# More on tame dynamical systems

Eli Glasner  Department of Mathematics, Tel-Aviv University, Ramat Aviv,
Israel<EMAIL_ADDRESS>http://www.math.tau.ac.il/∼glasner and  Michael
Megrelishvili  Department of Mathematics, Bar-Ilan University, 52900 Ramat-
Gan, Israel<EMAIL_ADDRESS>http://www.math.biu.ac.il/∼megereli

(Date: revised on 07/02/23)

###### Abstract.

In this work, on the one hand, we survey and amplify old results concerning
tame dynamical systems and, on the other, prove some new results and exhibit
new examples of such systems. In particular, we study tame symbolic systems
and establish a neat characterization of tame subshifts. We also provide
sufficient conditions which ensure that certain coding functions are tame.
Finally we discuss examples where certain universal dynamical systems
associated with some Polish groups are tame.

###### Key words and phrases:

Asplund space, entropy, enveloping semigroup, fragmented function, non-
sensitivity, null system, Rosenthal space, Sturmian sequence, subshift,
symbolic dynamical system, tame function, tame system

###### 2010 Mathematics Subject Classification:

Primary 37Bxx, 46-xx; Secondary 54H15, 26A45

This research was supported by a grant of Israel Science Foundation (ISF
668/13)

The first named author thanks the Hausdorff Institute at Bonn for the
opportunity to participate in the Program “Universality and Homogeneity” where
part of this work was written, November 2013.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Classes of dynamical systems
  4. 4 A characterization of tame symbolic systems
  5. 5 Entropy and null systems
  6. 6 Some examples of tame functions and systems
  7. 7 Remarks about order preserving systems
  8. 8 Tame minimal systems and topological groups

##  1\. Introduction

Tame dynamical systems were introduced by A. Köhler [53] in 1995 and their
theory developed during the last decade in a series of works by several
authors (see e.g. [24, 29, 31, 43, 52, 25, 67]). Recently, connections to
other areas of mathematics like: Banach spaces, circularly ordered systems,
substitutions and tilings, quasicrystals, cut and project schemes and even
model theory and logic were established (see e.g. [3, 47, 11] and the survey
[34] for more details).

Recall that for any topological group $G$ and any dynamical $G$-system $X$
(defined by a continuous homomorphism $j:G\to{\mathrm{H}}(X)$ into the group
of homeomorphisms of the compact space $X$) the corresponding enveloping
semigroup $E(X)$ was defined by Robert Ellis as the pointwise closure of the
subgroup $j(G)$ of ${\mathrm{H}}(X)$ in the product space $X^{X}$. $E(X)$ is a
compact right topological semigroup whose algebraic and topological structure
often reflects properties of $(G,X)$ like almost periodicity (AP), weak almost
periodicity (WAP), distality, hereditary nonsensitivity (HNS) and tameness, to
mention a few. In the domain of symbolic dynamics WAP (and even HNS) systems
are necessarily countable, and in these classes minimal tame subshifts are
necessarily finite. In contrast there are many interesting symbolic (both
minimal and non-minimal) tame systems which are not HNS. Sturmian subshifts is
an important class of such systems.

A metric dynamical $G$-system $X$ is tame if and only if every element $p\in
E(X)$ of the enveloping semigroup $E(X)$ is a limit of a sequence of elements
from $G$, [29, 37], if and only if its enveloping semigroup $E(X)$ has
cardinality at most $2^{\aleph_{0}}$ [29, 33]. For example, the enveloping
semigroup of a Sturmian system has the form
$E(X)={\mathbb{T}}_{{\mathbb{T}}}\cup{\mathbb{Z}}$, the union of the “double-
circle” ${\mathbb{T}}_{\mathbb{T}}$ and ${\mathbb{Z}}$. Thus its cardinality
is $2^{\aleph_{0}}$. Another interesting property of a Sturmian system $X$ is
that both $X$ and $E(X)$ are circularly ordered dynamical systems. As it turns
out all circularly ordered systems are tame, [36].

Another characterization of tameness of combinatorial nature, via the notion
of independence tuples, is due to Kerr and Li [52]. Finally, the metrizable
tame systems are exactly those systems which admit a representation on a
separable Rosenthal Banach space [31] (a Banach space is called _Rosenthal_ if
it does not contain an isomorphic copy of $l_{1}$). As a by-product of the
latter characterization we were able in [31] to show that, e.g. for a Sturmian
system, the corresponding representation must take place on a separable
Rosenthal space $V$ with a non-separable dual, thereby proving the existence
of such a Banach space. The question whether such Banach spaces exist was a
major open problem until the mid of 70’s (the first counterexamples were
constructed independently by James, and Lindenstrauss and Stegall). For a
survey of Banach representation theory for dynamical systems we refer the
reader to [34].

