic points the number of ergodic invariant measures of an orientation
preserving circle PWI cannot be larger than the number of elements of the
special partition. It is somewhat strange that having this answer the authors
even did not consider a question whether at least one invariant measure is
present. Observe that PWI maps are necessarily discontinuous and the classical
Krylov-Bogolyubov argument cannot be applied. The first step to overcome this
difficulty was done by B. Pires (2016) [20], where the existence was proven
under some (uncheckable) technical assumptions.222Basically this is a version
of the Krylov-Bogolyubov theorem, specialized for discontinuous maps under
study. Let us mention also two other papers [10, 22], where the analysis again
was based on a number of (uncheckable) technical assumptions. The first and
the only real result about the existence of invariant measures of an
orientation preserving circle PWI was obtained by S. Kryzhevich (2020) [14] by
means of rational approximations. We will discuss this approach in some detail
in Section 3.

##  3 Setup

Let $X$ be a subset of the Euclidean space ${\mathbb{R}}^{d},~{}d\geq 1$
equipped with a certain metric $\rho(\cdot,\cdot)$ belonging to the class of
$\tilde{\rho}_{p}(x,y):=(\sum_{i}|x_{i}-y_{i}|^{p})^{\frac{1}{p}},~{}p\geq 1$,
and let $\\{X_{i}\\}$ be a partition of $X$ into disjoint regions. By a region
we mean a convex set with a nonempty interior. The union of boundary points of
all regions we denote by $\Gamma$, which by definition is of zero
$d$-dimensional Lebesgue measure.

###### Definition 3.1

A piecewise isometry (PWI) is a map $T:X\to X$, satisfying the property that
its restriction to each region $X_{i}$ is an isometry. We will refer to
$\\{X_{i}\\}$ as a special partition, and to the maps $T_{i}|_{X_{i}}$ as
local maps.

###### Definition 3.2

A restriction $T_{|X_{i}}$ is said to be an extendable isometry if it can be
extended to an isometry $T_{i}$ on the entire $X$ such that
$T_{i}|_{X_{i}}\equiv T_{|X_{i}}$. Correspondingly a piecewise isometry $T$ is
said to be extendable if its restriction to each region $X_{i}$ is an
extendable isometry.

As we will see the innocent looking assumption of extendability is in fact
somewhat restricting. Collecting well known geometric facts (see e.g. [18, 1])
we get the following

###### Proposition 3.1

An isometry from a region $Y\subset{\mathbb{T}}^{d}$ to ${\mathbb{T}}^{d}$ is
extendable if and only if it can be represented as a finite superposition of
basic torus isometries: translations and coordinate flips or exchanges.

The main result of our study is the following theorem about extendable torus
piecewise isometries.

###### Theorem 3.1

Let $T$ be an extendable PWI of the unit torus $X$ with a finite special
partition $\\{X_{i}\\}$. Then there exists at least one probabilistic
$T$-invariant measure $\mu_{T}$. Additionally, if the boundary set $\Gamma$
contains no periodic points, this measure is non-atomic.

We expect that the extendability assumption can be avoided or significantly
weaken, but the methods used in this article make extensive use of this
assumption. In the one-dimensional orientation preserving case the claim of
Theorem 3.1 was proven in [14], but in the general (even one-dimensional)
setting this claim is new. Moreover, in [14] the non-atomic property of
invariant measures has been proven without the assumption of the absence of
periodic points at $\Gamma$. One of the key-points there was the demonstration
that a periodic trajectory of a PWI cannot be isolated.333A trajectory is
isolated if all nearby trajectories have other itineraries. It was expected
that even in the one-dimensional non orientation preserving case the situation
will be the same. This turns out to be wrong, and a counterexample will be
discussed in the last part of the proof of Proposition 7.3.

To emphasize that the claim of Theorem 3.1 is far from being obvious, we will
construct examples of extendable piecewise isometries without invariant
measures, albeit with a countable special partition (see Section 4).

A general scheme of the proof of Theorem 3.1 is as follows. First, we consider
a family of torus isometries $\\{g_{\theta}\\}$ satisfying the property that
any finite subcollection of such isometries generates a group
$G(g_{\theta_{1}},\dots,g_{\theta_{n}})$ having a finite number of elements.
We refer to this property as weak periodicity.444See Section 5 for discussion.
In Section 5 we will analyze properties of weakly periodic maps. In the case
of commuting generators the situation is rather trivial and an important point
here is that we are able to study some important cases when at least some of
the the generators $g_{\theta}$ do not commute.

On the 2-nd step we consider torus piecewise isometries $T$ such that the
restrictions $T_{X_{i}}$ can be extended to the torus isometries belonging to
the above family. In Section 5 we study properties of such maps and prove for
them the claim of Theorem 3.1.

Finally in Section 6 we approximate an arbitrary extendable torus piecewise
isometry by means of weakly periodic ones considered in Section 5, which will
allow to finish the proof of Theorem 3.1.

It is worth noting, that this result does not contradict to the possibility of
having attractors admitting singular invariant measures. Various numerical
results (see e.g. [11, 1]) give some evidence of the possibility of such
behavior. The point is that the corresponding measures might not be
represented as limits of invariant measures of maps discussed in Section 5 and
hence cannot be controlled by our construction.

A similar approach was used in [14], where a specialized version of the above
construction was applied for the case of orientation preserving circle
piecewise rotations. In that case the author used the approximation scheme
based on rational translations and special partitions with rational boundary
points. The last fact is a crucial point not allowing to use this
approximation in the multidimensional setting. Indeed, for $d>1$ the
boundaries of regions cannot consist of points with rational coordinates only.

##  4 Focusing phenomenon and absence of invariant measures

A natural approach to prove the existence of an invariant measure of a
piecewise continuous dynamical system $(T,{\mathbb{T}})$ is to consider a
sequence of images $\\{T^{n}m\\}_{n}\geq 0$ of the Lebesgue measure $m$ on the
torus ${\mathbb{T}}$. Let $\mu$ be a limit point of this sequence. If the
$\mu$-measure of a set $\Gamma$ of points of discontinuity of the map $T$ is
0, then under some simple technical assumptions the measure $\mu$ is
$T$-invariant. The opposite situation, when the images of the Lebesgue measure
concentrates on a small set we refer as the focusing phenomenon. To be precise
let us introduce the focusing set:

$S(T):=\liminf_{n\to\infty}T^{n}X,$

where $\liminf_{n\to\infty}A_{n}:=\cup_{n}\cap_{k\geq n}A_{k}$ is the set of
points belonging to all but finitely many sets $A_{n}$. Then the “size” of the
set $S(T)$ describes the focusing quantitatively.

The focusing or concentration phenomenon is well known for systems with
(locally) stable periodic orbits. The focusing occurs exactly to these
periodic orbits, and the main reason for this is the (locally) contracting
nature of the map $T$. In the situation with piecewise isometries we do not
observe the contraction anywhere even locally.

In this section we discuss how the focusing may occur for a PWI. In
distinction to other parts of this paper we allow the special partition to be
countable. The focusing possibilities for the case of the finite special
partition will be discussed in the end of the Section. The main idea under the
following two examples is that under the action of the maps under study any
measure converges weakly (focuses) to a measure $\mu$ supported by a very
small wandering set. Recall that the latter means that all its forward images
under dynamics do not intersect with the set. Therefore the measure $\mu$
cannot be invariant. Examples of this sort are known in the case of piecewise
contractive maps, where the phenomenon of the measure concentration is natural
(see e.g. [2]), but look somewhat surprising for the piecewise isometry.

###### Example 4.1

Let $X:=[0,1)^{2}$ and the special partition $\\{X_{n}\\}$ be defined as
follows: $X_{n}:=(1-2^{-n+1},1-2^{-n}]\times[0,1),~{}~{}n\geq 1$, while
$Tx:=(x_{1}+2^{-n-1},x_{2})$ if $x\in X_{n}$.

$0$$1$$1$$X_{1}$$X_{2}$$X_{3}$$\frac{1}{2}$$\frac{3}{4}$$\frac{7}{8}$$TX_{2}$$X_{n}:=(1-2^{-n+1},1-2^{-n}]\times[0,1)$$X:=\cup_{n\geq
1}X_{n}=[0,1)\times[0,1)$$Tx:=(x_{1}+2^{-n-1},x_{2})$ if $x\in
X_{n}$$T^{k}X\stackrel{{\scriptstyle
k\to\infty}}{{\longrightarrow}}\\{0\\}\times[0,1)$ Figure 3: Example of a
piecewise torus isometry without invariant measures. The element $X_{2}$ of
the special partition and its image are marked in yellow and green colors
correspondingly.

Observe that after exactly two iterations each element of the partition
$\\{X_{n}\\}$ will be mapped to the union of elements with larger indices. In
this situation obviously any measure converges under the action of $T$ to a
measure supported by the unit segment $\\{0\\}\times[0,1)$. This segment is an
wandering set, since its forward images under the action of $T$ do not
intersect with the segment, which proves the absence of invariant measures.

Nevertheless, in the Example 4.1 we do not have a complete concentration of
measure, namely the situation when the limit (but not invariant) measure is a
$\delta$-measure at a single point. To achieve this goal instead of the
basically one-dimensional example above, we consider a bit more involved
$d$-dimensional map.

###### Example 4.2

Let $X:=[0,1)^{d}$ and the special partition $\\{X_{n}\\}$ be defined as
follows:

$d(1-\frac{1}{n})\leq\sum_{i=1}^{d}x_{i}<d(1-\frac{1}{n+1}),$

while $Tx:=x+\frac{1}{2n(n+1)}\bar{1}\,({\rm mod\ }1)\,$ for $x\in X_{n}$,
where $\bar{1}$ is the $d$-dimensional vector with unit coordinates.

$X_{1}$$X_{2}$$TX_{2}$$X_{4}$$0$$1$$1$$X_{n}:=\\{x:~{}2(1-\frac{1}{n})\leq
x_{1}+x_{2}<2(1-\frac{1}{n+1})\\}$$X:=\cup_{n\geq
1}X_{n}=[0,1)\times[0,1)$$Tx:=x+\frac{1}{2n(n+1)}\bar{1}\,({\rm mod\ }1)\,$ if
$x\in X_{n}$$T^{k}X\stackrel{{\scriptstyle
k\to\infty}}{{\longrightarrow}}\\{1\\}\equiv\\{0\\}$ Figure 4: Example of a
piecewise torus isometry converging to $\bar{1}$. The element $X_{2}$ of the
special partition and its image are marked in yellow and green colors
correspondingly.

In distinction to the previous example the image of $X_{n}$ may intersect with
elements $X_{k}$ with smaller indices $k<n$. Since the diameters of the
elements of the partition decrease when their indices grow, after the
implementation of the operation $\,({\rm mod\ }1)\,$ peripheral parts of the
image may be located below the central part and intersect with the elements
with lower indices. In fact, $TX_{n}\cap X_{1}\neq\emptyset~{}\forall n$.
Nevertheless the Hausdorff distance from the set $T^{k}X_{n}$ to the point
$\bar{1}$ goes to 0 when $k\to\infty$. Therefore under the action of $T$ any
measure weakly converges to $\delta_{\bar{1}}$, while the point $\bar{1}\equiv
0$ is again an wandering set.

If the special partition is finite we cannot provide examples with the
focusing to the set of zero Lebesgue measure. Instead we try to estimate the
“size” of the focusing set from bellow.

###### Example 4.3

Let $X:=[0,1)$ and the special partition $\\{X_{n}\\}$ consists of $N$
intervals of equal length. Let $\alpha:=\\{\alpha_{i}\\}_{i=1}^{n}$ be a
collection of numbers and let $T_{\alpha}x:=x+\alpha_{i}\,({\rm mod\ }1)\,$
for $x\in X_{i}$. We are interested in the focusing for this family of maps,
namely in

$\kappa:=\inf_{\alpha}m(S(T_{\alpha})).$

For a given $\alpha$, the set $T_{\alpha}$ is a typical one-dimensional
orientation preserving piecewise isometry. Siince the images of the elements
of the special partition may intersect, the value of $\kappa$ needs not to be
close to one. However, since the map is locally isometric, by the same
telescoping construction as in the examples 4.1, 4.2 one can achieve
$\kappa=\frac{1}{N}$. From the first sight it seems that this estimate cannot
be significantly improved. To demonstrate that this is absolutely not the case
choose an arbitrary small parameter $\varepsilon>0$ and set
$\alpha_{1}:=\varepsilon,\alpha_{2}:=-\varepsilon,\alpha_{n}:=-\frac{1}{N}$
for $n>2$. In other words, $T_{\alpha}X_{n+1}=X_{n}$ for $n>3$ (telescoping),
but the images of the first two elements intersect by the set of length
$2\varepsilon$. It is easy to calculate that
$S(T_{\alpha}):=[\frac{1}{N}-\varepsilon,\frac{1}{N}+\varepsilon]$. Since
$\varepsilon>0$ is arbitrary, we get $\kappa=0$.

##  5 Weakly periodic semigroups of maps

The main aim of this Section is to introduce the class of weakly periodic
torus piecewise isometries and to prove Theorem 3.1 for them.

###### Definition 5.1

A collection of maps $\\{g_{i}\\}_{i=1}^{n}$ from a set $X$ into itself is
said to be weakly periodic (notation WPer) if the number of elements in the
semigroup generated by them is finite, i.e. $\\#G(g_{1},\dots,g_{n})<\infty$.

The reason under this terminology is that a periodic homeomorphism $T$ (i.e.
$T^{n}=Id$) generates a semigroup with a finite number of elements, i.e.
$T\in$ WPer.

The simplest (and very instructive) example of a weakly periodic semigroup of
maps is a semigroup generated by a finite number of rational circle
translations.

An important observation about this property gives the following result.

###### Proposition 5.1

Let $g_{i}\circ g_{j}=g_{j}\circ g_{i}~{}\forall i,j$, then
$\sum_{i}\\#G(g_{i})<\infty$ implies $\\#G(g_{1},\dots,g_{n})<\infty$.
Otherwise there exists a pair of non-commuting maps $g_{1},g_{2}$ such that
$\sum_{i=1}^{2}\\#G(g_{i})=4$, but $\\#G(g_{1},g_{2})=\infty$.