In Sections 2 and 3 we review and amplify some basic results concerning tame
systems, fragmentability, and independent families of functions. In section 4
we provide a new characterization of tame symbolic dynamical systems, Theorem
4.7, and a combinatorial characterization of tame subsets
$D\subset{\mathbb{Z}}$ (i.e., subsets $D$ such that the associated subshift
$X_{D}\subset\\{0,1\\}^{{\mathbb{Z}}}$ is tame), Theorem 4.10. In Section 5 we
briefly review results relating tameness to independence and entropy.

In Section 6 we study coding functions that yield tame dynamical systems. A
closely related task is to produce invariant families of real valued functions
which do not contain independent infinite sequences (_tame families_). Theorem
6.8 gives some useful sufficient conditions for the tameness of families. For
instance, as a corollary of this theorem, we show in Theorem 6.10.3 that if
$G$ is a compact metric group and $f:X\to{\mathbb{R}}$ is a bounded function
with finitely many discontinuities, then $fG$ is a tame family.

We also describe some old and new interesting examples of symbolic tame
systems. E.g. in Theorem 6.21 we present a Sturmian-like extension of a
rotation on ${\mathbb{T}}^{d}$ whose enveloping semigroup has the form
$E(X)={\mathbb{Z}}\cup({\mathbb{T}}^{d}\times\mathcal{F})$, where
$\mathcal{F}$ is the collection of ordered orthonormal bases for
${\mathbb{R}}^{d}$.

In Section 7 we consider dynamical properties which are related to order
preservation. If $X$ is a compact space equipped with some kind of order
$\leq$, then subgroups of ${\mathrm{H}}_{+}(X,\leq)$, the group of order
preserving homeomorphisms of $X$, often have some special properties. E.g.
${\mathrm{H}}_{+}({\mathbb{T}})$, the group of orientation preserving
homeomorphisms of the circle ${\mathbb{T}}$, is Rosenthal representable,
Theorem 7.4, and we observe in Theorem 7.2 that it is Roelcke precompact. The
recipe described in Theorem 7.7 yields many tame coding functions for
subgroups of ${\mathrm{H}}({\mathbb{T}})$. Considering ${\mathbb{Z}}^{k}$ or
${\mathrm{PSL}}_{2}({\mathbb{R}})$ as subgroups of
${\mathrm{H}}({\mathbb{T}})$ we obtain in this way tame Sturmian like
${\mathbb{Z}}^{k}$ and ${\mathrm{PSL}}_{2}({\mathbb{Z}})$ dynamical systems.

Every topological group $G$ has a universal minimal system $M(G)$, a universal
minimal tame system $M_{t}(G)$, which is the largest tame $G$-factor of
$M(G)$, and also a universal irreducible affine $G$-system $I\\!A(G)$. In the
final Section 8 we discuss some examples, where $M(G)$ and $I\\!A(G)$ are
tame. When $M(G)$ is tame, so that $M(G)=M_{t}(G)$, we say that $G$ is
intrinsically tame. Of course every extremely amenable group (i.e. a group
with trivial $M(G)$) is intrinsically tame, and in Theorem 8.4 we show that
the Polish groups $G={\mathrm{H}}_{+}({\mathbb{T}})$ as well as the groups
${\mathrm{A}ut}\,(\mathbf{S}(2))$ and ${\mathrm{A}ut}\,(\mathbf{S}(3))$, of
automorphisms of the circular directed graphs $\mathbf{S}(2)$ and
$\mathbf{S}(3)$ respectively, are all intrinsically tame (but have nontrivial
$M(G)$). When the universal system $I\\!A(G)$ is tame we say that the group
$G$ is convexly intrinsically tame. Trivially every amenable group is convexly
intrinsically tame, and every intrinsically tame group is convexly
intrinsically tame. We show here that the group
${\mathrm{H}}_{+}({\mathbb{T}})$ is a nonamenable convexly intrinsically tame
topological group. Also, every semisimple Lie group $G$ with finite center and
no compact factors (e.g., $SL_{n}({\mathbb{R}})$) is convexly intrinsically
tame (but not intrinsically tame) 111An earlier version of this work is posted
on the Arxiv (arXiv:1405.2588).

##  2\. Preliminaries

By a topological space we mean a Tychonoff (completely regular Hausdorff)
space. The closure operator in topological spaces will be denoted by
${\rm{cls\,}}$. A function $f:X\to Y$ is _Baire class 1 function_ if the
inverse image $f^{-1}(O)$ of every open set $O\subset Y$ is $F_{\sigma}$ in
$X$, [48]. For a pair of topological spaces $X$ and $Y$, $C(X,Y)$ is the set
of continuous functions from $X$ into $Y$. We denote by $C(X)$ the Banach
algebra of bounded continuous real valued functions even when $X$ is not
necessarily compact.

All semigroups $S$ are assumed to be monoids, i.e., semigroups with a neutral
element which will be denoted by $e$. A (left) action of $S$ on a space $X$ is
a map $\pi:S\times X\to X$ such that $\pi(e,x)=x$ and
$\pi(st,x)=\pi(s,\pi(t,x))$ for every $s,t\in S$ and $x\in X$. We usually
write simply $sx$ for $\pi(s,x)$.

An $S$-_space_ is a topological space $X$ equipped with a continuous action
$\pi:S\times X\to X$ of a topological semigroup $S$ on the space $X$. A
compact $S$-space $X$ is called a _dynamical $S$-system_ and is denoted by
$(S,X)$. Note that in [31] and [33] we deal with the more general case of
separately continuous actions. We reserve the symbol $G$ for the case where
$S$ is a topological group. As usual, a continuous map $\alpha:X\to Y$ between
two $S$-systems is called an $S$-map or a homomorphism when
$\alpha(sx)=s\alpha(x)$ for every $(s,x)\in S\times X$. For every function
$f:X\to{\mathbb{R}}$ and $s\in S$ denote by $fs$ the composition
$f\circ\tilde{s}$. That is, $(fs)(x):=f(sx)$.

For every $S$-system $X$ we have a monoid homomorphism $j:S\to C(X,X)$,
$j(s)=\tilde{s}$, where $\tilde{s}:X\to X,x\mapsto sx=\pi(s,x)$ is the
$s$-translation ($s\in S$). The action is said to be _effective_
(_topologically effective_) if $j$ is an injection (respectively, a
topological embedding).

The enveloping semigroup $E(S,X)$ (or just $E(X)$) for a compact $S$-system
$X$ is defined as the pointwise closure ${{\rm{cls\,}}}(j(S))$ of
$\tilde{S}=j(S)$ in $X^{X}$. Then $E(S,X)$ is a right topological compact
monoid; i.e. for each $p\in E(X)$ right multiplication by $p$ is a continuous
map.

By a _cascade_ on a compact space $X$ we mean a ${\mathbb{Z}}$-action
${\mathbb{Z}}\times X\to X$. When dealing with cascades we usually write
$(T,X)$ instead of $({\mathbb{Z}},X)$, where $T$ is the $s$-translation $X\to
X$ corresponding to $s=1\in{\mathbb{Z}}$ ($0$ acts as the identity).

###  2.1. Background on fragmentability and tame families

The following definitions provide natural generalizations of the
fragmentability concept [45].

######  Definition 2.1.

Let $(X,\tau)$ be a topological space and $(Y,\mu)$ a uniform space.

  1. (1)

[44, 56] $X$ is $(\tau,\mu)$-fragmented by a (typically, not continuous)
function $f:X\to Y$ if for every nonempty subset $A$ of $X$ and every
${\varepsilon}\in\mu$ there exists an open subset $O$ of $X$ such that $O\cap
A$ is nonempty and the set $f(O\cap A)$ is ${\varepsilon}$-small in $Y$. We
also say in that case that the function $f$ is fragmented. Notation:
$f\in{\mathcal{F}}(X,Y)$, whenever the uniformity $\mu$ is understood. If
$Y={\mathbb{R}}$ then we write simply ${\mathcal{F}}(X)$.

  2. (2)

[29] We say that a family of functions $F=\\{f:(X,\tau)\to(Y,\mu)\\}$ is
fragmented if condition (1) holds simultaneously for all $f\in F$. That is,
$f(O\cap A)$ is ${\varepsilon}$-small for every $f\in F$.

  3. (3)

[31] We say that $F$ is an _eventually fragmented family_ if every infinite
subfamily $C\subset F$ contains an infinite fragmented subfamily $K\subset C$.