Proof. Each element $T\in G(g_{1},\dots,g_{n})$ may be represented in the form
of a finite superposition of the generators $g_{i}$, and due to their
commutativity we get

$Tx=g_{1}^{k_{1}}\circ\dots\circ g_{n}^{k_{n}}x,~{}~{}0\leq
k_{i}=k_{i}(x)<\infty~{}~{}\forall x\in X.$

By the assumption $\exists N<\infty$ such that $\\#G(g_{i})<N~{}\forall i$.
Therefore,

$\\#(\cup_{T\in G}Tx)\leq N^{n}<\infty~{}~{}\forall x\in X,$

which implies the first claim.

The 2nd claim is not so obvious as one might expect. In particular, in the
one-dimensional case it is wrong for orientation preserving maps. The latter
property turns out to be crucial here. Let $X:=[0,1),~{}g_{1}(x):=-x\,({\rm
mod\ }1)\,,~{}g_{2}(x):=-x+\alpha\,({\rm mod\
}1)\,,~{}\alpha\notin{\mathbb{Q}}$. Observe that $\\#G(g_{i})=2,i=1,2$, but
$(g_{1}g_{2})^{n}x=x-n\alpha\,({\rm mod\ }1)\,$, which implies
$\\#G(g_{1},g_{2})=\infty$ since $\alpha\notin{\mathbb{Q}}$.
$\sqcup\\!\\!\\!\\!\sqcap$

The definition of the weakly periodic maps implies the following simple
technical result.

###### Proposition 5.2

Let $\\{g_{i}\\}_{i=1}^{n}\in$ WPer and let a PWI map $T:X\to X$ with a finite
special partition $\\{X_{i}\\}_{i=1}^{n}$ such that $T|_{X_{i}}\equiv
g_{i}|_{X_{i}}~{}\forall i$. Then $\forall x\in X$ the trajectory
$\\{T^{n}x\\}_{n\geq 0}$ starting at $x$ is eventually periodic.

Proof. The claim follows immediately from the observation that the total
number of different points obtained by succesive applications of the
generators $g_{i}$ is finite. $\sqcup\\!\\!\\!\\!\sqcap$

Despite that according to Proposition 5.1 in general the WPer property for
individual maps does not imply this property for a collection of them, the
situation becomes more optimistic when we restrict ourselves to torus
isometries.

As we already noted in Proposition 3.1, any $d$-dimensional torus isometry may
be represented by a superposition of basic isometries of the following 3
types:

  * (i)

translation: $x\to x+v\,({\rm mod\ }1)\,,~{}~{}v\in{\mathbb{R}}^{d}$

  * (ii)

coordinate flip: $x_{i}\to-x_{i}$

  * (iii)

coordinate exchange: $x_{i}\leftrightarrow x_{j}$

The maps belonging to different types do not commute, but at least the maps
from the first two types commute with the maps of the same type, however the
maps of type (iii) do not commute even between themselves. Weak periodicity of
the maps from the first type is equivalent to their periodicity, while the
maps of the other two types are always 2-periodic.

To apply the machinery developed in Proposition 5.4 one needs to show that a
PWI generated by a finite collection of periodic basic isometries satisfy the
weak periodicity property.

###### Theorem 5.1

Let $T$ be a torus PWI such that $T|_{X_{i}}$ is a finite composition of
periodic basic isometries $\forall i$. Then the map $T\in$ WPer.

The proof of this intuitively obvious claim (to which we devote the remaining
part of this Section) turns out to be surprisingly cumbersome. In particular,
we are not able to perform the proof directly for the map $T$ itself and are
instead embed it to a certain semigroup generated by symmetrized torus
extensions of the local isometries.

###### Definition 5.2

By a symmetrized semigroup $G^{*}(g_{1},\dots,g_{n})$ generated by basic torus
isometries $\\{g_{i}\\}$ we mean a semigroup generated by the maps
$\\{g_{i}\\}$, where each translation map $x\to x+v\,({\rm mod\ }1)\,$ is
considered together with all permutations and sign changes of the coordinates
of the vector $v$, similarly each coordinate flip or exchange is considered
together with all other coordinate flips or exchanges respectively. In the
same manner by $(\\{g_{i}\\})^{*}$ we mean a symmetrized collection of basic
torus isometries $\\{g_{i}\\}$.

Fortunately to the finite dimension of the torus, the number of additional
generators in the symmetrized semigroup is finite as well.555It cannot be
enlarged in more than $(d!)^{3}$ times.

###### Proposition 5.3

Let $\\{g_{i}\\}_{i=1}^{n}$ be a collection of periodic basic torus
isometries. Then
$\\#G^{*}(g_{1},g_{2},\dots,g_{n})\leq\prod_{i=1}^{n}\\#G^{*}(g_{i})<\infty$.

Proof. We proceed by induction. The claim for $n=1$ is trivial, except for its
finiteness. If $g_{i}$ belongs to (i) or (ii) types, then the finiteness
follows from the periodicity assumption together with the commutation
property. If $g_{i}$ is of the 1st type, the last argument cannot be applied,
but instead one gets a direct estimate $G^{*}(g_{i})\leq d!$ since all
possible coordinate permutations are taken into account.

Let us make the induction step from $n$ to $n+1$. Observe that the application
of the semigroup $G^{*}(g_{1},g_{2},\dots,g_{n})$ to a given point $x\in X$,
i.e. the set $G^{*}(g_{1},g_{2},\dots,g_{n})x$ is invariant with respect to
each of the maps $(\\{g_{i}\\}_{i=1}^{n})^{*}$. The latter means that

$g(G^{*}(g_{1},g_{2},\dots,g_{n})x)=G^{*}(g_{1},g_{2},\dots,g_{n})x\qquad\forall
g\in(\\{g_{i}\\}_{i=1}^{n})^{*}.$

We will show that for any periodic basic isometry $g_{n+1}$ the set
$G^{*}(g_{n+1})G^{*}(g_{1},g_{2},\dots,g_{n})x$ is invariant with respect to
each of the maps $(\\{g_{i}\\}_{i=1}^{n+1})^{*}$, which implies the claim.

Denote
$A:=G^{*}(g_{1},g_{2},\dots,g_{n})x,~{}B:=G^{*}(g_{n+1})G(g_{1},g_{2},\dots,g_{n})x$
and consider all possibilities:

  * •

$g_{n+1}$ is a translation. If $A$ is invariant with respect to some
translation $g_{i}$, then the the invariance of $B$ with respect to $g_{i}$
follows from the commutation property. The invariance of $A$ with respect to a
coordinate flip is equivalent to the fact that the set $A$ is a uniform
lattice on the torus. The latter property cannot be destroyed by a torus
translation. If $A$ is invariant with respect to a coordinate exchange, then
this set is characterized by the symmetry with respect to the corresponding
unit cube diagonal. Unfortunately the translation $g_{n+1}$ may destroy this
invariance for $B$. Indeed a symmetric pair $(a,b),(b,a)$ may by mapped to
$(a+c,b),(b+c,a)$. This is exactly the point, where we need the
symmetrization, because $G^{*}(g_{n+1})A$ clearly preserves the symmetry under
question.