In Definition 2.1.1 when $Y=X,f={id}_{X}$ and $\mu$ is a metric uniformity, we
retrieve the usual definition of fragmentability (more precisely,
$(\tau,\mu)$-fragmentability) in the sense of Jayne and Rogers [45].
Implicitly it already appears in a paper of Namioka and Phelps [62].

######  Lemma 2.2.

[29, 31]

  1. (1)

It is enough to check the conditions of Definition 2.1 only for
$\varepsilon\in\gamma$ from a subbase $\gamma$ of $\mu$ and for closed
nonempty subsets $A\subset X$.

  2. (2)

If $f:(X,\tau)\to(Y,\mu)$ has _a point of continuity property_ PCP (i.e., for
every closed nonempty $A\subset X$ the restriction $f_{|A}:A\to Y$ has a
continuity point) then it is fragmented. If $(X,\tau)$ is hereditarily Baire
(e.g., compact, or Polish) and $(Y,\mu)$ is a pseudometrizable uniform space
then $f$ is fragmented if and only if $f$ has PCP. So, in particular, for
compact $X$, the set ${\mathcal{F}}(X)$ is exactly $B^{\prime}_{r}(X)$ in the
notation of [71].

  3. (3)

If $X$ is Polish and $Y$ is a separable metric space then $f:X\to Y$ is
fragmented iff $f$ is a Baire class 1 function (i.e., the inverse image of
every open set is $F_{\sigma}$).

  4. (4)

Let $(X,\tau)$ be a separable metrizable space and $(Y,\rho)$ a pseudometric
space. Suppose that $f:X\to Y$ is a fragmented onto map. Then $Y$ is
separable.

For other properties of fragmented maps and fragmented families we refer to
[61, 44, 50, 56, 57, 29, 31, 35]. Basic properties and applications of
fragmentability in topological dynamics can be found in [31, 34, 35].

###  2.2. Independent sequences of functions

Let $\\{f_{n}:X\to{\mathbb{R}}\\}_{n\in{\mathbb{N}}}$ be a uniformly bounded
sequence of functions on a _set_ $X$. Following Rosenthal [68] we say that
this sequence is an _$l_{1}$ -sequence_ on $X$ if there exists a real constant
$a>0$ such that for all $n\in{\mathbb{N}}$ and choices of real scalars
$c_{1},\dots,c_{n}$ we have

$a\cdot\sum_{i=1}^{n}|c_{i}|\leq||\sum_{i=1}^{n}c_{i}f_{i}||_{\infty}.$

A Banach space $V$ is said to be Rosenthal if it does not contain an
isomorphic copy of $l_{1}$, or equivalently, if $V$ does not contain a
sequence which is equivalent to an $l_{1}$-sequence.

A Banach space $V$ is an Asplund space if the dual of every separable Banach
subspace is separable. Every Asplund space is Rosenthal and every reflexive
space is Asplund.

A sequence $f_{n}$ of real valued functions on a set $X$ is said to be
_independent_ (see [68, 71]) if there exist real numbers $a<b$ such that

$\bigcap_{n\in P}f_{n}^{-1}(-\infty,a)\cap\bigcap_{n\in
M}f_{n}^{-1}(b,\infty)\neq\emptyset$

for all finite disjoint subsets $P,M$ of ${\mathbb{N}}$.

######  Definition 2.3.

We say that a bounded family $F$ of real valued (not necessarily, continuous)
functions on a set $X$ is tame if $F$ does not contain an independent
sequence.

Every bounded independent sequence is an $l_{1}$-sequence [68]. The sequence
of projections on the Cantor cube

$\\{\pi_{n}:\\{0,1\\}^{{\mathbb{N}}}\to\\{0,1\\}\\}_{n\in{\mathbb{N}}}$

and the sequence of Rademacher functions

$\\{r_{n}:[0,1]\to{\mathbb{R}}\\}_{n\in{\mathbb{N}}},\ \
r_{n}(x):={\rm{sgn}}(\sin(2^{n}\pi x))$

both are independent (hence, nontame). The following useful theorem
synthesizes some known results. It mainly is based on results of Rosenthal and
Talagrand. The equivalence of (1), (3) and (4) is a part of [71, Theorem
14.1.7] For the case (1) $\Leftrightarrow$ (2) note that every bounded
independent sequence $\\{f_{n}:X\to{\mathbb{R}}\\}_{n\in{\mathbb{N}}}$ is an
$l_{1}$-sequence (in the $\sup$-norm), [68, Prop. 4]. On the other hand, as
the proof of [68, Theorem 1] shows, if $\\{f_{n}\\}_{n\in{\mathbb{N}}}$ has no
independent subsequence then it has a pointwise convergent subsequence.
Bounded pointwise-Cauchy sequences in $C(X)$ (for compact $X$) are weak-Cauchy
as it follows by Lebesgue’s theorem. Now Rosenthal’s dichotomy theorem [68,
Main Theorem] asserts that $\\{f_{n}\\}$ has no $l_{1}$-sequence. In [31,
Sect. 4] we show why eventual fragmentability of $F$ can be included in this
list (item (5)).

######  Theorem 2.4.

Let $X$ be a compact space and $F\subset C(X)$ a bounded subset. The following
conditions are equivalent:

  1. (1)

$F$ does not contain an $l_{1}$-sequence.

  2. (2)

$F$ is a tame family (does not contain an independent sequence).

  3. (3)

Each sequence in $F$ has a pointwise convergent subsequence in
${\mathbb{R}}^{X}$.

  4. (4)

The pointwise closure ${{\rm{cls\,}}}(F)$ of $F$ in ${\mathbb{R}}^{X}$
consists of fragmented maps, that is,
${{\rm{cls\,}}}(F)\subset{\mathcal{F}}(X).$

  5. (5)

$F$ is an eventually fragmented family.

Let $X$ be a topological space and $F\subset l_{\infty}(X)$ be a norm bounded
family. Recall that $F$ has Grothendieck’s Double Limit Property (DLP) on $X$
if for every sequence $\\{f_{n}\\}\subset F$ and every sequence
$\\{x_{m}\\}\subset X$ the limits

$\lim_{n}\lim_{m}f_{n}(x_{m})\ \ \ \text{and}\ \ \
\lim_{m}\lim_{n}f_{n}(x_{m})$

are equal whenever they both exist.

The following examples are mostly reformulations of known results; the details
can be found in [31, 35, 59].

######  Examples 2.5.

  1. (1)

A Banach space $V$ is Rosenthal iff every bounded subset $F\subset V$ is tame
(as a family of functions) on every bounded subset $Y\subset V^{*}$ of the
dual space $V^{*}$, iff $F$ is eventually fragmented on $Y$.

  2. (2)

A Banach space $V$ is Asplund iff every bounded subset $F\subset V$ is a
fragmented family of functions on every bounded subset $Y\subset V^{*}$.

  3. (3)

A Banach space is reflexive iff every bounded subset $F\subset V$ has DLP on
every bounded subset $X\subset V^{*}$.

  4. (4)

((DLP) $\Rightarrow$ Tame) Let $F$ be a bounded family of real valued (not
necessarily continuous) functions on a set $X$ such that $F$ has DLP. Then $F$
is tame.

  5. (5)

The family ${\mathrm{Homeo}}\,[0,1]$, of all self homeomorphisms of $[0,1]$,
is tame (but does not have the DLP on $[0,1]$).

  6. (6)

Let $X$ be a circularly (e.g., linearly) ordered set. Then any bounded family
$F$ of real functions with bounded total variation is tame.

Note that in (1), (2) and (3) the converse statements are true; as it follows
from results of [35] every bounded tame (DLP) family $F$ on $X$ can be
represented on a Rosenthal (Asplund, reflexive) Banach space. Recall that a
representation of $F$ on a Banach space $V$ consists of a pair $(\nu,\alpha)$
of bounded maps $\nu:F\to V,\ \alpha:X\to V^{*}$ (with weak-star continuous
$\alpha$) such that

$f(x)=\langle\nu(f),\alpha(x)\rangle\ \ \ \forall\ f\in F,\ \ \forall\ x\in
X.$

In other words, the following diagram commutes

$\textstyle{F\ignorespaces\ignorespaces\ignorespaces\ignorespaces\times
X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nu}$$\scriptstyle{\alpha}$$\textstyle{{\mathbb{R}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{id}$$\textstyle{V\times
V^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{R}}$

###  2.3. More properties of fragmented families

Here we discuss a general principle: the fragmentability of a family of
continuous maps defined on a compact space is “countably-determined”. The
following theorem is inspired by results of Namioka and can be deduced, after
some reformulations, from [61, Theorems 3.4 and 3.6]. See also [9, Theorem
2.1].

######  Theorem 2.6.

Let $F=\\{f_{i}:X\to Y\\}_{i\in I}$ be a bounded family of continuous maps
from a compact (not necessarily metrizable) space $(X,\tau)$ into a
pseudometric space $(Y,d)$. The following conditions are equivalent:

  1. (1)

$F$ is a fragmented family of functions on $X$.