  * •

$g_{n+1}$ is a coordinate flip. If $A$ was invariant with respect to some
translation, $B:=G^{*}(g_{n+1})A$ preserves this invariance due to the
symmetrization. The invariance of $A$ with respect to a coordinate flip
$g_{i}$ implies the same property for $B$, since coordinate flips commute. The
invariance of $A$ with respect to a coordinate exchange is inherited by $B$
again helps to the symmetrization with respect to the coordinate flips.

  * •

$g_{n+1}$ is a coordinate exchange $x_{i}\leftrightarrow x_{j}$. This
situation can be analyzed in exactly the same manner and we skip it.

The induction step is proven. $\sqcup\\!\\!\\!\\!\sqcap$

This result finishes the proof of Theorem 5.1. Indeed, denote the torus
extensions of the local isometries $T|_{X_{i}}$ by $g_{i}$ and assume that the
maps $g_{i}$ are finite superpositions of periodic basic torus isometries.
Then the trajectory of length $n$ under the action of $T$ of any given point
$x\in X$ is a subset of $G^{*}(g_{1},\dots,g_{n})x$, which is uniformly finite
on $x$ and $n$ by Proposition 5.3. This proves the weak periodicity of the map
$T$. $\sqcup\\!\\!\\!\\!\sqcap$

Additionally from the weak periodicity of the map $T$ in this setting we get
some important information about its invariant measures.

###### Proposition 5.4

Let $\\{g_{i}\\}_{i=1}^{n}$ be a collection of torus diffeomorphisms and let
$\\#G(g_{1},\dots,g_{n})<\infty$. Let $T$ be a piecewise isometry of the unit
torus $X$ with the finite partition $\\{X_{i}\\}$ and such that each
$T|_{X_{i}}$ can be extended to a torus map $T_{i}\in G(g_{1},\dots,g_{n})$.
Then there exists at least one absolutely continuous invariant measure, being
a restriction of the Lebesgue measure to a certain region.

Proof. Denote by $\tilde{\Gamma}:=\cup_{g\in
G(g_{1},\dots,g_{n})}g^{-1}\Gamma$ – the union of all pre-images under the
action of all maps $g\in G(g_{1},\dots,g_{n})$ of the boundary $\Gamma$ of the
elements of the partition $\\{X_{i}\\}$. By the weak periodicity assumption
$\tilde{\Gamma}$ divides the torus $X$ into a finite number of regions
$\\{Y_{j}\\}$ of positive Lebesgue measure, such that points belonging to the
same region $\\{Y_{j}\\}$ have the same itineraries under the action of the
map $T$. Therefore each of the regions $Y_{i}$ is either periodic under the
action of $T$, or is wandering. The latter case is out of interest for us,
while a collection of periodically exchanging regions
$Y_{i_{1}},\dots,Y_{i_{k}}$ supports the Lebesgue measure on them being
invariant with respect to the map $T$. Moreover, all absolutely continuous
$T$-invariant measures may be described this way. $\sqcup\\!\\!\\!\\!\sqcap$

##  6 Approximation

In what follows to control the rate of recurrence we will need the following
abstract result.

###### Lemma 6.1

Let $(T,X,{\cal B},\mu)$ be a measurable dynamical system, let $A\in{\cal
B},~{}\mu(A)>0$, and let $\tau_{A}(x)$ be the first return time for a point
$x\in A$ to return to the set $A$ under the action of $T$. Then $\mu(\\{x\in
A:~{}\tau_{A}(x)\leq\frac{1}{\mu(A)}\\})>0$.

Before to prove this result, observe that according to the famous Kac Lemma
(see e.g. [8]) in the ergodic case the mathematical expectation of the first
return time is exactly equal to $1/\mu(A)$. On the other hand, in our setting
we cannot a priori assume ergodicity without which the claim of Lemma 6.1 is
as best as possible.

Proof. By the Poincare Recurrence Theorem $\mu$-a.a. points return to $A$
under the action of $T$. Let $A_{n}:=\\{x\in A:~{}n\leq\tau_{A}(x)<\infty\\}$
be the set of points returning to $A$ at least after $n\geq 0$ iterations, and
suppose $A\setminus A_{n}=\emptyset\Longrightarrow\mu(A)=\mu(A_{n})$. Since
$T^{-k}A_{n}\cap A=\emptyset~{}~{}\forall k<n$, we get $T^{-i}A_{n}\cap
T^{-j}A_{n}=\emptyset~{}~{}\forall i\neq j<n$ (otherwise the points from the
intersection above will not return to A). Hence

$1\geq\mu(\cup_{k=0}^{n-1}T^{-k}A_{n})=\sum_{k=0}^{n-1}\mu(T^{-k}A_{n})=n\mu(A)$

and thus $n\leq 1/\mu(A)$. $\sqcup\\!\\!\\!\\!\sqcap$

Let $\\{T^{(n)}\\}$ be a sequence of piecewise torus isometries with the
common finite special partition $\\{X_{i}\\},~{}\Gamma:=\cup_{i}\partial
X_{i}$, satisfying the assumptions of Proposition 5.4, and let $\\{\mu_{n}\\}$
be a sequence of their absolutely continuous invariant measures.

###### Lemma 6.2

Let $z\not\in{\rm Per}(T)$. Then
$\forall\varepsilon>0~{}\exists\delta>0~{}\limsup\limits_{n\to\infty}\mu_{n}(B_{\delta}(z))<\varepsilon$.

Proof. Assume from the contrary that there is $\varepsilon>0$ and a pair of
sequences $\\{n_{k}\\}\stackrel{{\scriptstyle
k\to\infty}}{{\longrightarrow}}\infty$ and
$\\{\delta_{k}\\}\stackrel{{\scriptstyle k\to\infty}}{{\longrightarrow}}0$
such that $\mu_{n_{k}}(B_{\delta_{k}}(z))\geq\varepsilon$. Then by Lemma 6.1

$\exists N<1/\varepsilon:~{}~{}(T^{(n_{k})})^{N}B_{\delta_{k}}(z)\cap
B_{\delta_{k}}(z)\neq\emptyset,$

which implies that $z\in{\rm Per}(T)$. We came to the contradiction.
$\sqcup\\!\\!\\!\\!\sqcap$

Since we deal with a compact phase space, the sequence of the above mentioned
absolutely continuous invariant measures of the approximating weakly periodic
maps has a limit point. Denote this limit point by $\tilde{\mu}$ and choose a
sequence of the measures $\\{\mu_{n}\\}$ converging to it in the weak
topology.