  2. (2)

Every _countable_ subfamily $K$ of $F$ is fragmented.

  3. (3)

For every countable subfamily $K$ of $F$ the pseudometric space
$(X,\rho_{K,d})$ is separable, where $\rho_{K,d}(x_{1},x_{2}):=\sup_{f\in
K}d(f(x_{1}),f(x_{2})).$

###### Proof.

(1) $\Rightarrow$ (2) is trivial.

(2) $\Rightarrow$ (3): Let $K$ be a countable subfamily of $F$. Consider the
natural map

$\pi:X\to Y^{K},\pi(x)(f):=f(x).$

By (2), $K$ is a fragmented family. This means (see [29, Def. 6.8]) that the
map $\pi$ is $(\tau,\mu_{K})$-fragmented, where $\mu_{K}$ is the uniformity of
$d$-uniform convergence on $Y^{K}:=\\{f:K\to(Y,d)\\}$. Then the map $\pi$ is
also $(\tau,d_{K})$-fragmented, where $d_{K}$ is the pseudometric on $Y^{K}$
defined by

$d_{K}(z_{1},z_{2}):=\sup_{f\in K}d(z_{1}(f),z_{2}(f)).$

Since $d$ is bounded, $d_{K}(z_{1},z_{2})$ is finite and $d_{K}$ is well-
defined. Denote by $(X_{K},\tau_{p})$ the subspace $\pi(X)\subset Y^{K}$ in
pointwise topology. Since $K\subset C(X)$, the induced map $\pi_{0}:X\to
X_{K}$ is a continuous map onto the compact space $(X_{K},\tau_{p})$. Denote
by $i:(X_{K},\tau_{p})\to(Y^{K},d_{K})$ the inclusion map. So,
$\pi=i\circ\pi_{0}$, where the map $\pi$ is $(\tau,d_{K})$-fragmented. This
easily implies (see [31, Lemma 2.3.5]) that $i$ is
$(\tau_{p},d_{K})$-fragmented. It immediately follows that the identity map
$id:(X_{K},\tau_{p})\to(X_{K},d_{K})$ is $(\tau_{p},d_{K})$-fragmented.

Since $K$ is countable, $(X_{K},\tau_{p})\subset Y^{K}$ is metrizable.
Therefore, $(X_{K},\tau_{p})$ is second countable (being a metrizable
compactum). Now, since $d_{K}$ is a pseudometric on $Y^{K}$, and
$id:(X_{K},\tau_{p})\to(X_{K},d_{K})$ is $(\tau_{p},d_{K})$-fragmented, we can
apply Lemma 2.2.4. It directly implies that the set $X_{K}$ is a separable
subset of $(Y^{K},d_{K})$. This means that $(X,\rho_{K,d})$ is separable.

(3) $\Rightarrow$ (1) : Suppose that $F$ is not fragmented. Thus, there exists
a non-empty closed subset $A\subset X$ and an ${\varepsilon}>0$ such that for
each non-empty open subset $O\subset X$ with $O\cap A\neq\emptyset$ there is
some $f\in F$ such that $f(O\cap A)$ is not ${\varepsilon}$-small in $(Y,d)$.
Let $V_{1}$ be an arbitrary non-empty relatively open subset in $A$. There are
$a,b\in V_{1}$ and $f_{1}\in F$ such that
$d(f_{1}(a),f_{1}(b))>{\varepsilon}$. Since $f_{1}$ is continuous we can
choose relatively open subsets $V_{2},V_{3}$ in $A$ with
${\rm{cls\,}}(V_{2}\cup V_{3})\subset V_{1}$ such that
$d(f_{1}(x),f_{1}(y))>{\varepsilon}$ for every $(x,y)\in V_{2}\times V_{3}$.

By induction we can construct a sequence $\\{V_{n}\\}_{n\in{\mathbb{N}}}$ of
non-empty relatively open subsets in $A$ and a sequence
$K:=\\{f_{n}\\}_{n\in{\mathbb{N}}}$ in $F$ such that:

  * (i)

$V_{2n}\cup V_{2n+1}\subset V_{n}$ for each $n\in{\mathbb{N}}$;

  * (ii)

$d(f_{n}(x),f_{n}(y))>{\varepsilon}$ for every $(x,y)\in V_{2n}\times
V_{2n+1}$.