###### Lemma 6.3

Let $\Gamma\cap{\rm Per}(T)=\emptyset$. Then $\tilde{\mu}(\Gamma)=0$ and
$\forall k,\varphi\in C^{0}(X,{\mathbb{R}})$ we have

$\lim_{n\to\infty}\mu_{n}(1_{{\rm
Clos}(X_{k})}\cdot\varphi)=\tilde{\mu}(1_{{\rm Clos}(X_{k})}\cdot\varphi).$

Proof. The first statement follows from Lemma 6.2. To prove the second
statement without loss of generality we assume that $|\varphi(x)|\leq 1$.
Using again Lemma 6.2 we get: $\forall\varepsilon>0~{}\exists\delta>0$ such
that $\mu_{n}(B_{\delta}(\partial X_{k}))<\varepsilon$. Choose now a
nonnegative function $h\in C^{0}(X,[0,1])$ such that $h_{|X_{k}\setminus
B_{\delta}(\partial X_{k})}\equiv 1$ and $h_{|X\setminus B_{\delta}(\partial
X_{k})}\equiv 0$. Then
$\lim_{n\to\infty}\mu_{n}(h\varphi)=\tilde{\mu}(h\varphi)$. by the definition
of the weak convergence of measures.

Therefore

$\lim_{n\to\infty}|\mu_{n}(\varphi)-\tilde{\mu}(\varphi)|\leq
m(B_{\delta}(\Gamma)),$

which implies the result since the value of $\varepsilon>0$ can be taken
arbitrarily small. $\sqcup\\!\\!\\!\\!\sqcap$

###### Lemma 6.4

Let $\Gamma\cap{\rm Per}(T)=\emptyset$ and let $\\{T^{(n)}\\}$ be a sequence
of approximating weakly periodic maps, whose invariant measures
$\mu_{n}\stackrel{{\scriptstyle n\to\infty}}{{\longrightarrow}}\tilde{\mu}$.
Then

$\mu_{n}(\varphi\circ T^{(n)}-\varphi\circ T)\stackrel{{\scriptstyle
n\to\infty}}{{\longrightarrow}}0~{}~{}\forall\varphi\in
C^{0}(X,{\mathbb{R}}).$

Proof. Given $\varepsilon>0$ by Lemma 6.2 we can choose $\delta>0$ such that
$\mu_{n}(B_{\delta}(\Gamma))<\varepsilon$.666Note that the convexity of the
elements of the special partition is important at this point. Otherwise one
needs to assume that the boundaries of those elements are smooth enough. From
the approximation assumption $\exists n_{0}$ such that $|\varphi\circ
T^{(n)}(x)-\varphi\circ T(x)|<0$ $\forall n>n_{0}$ and $x\not\in
B_{\delta}(\Gamma)$. This implies the claim. $\sqcup\\!\\!\\!\\!\sqcap$

Now we are ready to finish the proof of Theorem 3.1.

First, observe that each map, satisfying the assumption of Theorem 3.1, can be
approximated arbitrary well by weakly periodic ones. Indeed, by since
$T|_{X_{i}}$ is extendable it can be represented by a finite superposition
basic isometries. If there are isometries of type (i) among them, they can be
approximated arbitrary well by rational translations (which are periodic).
Basic isometries of other types are periodic from the very beginning.

The second step is to check that “good” invariant measures of the weakly
periodic approximations, existing by Theorem 5.1 and Proposition 5.4, converge
to a $T$-invariant measure.

###### Lemma 6.5

If $\exists\gamma\in\Gamma\cap{\rm Per}(T)$, then the measure uniformly
distributed on the trajectory of $\gamma$ is $T$-invariant. Otherwise, if
$\Gamma\cap{\rm Per}(T)=\emptyset$ the measure $\tilde{\mu}$ constructed in
Lemmas above is $T$-invariant.

Proof.

$|\tilde{\mu}(\varphi-\varphi\circ T)|\leq|\tilde{\mu}(\varphi-\varphi\circ
T)-\mu_{n}(\varphi-\varphi\circ T)|$ $\hskip
133.72786pt+~{}|\mu_{n}(\varphi-\varphi\circ T^{(n)})|+|\mu_{n}(\varphi\circ
T^{(n)}-\varphi\circ T)|.$

The first term in the rhs goes to 0 as $n\to\infty$ by Lemma 6.3, the second
term vanishes for each $n$ since the measures $\mu_{n}$ are
$T^{(n)}$-invariant, while the third term goes to 0 by Lemma 6.4. Hence
$\tilde{\mu}(\varphi)=\tilde{\mu}(\varphi\circ T)$, which implies the claim.
$\sqcup\\!\\!\\!\\!\sqcap$

##  7 Torus restriction of a plane isometry

For the family of torus metrics $\rho_{p}(\cdot,\cdot)$ for all $p\neq 2$ the
list of extendable isometries (translations, coordinate flips and coordinate
exchanges) coincides with all possible isometries. However in the important
Euclidean case $\rho_{2}(\cdot,\cdot)$ there is an additional isometry –
rotation, which is not extendable. The techniques developed in this paper does
not allow to consider local rotations, because we do not have their weakly
periodic approximations.

In this section we consider actions of plane isometries restricted to the unit
two-dimensional torus $X:={\mathbb{T}}^{2}$ as the simplest models local
rotations. Some results in this direction (mainly about topological
properties) may be found in [15] (see further references therein). Recall that
a general plane isometry can be written as $x\to{\cal R}x+b$, where
$x,b\in{\mathbb{R}}^{2}$ and ${\cal R}$ is an orthogonal matrix. Restricting
this map to the torus we get the torus map $Tx:={\cal R}x+b\,({\rm mod\
}1)\,:{\mathbb{T}}^{2}\to{\mathbb{T}}^{2}$.

###### Example 7.1

Let ${\cal R}={\cal R}(\varphi)$ be the rotation around the origin by the
angle $\varphi\in[0,2\pi)$ and let $b=0$.

Having in mind that that each trajectory of the plane rotation belongs to a
single circle around the origin, while a typical torus translation $x\to
x+b\,({\rm mod\ }1)\,$ fills the torus densely, one naively expects something
of the same sort for the map $T$. It turns out that if the rotation angle
$\varphi\neq 0$ the situation is completely different (see Fig. 1).

The map $T$ defined above, considered as a map from the unit torus
$X:={\mathbb{T}}^{2}$ into itself is continuous everywhere inside the open
unit square $\tilde{X}:=(0,1)^{2}$ and the set of discontinuities is
represented by the boundary of this set $\partial\tilde{X}$. On the other
hand, the same map may be considered as a map from the unit square
$\hat{X}:=[0,1]^{2}$ into itself. In the latter case the set of discontinuous
is much more involved (see Fig. 5).

$0$$1$$1$$\varphi$$\varphi=\frac{\pi}{e}$ Figure 5: Thin blue/green lines –
the boundaries of forward/backward images of the boundaries of the unit square
under the rotation by the angle $\pi/e$ around the origin. Thick black lines –
boundaries of the special partition.