We claim that $(X,\rho_{K,d})$ is not separable, where

$\rho_{K,d}(x_{1},x_{2}):=\sup_{f\in K}d(f(x_{1}),f(x_{2})).$

In fact, for each _branch_

$\alpha:=V_{1}\supset V_{n_{1}}\supset V_{n_{2}}\supset\cdots$

where for each $i,n_{i+1}=2n_{i}$ or $2n_{i}+1$, by compactness of $X$ one can
choose an element

$x_{\alpha}\in\bigcap_{i\in{\mathbb{N}}}{{\rm{cls\,}}}(V_{n_{i}}).$

If $x=x_{\alpha}$ and $y=x_{\beta}$ come from different branches, then there
is an $n\in{\mathbb{N}}$ such that $x\in{{\rm{cls\,}}}(V_{2n})$ and
$y\in{{\rm{cls\,}}}(V_{2n+1})$ (or vice versa). In any case it follows from
(ii) and the continuity of $f_{n}$ that
$d(f_{n}(x),f_{n}(y))\geq{\varepsilon}$, hence
$\rho_{K,d}(x,y)\geq{\varepsilon}$. Since there are uncountably many branches
we conclude that $A$ and hence also $X$ are not $\rho_{K,d}$-separable. ∎

######  Definition 2.7.

[17, 57] Let $X$ be a compact space and $F\subset C(X)$ a norm bounded family
of continuous real valued functions on $X$. Then $F$ is said to be an _Asplund
family for_ $X$ if for every countable subfamily $K$ of $F$ the pseudometric
space $(X,\rho_{K,d})$ is separable, where

$\rho_{K,d}(x_{1},x_{2}):=\sup_{f\in K}|f(x_{1})-f(x_{2})|.$

######  Corollary 2.8.

Let $X$ be a compact space and $F\subset C(X)$ a norm bounded family of
continuous real valued functions on $X$. Then $F$ is fragmented if and only if
$F$ is an Asplund family for $X$.

######  Theorem 2.9.

Let $F=\\{f_{i}:X\to Y\\}_{i\in I}$ be a family of continuous maps from a
compact (not necessarily metrizable) space $(X,\tau)$ into a uniform space
$(Y,\mu)$. Then $F$ is fragmented if and only if every countable subfamily
$A\subset F$ is fragmented.

###### Proof.

The proof can be reduced to Theorem 2.6. Every uniform space can be uniformly
approximated by pseudometric spaces. Using Lemma 2.2.1 we can assume that
$(Y,\mu)$ is pseudometrizable; i.e. there exists a pseudometric $d$ such that
$\mathrm{unif}(d)=\mu$. Moreover, replacing $d$ by the uniformly equivalent
pseudometric $\frac{d}{1+d}$ we can assume that $d\leq 1$. ∎

##  3\. Classes of dynamical systems

######  Definition 3.1.

A compact dynamical $S$-system $X$ is said to be _tame_ if one of the
following equivalent conditions are satisfied:

  1. (1)

for every $f\in C(X)$ the family $fS:=\\{fs:s\in S\\}$ has no independent
subsequence.

  2. (2)

every $p\in E(X)$ is a fragmented map $X\to X$.

  3. (3)

for every $p\in E(X)$ and every $f\in C(X)$ the composition
$fp:X\to{\mathbb{R}}$ has PCP.

The following principal result is a dynamical analog of the Bourgain-Fremlin-
Talagrand dichotomy [8, 72].

######  Theorem 3.2.

[29] _(A dynamical version of BFT dichotomy)_ Let $X$ be a compact metric
dynamical $S$-system and let $E=E(X)$ be its enveloping semigroup. Either

  1. (1)

$E$ is a separable Rosenthal compact space (hence $E$ is Fréchet and
${card}\;{E}\leq 2^{\aleph_{0}}$); or

  2. (2)

the compact space $E$ contains a homeomorphic copy of $\beta{\mathbb{N}}$
(hence ${card}\;{E}=2^{2^{\aleph_{0}}}$).

The first possibility holds iff $X$ is a tame $S$-system.

Thus, a metrizable dynamical system is tame iff $\rm{card\,}(E(X))\leq
2^{\aleph_{0}}$ iff $E(X)$ is a Rosenthal compactum (or a Fréchet space).
Moreover, by [37] a metric $S$-system is tame iff every $p\in E(X)$ is a Baire
class 1 map $p:X\to X$.