If $\det(I-{\cal R})\neq 0$ there is at least one fixed point of the map
$Tx:={\cal R}x+b\,({\rm mod\ }1)\,$. Here and in the sequel by $I$ we denote
the unit matrix. Indeed, making the change of variables $x=y+a$, we rewrite
the equation for the fixed point of the map $T$ as $y+a={\cal R}(y+a)+b\,({\rm
mod\ }1)\,$, which gives a solution $y:=0,~{}a:=(I-{\cal R})^{-1}b\,({\rm mod\
}1)\,$. Thus $z:=(I-{\cal R})^{-1}b\,({\rm mod\ }1)\,$ is the fixed point. If
additionally, $b=0$ (case of a pure rotation) the origin is always the fixed
point even without the assumption about the absence of the degeneracy of the
matrix $I-{\cal R}$

Therefore the main question is about the existence of nontrivial periodic
trajectories. Our numerical observations allow to formulate the following

Hypothesis. Let $Tx:=j{\cal R}(\varphi)x+b\,({\rm mod\ }1)\,$, where $j=\pm
1,~{}\varphi\in[0,2\pi),x,b\in{\mathbb{R}}^{2}$. Then $\forall
j,\varphi,b~{}~{}\exists N=N(j,\varphi,b)\in{\mathbb{Z}}_{+}$ and
$z=z(j,\varphi,b,N)$ such that $T^{N}z=z$. Moreover for a given triple
$j,\varphi,b$ one may find arbitrary large $N$ satisfying the above property.

An idea of the possible proof. We consider only the case of the pure rotation
(i.e. $j=1,~{}b=0$) by the irrational angle (i.e.
$\varphi=\gamma\pi,~{}\gamma\notin{\mathbb{Q}})$, leaving the general case
(where a number of cumbersome combinatorial type calculations are necessary)
for the future.

Observe that $Tx={\cal R}x\,({\rm mod\ }1)\,={\cal R}x+\xi(x)$, where
$\xi(x)\in{\mathbb{Z}}^{2}$. Therefore to find a periodic point of period $1$
one needs to solve the equation:

$z={\cal R}z+\xi_{1},$

and for the period $2$:

$z={\cal R}({\cal R}z+\xi_{1})+\xi_{2}={\cal R}^{2}z+{\cal R}\xi_{1}+\xi_{2}$

Continuing this, for a point of period $n$ we have

$z={\cal R}^{n}z+\sum_{j=1}^{n}{\cal R}^{n-j}\xi_{j},$

which is equivalent to

$(I-{\cal R}^{n})z=\sum_{j=1}^{n}{\cal R}^{n-j}\xi_{j}.$

If the angle $\varphi$ is irrational, then the inverse matrix $(I-{\cal
R}^{n})^{-1}$ is well defined. Thus we have an explicit solution

$z:=(I-{\cal R}^{n})^{-1}\sum_{k=1}^{n}{\cal R}^{n-k}\xi_{k}.$

Unfortunately this is not the end of the story, since in fact the integer
vectors $\xi_{i}$ depend on $z$ and should be chosen in such a way that ${\cal
R}^{k}z+\sum_{j=1}^{n}{\cal R}^{n-j}\xi_{j}$ belong to the unit square for
each $1\leq k\leq n$, otherwise further corrections need to be taken into
account.

Let us formulate and prove a partial result leaving the general case for
future studies.

###### Proposition 7.1

Let $\varphi\notin[-\varphi_{0},\varphi_{0}]$, where
$\tan\frac{\varphi_{0}}{2}=\frac{1}{2}$. Then the equation $A(\varphi)x=x$
always has a nontrivial solution.

Proof. Consider a point $x$ lying on a vertical line with the 1st coordinate
being equal to $\frac{1}{2}$, i.e. $x=(\frac{1}{2},x_{2})$. Then under the
rotation around the origin by the angle $\varphi:=-2\arctan(\frac{1}{2x_{2}})$
we get a point $\tilde{x}:=(-\frac{1}{2},x_{2})$, which coincides with
$x=(\frac{1}{2},x_{2})$ modulo 1. The assumption
$\varphi\notin[-\varphi_{0},\varphi_{0}]$ is explained by the choice of the
point $x$. $\sqcup\\!\\!\\!\\!\sqcap$

###### Proposition 7.2

Let $z$ be a $N$-periodic point and let $R:=\inf_{k}{\rm dist}(T^{k}z,\partial
X)>0$. Then $S(r):=\cup_{i=0}^{N-1}B_{r}(z)$ is a forward invariant set
$\forall r\in(0,R)$.

Proof. Consider any point $x$ close enough to $z$. Under the action of the map
$T$ the $\rho_{2}(\cdot,\cdot)$ distance between $T^{n}x$ and $T^{n}z$ is
preserved. Therefore each circle centered at $z$ with a small radius $r$ is
mapped into itself. The value of the radius $r$ is bounded from above by the
condition that the rotated circle does not intersect with the boundary of the
unit square. If additionally $\varphi/\pi$ is an irrational number, then the
trajectory of the point $x$ fills densely the corresponding circle, otherwise,
it is periodic. $\sqcup\\!\\!\\!\\!\sqcap$

Observe that the number of ergodic invariant measures is always countable in
distinction to the one-dimensional case, where it is known that in a “typical”
situation the number of ergodic measures is bounded from above by the number
of elements of the special partition.

A substantially more general result about the dynamics of a general PWI in a
neighborhood of a given trajectory based on the Proposition 7.2 may be
formulated as follows.

Let $T$ be a PWI of $X\in{\mathbb{R}}^{d}$ with respect to a metric
$\rho(\cdot,\cdot)$ and a special partition $\\{X_{i}\\}$.

###### Proposition 7.3

Let a trajectory of a point $z\in X$ be separated fromcthe boundary of the
special partition $\Gamma$, i.e. $\inf_{n\geq 0}\rho(T^{n}z,\Gamma):=R>0$.
Then $\forall x\in B_{R}(z)$ the itineraries of the trajectories started at
the points $x,z$ coincide, i.e. $T^{n}x\in X_{i}\Longleftrightarrow T^{n}z\in
X_{i}~{}\forall n\geq 0$. Moreover, $\forall r\in(0,R)$ the spheres
$S_{r}(T^{n}z)$ of radius $r$ centered at $T^{n}z$ are mapped one to another
under dynamics, i.e. $TS_{r}(T^{n}z)=S_{r}(T^{n+1}z)$. If the separation
property breaks down the trajectory stated at $z$ may be completely isolated
in the sense that $\lim\inf_{n\to\infty}\rho(T^{n}x,T^{n}z)>0$ whenever $x\neq
z$.

Proof. The separation assumption means that the ball $B_{R}(z)$ at each
iteration of the map $T$ falls entirely to a single element of the special
partition, which implies the claim about the itineraries. Now using the same
argument as in the proof of Proposition 7.2 we see that the sequence of balls
$B_{R}(T^{n}z)$ splits into the continuous on-parametric family of concentric
spheres $S_{r}(T^{n}z)$ mapped one to another under dynamics.

Observe that in the case of a periodic trajectory the separation property
becomes especially simple and trivially checkable: $\cup_{n\geq
0}T^{n}z\setminus\Gamma=\emptyset$.

To demonstrate the last claim we return to the pure rotation of the
2-dimensional torus around the origin by the angle $\pi/4$. Clearly, the point
$z=0$ is fixed under dynamics. Observe that a sufficiently small neighborhood
$O_{z}$ of this point on the torus consists of 4 triangular sectors at the
corners of the unit square and that each of these sectors belong to at most 2
elements of the special partition. This allows to use a simple numerical
simulations to check that $T^{n}(X\setminus\\{z\\})\cap
O_{z}=\emptyset~{}~{}\forall n\gg 1$, which finishes the proof.

Our numerical simulations indicate that the last claim remains valid for any
rotation angle $\varphi\neq 0$ and any shift $b$, but this is not proven yet.
Instead we consider an even more unexpected situation when a periodic
trajectory of an one-dimensional non orientation preserving PWI is isolated.

###### Example 7.2

Consider a PWI $T:X\to X:=[0,1)$ with the special partition into 4 intervals
$X_{1}:=[0,\frac{1}{4}),X_{2}:=[\frac{1}{4},\frac{1}{2}),X_{3}:=[\frac{1}{2},\frac{3}{4}],X_{2}:=(\frac{3}{4},1)$,
where the map is defined as follows
$TX_{1}:=X_{4},TX_{2}:=-X_{3},TX_{3}:=X_{1},TX_{4}=X_{4}$. The minus in
$TX_{2}:=-X_{3}$ means the flip (orientation change) of the of the interval
$X_{3}$, i.e. $T\frac{1}{4}=\frac{3}{4},T\frac{1}{2}=\frac{1}{4}$.

$0$$1$$1$$\frac{1}{4}$$\frac{3}{4}$$X_{1}:=[0,\frac{1}{4}),X_{2}:=[\frac{1}{4},\frac{1}{2}),X_{3}:=[\frac{1}{2},\frac{3}{4}],X_{4}:=(\frac{3}{4},1)$$X:=\cup_{i}X_{i}=[0,1),~{}~{}T\frac{1}{4}=\frac{3}{4},~{}T\frac{3}{4}=\frac{1}{4}$
Figure 6: A non orientation preserving circle PWI with an isolated periodic
trajectory $1/2,3/4$.

Observe that in this example the trajectory started at the point $\frac{1}{4}$
is 2-periodic, while all points from the surrounding intervals
$(0,\frac{1}{4})$ and $(\frac{1}{4},\frac{3}{4})$ in 2 iterations are mapped
to the 4-th interval $(\frac{3}{4},1)$ and never return back.
$\sqcup\\!\\!\\!\\!\sqcap$

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11institutetext: Big Data Institute, University of Oxford, England
22institutetext: Department of Oncology, University of Oxford, England
33institutetext: Department of Medical Physics and Clinical Engineering,
Oxford University Hospitals NHS FT, Oxford, England. 33email:
<EMAIL_ADDRESS>

# Autopet Challenge 2023: nnUNet-based whole-body 3D PET-CT Tumour
Segmentation

Anissa Alloula 11 0000-0003-1525-3994 Daniel R McGowan 2233
0000-0002-6880-5687 Bartłomiej W. Papież 11 0000-0002-8432-2511

###### Abstract

Fluorodeoxyglucose Positron Emission Tomography (FDG-PET) combined with
Computed Tomography (CT) scans are critical in oncology to the identification
of solid tumours and the monitoring of their progression. However, precise and
consistent lesion segmentation remains challenging, as manual segmentation is
time-consuming and subject to intra- and inter-observer variability. Despite
their promise, automated segmentation methods often struggle with false
positive segmentation of regions of healthy metabolic activity, particularly
when presented with such a complex range of tumours across the whole body. In
this paper, we explore the application of the nnUNet to tumour segmentation of
whole-body PET-CT scans and conduct different experiments on optimal training
and post-processing strategies. Our best model obtains a Dice score of 69% and
a false negative and false positive volume of 6.27 and 5.78 mL respectively,
on our internal test set. This model is submitted as part of the autoPET 2023
challenge. Our code is available at:
https://github.com/anissa218/autopet_nnunet

##  1 Introduction

Positron Emission Tomography / Computed Tomography (PET/CT) imaging is a vital
element of the diagnostic process and clinical management for a wide range of
solid tumours [1, 2]. PET imaging, usually conducted with Fluorodeoxyglucose
(FDG) radiotracer, a glucose analogue, provides metabolic information of areas
with high glucose consumption, and these can be indicative of the presence of
tumours [2, 3]. On the other hand, CT images provide anatomic information,
which can help a clinician determine whether abnormal glucose uptake
corresponds to a normal healthy metabolic organ or a malignant tumour [1]. The
accurate detection and quantification of tumours is crucial for diagnosis,
tumour staging and characterisation, and monitoring of disease evolution.
Manual tumour segmentation is a costly and long process, which may be subject
to intra- and inter- clinician variability [4]. For these reasons, automated
segmentation of these tumours is of particular importance.

Previous research has shown that it is possible to accurately segment tumours
with a variety of deep learning algorithms such as convolutional neural
networks like the U-Net or transformer-based architectures [5, 6, 7, 8].
However, whole-body PET/CT tumour segmentation is a particularly challenging
problem given the inherent multi-modality of the data as well as the extent of
anatomical coverage and morphological variability of the tumours that can be
present across the body[9]. Notably, last year’s challenge results showed
accurate segmentation of tumours in whole-body PET/CT scans, with a Dice score
of 0.79 for the winning model [9]. However, performance of the algorithms on
images acquired in a different hospital to the ones they were trained on was
substantially inferior, particularly in terms of Dice accuracy [9]. This
year’s challenge, Autopet-ii, aims to extend this work by focusing on
generalisation of the model to other acquisition protocols and sites.
Generalisation beyond a single scanner or acquisition site is challenging
because of domain shift, for instance due to different image resolutions,
varying levels of noise, and spatial variations [10, 11]. This is also
hindered by the lack of publicly available segmented PET/CT datasets on which
robust models can be trained. To this end, the Autopet-ii organisers provide
an extensive dataset of whole-body PET/CT scans of patients with and without
tumours [12].

In this paper, we extend nnUNet, a semantic segmentation method which
automatically adapts to a given dataset, and which has shown state-of-the art
results across a wide range of medical image segmentation tasks [13]. We also
investigate different post-processing methods to the improve final prediction.
Indeed, nnU-Net based methods consistently outperformed other architectures in
last year’s challenge [9].

##  2 Methods

###  2.1 Data

The challenge provided a set of 1016 whole-body PET/CT scans publicly
available on The Cancer Imaging Archive (TCIA) as well as the mask as
segmented by two radiologists [12]. These were acquired from University
Hospital Tübingen and University Hospital of the LMU with Siemens Biograph
mCT, mCT Flow and Biograph 64, GE Discovery 690 PET/CT scanners. 900 patients
were involved; approximately half had no cancer, and the other half presented
with histologically-proven malignant melanoma, lymphoma, or lung cancer.

In addition, a hidden test-set of 205 images were used to evaluate the models,
which was drawn in part from the same source distribution (1/4) and in part
(3/4) from a different distribution. 5 of these images were used for
preliminary testing, and the remainder will be used for final evaluation at
the end of the challenge (not available to the authors at the time of
submission).

One of the submitted models was also trained on a dataset including both the
TCIA PET/CT scans and an additional 200 PET/CT head and neck scans from the
Head and Neck Tumour Segmentation and Outcome Prediction in PET/CT 2022
challenge (HECKTOR) [14]. The aim was increase the diversity of the training
data in order to improve model generalisability.

###  2.2 Pre-processing

DICOM files were resampled (CT to PET imaging resolution, ranging from
200-677x400x400) and normalised. This involves Z-score intensity normalisation
for PET images and global dataset percentile clipping and Z-score
normalisation for CT. Subsequent pre-processing was done according to standard
nnUNet procedure [13]. The PET and CT images were concatenated into 2-channel
3D images with a median size of 2x236x400x400, and 90% were kept for training
and cross-validation, 10% were kept as a held-out internal test set.

###  2.3 Architecture and training

A standard 3D full resolution U-Net was trained and cross-validated on a Tesla
P100-SXM2-16GB GPU for 5 folds with the specifications shown in Table 1. PET-
CT channels were concatenated and cropped to patches as input. For each batch
of 2 patches, oversampling was implemented so that at least one of the two
contained a positive label in the ground truth segmentation. This was done to
ensure the model was trained with enough examples of lesions.

Table 1: Final training strategy Training strategy | Final Implementation
---|---
Number of epochs | 1500
Iterations per epoch | 250
Batch size | 2
Patch size | 128x128x128
Foreground batch oversampling |  $>$50%
Learning rate | 0.0001
Learning scheduler | Poly learning rate schedule
Loss function | Soft dice loss and cross entropy loss

Standard nnUNet data augmentation was used during training. This includes
random rotation, scaling, Gaussian noise and blur, brightness, contrast
augmentation, gamma correction, simulation of low resolution, and mirroring.

###  2.4 Post-processing

After training, different methods of post-processing were evaluated, based on
the removal of predicted tumour areas of low size. This was done through
connected component analysis of positive voxels [15].

###  2.5 Evaluation

Inference was conducted on the held-out set of 101 images and all of the
models were evaluated based on three metrics. At a later stage, the
preliminary test set (released by the organisers) of 5 images was also used
for evaluation. For the images where tumours were present, Dice overlap score
of the segmented lesions, as well as volume of false positive connected
components that do not overlap with true positives, were measured. In all
images, false negative volume was also evaluated. This corresponds to the
volume of positive connected components in the ground truth label that did not
overlap with any positive segmented mask.

##  3 Results

Models were trained with a variety of nnUNet parameters in order to determine
the optimal configuration, and results are presented in Table 2. These results
suggest that increasing training duration improved the model’s performance on
both the cross-validation and testing set. Removing mirroring as a data
augmentation step also showed similar benefits, perhaps as there are important
specifics to the organs on the left and right side of the PET/CT scans.
Moreover, increasing the input patch size and maximum number of feature maps
used in the model’s architecture did not translate to better performance in
the test set, perhaps due to over-fitting of the model. Finally, reducing the
number of positive images presented to the network at each batch did not
improve performance.

Table 2: Dice, false negative, and false positive scores on cross-validation (CV) images across all folds, as well as on 101 held-out internal test images. Removal of segmented regions of less than 10 connected voxels was also performed on the test predictions and metrics are shown in parentheses. Inference was not performed for the final g model due to time constraints.  |  | Cross-validation data | Internal testing data
---|---|---|---
Model |  Changes relative to baseline | Dice | FN | FP | Dice | FN | FP
baseline |  | 0.716 | 8.345 | 11.916 | 0.664 | 7.651 | 11.883
h |  1500 epochs | 0.734 | 7.430 | 10.258 | 0.680 | 7.376 | 8.043
a |  1500 epochs, no mirroring | 0.732 | 8.265 | 8.577 | 0.685 | 6.270 | 5.778
e |  1500 epochs, 0.01 lr | 0.670 | 9.926 | 22.255 | 0.611 | 14.878 | 13.408
c |  1500 epochs, 192 patch size, 512 features, 0.01 lr | 0.743 | 13.960 | 6.746 | 0.673 | 13.807 | 3.788
d |  192 patch size, 512 features | 0.716 | 8.345 | 11.916 | 0.682 | 8.923 | 3.239
f |  33% oversampling | 0.714 | 7.93 | 13.2 | 0.660 | 8.190 | 12.412
g |  10% oversampling | 0.722 | 7.227 | 23.79 |  |  |

The examination of the model’s prediction showed that a high proportion of the
segmentations were of small size (see Figure 1). As can be seen in Table 3,
removing these tumours caused a minor reduction in false positive volumes
compared to no removal (min size of 0). A threshold of 10 appeared as optimal,
maintaining a high Dice score while slightly lowering false positive scores.

Figure 1: Distribution of median positive connected component size across all training images. The purple bars show the sizes of the components in the ground truth labels while the red bars show the median sizes predicted by model ”a”. Table 3: Effect of the removal of segmented lesions below a certain threshold size of connected components on the three challenge metrics in the test set. Min size threshold | Dice | FP | FN
---|---|---|---
0 | 0.685 | 5.778 | 6.270
5 | 0.685 | 5.763 | 6.510
10 | 0.686 | 5.744 | 6.723
20 | 0.686 | 5.679 | 7.909
40 | 0.653 | 5.576 | 9.195
80 | 0.621 | 5.312 | 11.623

##  4 Discussion

As model ”a” obtained the best results on the internal test set, it was
submitted as our final model to the Autopet-2023 challenge.

However, in the future, many aspects remain to be investigated. Firstly, the
model outputs high volumes of false positive and false negative components,
despite having a relatively high Dice overlap score. Ensembling the
predictions of multiple models of different architectures and trained on
different modalities of input data (for instance just PET, just CT, or both)
may help maximise robustness and minimise false predictions. Indeed, in last
year’s challenge and other similar medical image segmentation tasks, ensembles
of multiple diverse models often outperform any individual model [9, 16, 17].

This includes developing an end-to-end model which predicts fewer very small
false positive connected components and therefore does not require post-
processing. Modifying the loss function, for instance with generalised Dice
overlap, which weighs the contribution of each label by the inverse of its
volume, may be a way to target this by penalising false small predictions
[18].

Moreover, in order to build a more generalisable model, it would be of
interest to quantify the uncertainty associated with the model’s predictions.
This would give an indication of how trustworthy each prediction is, which
would be particularly useful when doing inference on out-of-distribution
images [19, 20, 21]. Incorporating uncertainty during model training could
also be beneficial by maximising the learning of examples which the model is
uncertain about [20].

This work highlights the strength and adaptability of the nnUNet, which, with
very little parameter tuning, accurately segmented lesions. This work and the
Autopet-ii challenge represent crucial steps towards the development of
reliable and robust PET/CT segmentation algorithms, with significant potential
for valuable clinical application.

##  5 Acknowledgements

This work was supported by the EPSRC grant number EP/S024093/1 and the Centre
for Doctoral Training in Sustainable Approaches to Biomedical Science:
Responsible and Reproducible Research (SABS: R3) Doctoral Training Centre,
University of Oxford. The authors acknowledge the AUTOPET challenge for the
free publicly available PET/CT images used in this study. The computational
aspects of this research were supported by the Wellcome Trust Core Award Grant
Number 203141/Z/16/Z and the NIHR Oxford BRC. The views expressed are those of
the author(s) and not necessarily those of the NHS, the NIHR or the Department
of Health.

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