The class of tame dynamical systems is quite large. It is closed under
subsystems, products and factors. Recall that an $S$-dynamical system $X$ is
weakly almost periodic ($\mathrm{WAP}$) if and only if every $p\in E(X)$ is a
continuous map. As every continuous map $X\to X$ is fragmented it follows that
every WAP system is tame. A metrizable $S$-system $X$ is WAP iff $(S,X)$ is
representable on a reflexive Banach space. The class of hereditarily
nonsensitive systems ($\mathrm{HNS}$) is an intermediate class of systems,
[34]. The property HNS admits a reformulation in terms of enveloping
semigroup: $(S,X)$ is HNS iff $E(S,X)$ (equivalently, $\tilde{S}$) is a
fragmented family. Of course, this implies that every $p\in E(X)$ is
fragmented. So, indeed, $\mathrm{WAP}\subset\mathrm{HNS}\subset\mathrm{Tame}$.
A metrizable $S$-system $X$ is HNS iff $E(X)$ is metrizable iff $(S,X)$ is
Asplund representable (RN, in another terminology), [29, 37].

###  3.1. Some classes of functions

A _compactification_ of $X$ is a continuous map $\gamma:X\to Y$ with a dense
range where $Y$ is compact. When $X$ and $Y$ are $S$-spaces and $\gamma$ is an
$S$-map we say that $\gamma$ is an _$S$ -compactification_.

A function $f\in C(X)$ on an $S$-space $X$ is said to be Right Uniformly
Continuous if the induced right action $C(X)\times S\to C(X)$ is continuous at
the points $(f,s)$, where $s\in S$. Notation: $f\in\operatorname{RUC}(X)$. If
$X$ is a compact $S$-space then $\operatorname{RUC}(X)=C(X)$. Note that
$f\in\operatorname{RUC}(X)$ if and only if there exists an
$S$-compactification $\gamma:X\to Y$ such that $f=\tilde{f}\circ\gamma$ for
some $\tilde{f}\in C(Y)$. In this case we say that $f$ _comes_ from the
$S$-compactification $\gamma:X\to Y$.

The function $f$ is said to be: a) _WAP_ ; b) _Asplund_ ; c) _tame_ if $f$
comes from an $S$-compactification $\gamma:X\to Y$ such that $(S,Y)$ is: WAP,
HNS or tame respectively. For the corresponding classes of functions we use
the notation: $\operatorname{WAP}(X)$, $\operatorname{Asp}(X),$
$\mathrm{Tame}(X)$, respectively. Each of these is a norm closed $S$-invariant
subalgebra of the $S$-invariant algebra $\operatorname{RUC}(X)$ and
$\operatorname{WAP}(X)\subset\operatorname{Asp}(X)\subset\mathrm{Tame}(X).$
For more details see [33, 34]. As a particular case we have defined the
algebras $\operatorname{WAP}(S)$, $\operatorname{Asp}(S)$, $\mathrm{Tame}(S)$
corresponding to the left action of $S$ on itself.

The $S$-invariant subalgebra $\operatorname{Tame}(S)$ of
$\operatorname{RUC}(S)$ induces an $S$-compactification of $S$ which we denote
by $S\to S^{\operatorname{Tame}}$. Recall that it is a semigroup
compactification of $S$ and that $S^{\operatorname{Tame}}$ is a compact right
topological semigroup, [33]. Similarly, one defines the compactifications
$S^{\operatorname{AP}},S^{\operatorname{WAP}},S^{\operatorname{Asp}}$. Here AP
means almost periodic. AP compact $G$-systems (for groups $S:=G$) are just
equicontinuous systems.

###  3.2. Cyclic $S$-compactifications

Let $X$ be an $S$-space. For every $f\in\mathrm{RUC}(X)$ define the following
pointwise continuous natural $S$-map

$\delta_{f}:X\to\operatorname{RUC}(S),\ \ \delta_{f}(x)(g):=f(gx).$

It induces an $S$-compactification $\delta_{f}:X\to X_{f}$, where $X_{f}$ is
the pointwise closure of $\delta_{f}(X)$ in $\operatorname{RUC}(S)$. Denote by
${\mathcal{A}}_{f}:=\langle fS\rangle$ the smallest $S$-invariant unital
Banach subalgebra of $\mathrm{RUC}(X)$ which contains $f$. The corresponding
Gelfand $S$-compactification is equivalent to $\delta_{f}:X\to X_{f}$. Let
$\widetilde{f}:=\widehat{e}|_{X_{f}}$, where $\widehat{e}$ is the evaluation
at $e$ functional on $\operatorname{RUC}(S)$. Then $f$ comes from the
$S$-system $X_{f}$. Moreover, $\widetilde{f}S$ separates points of $X_{f}$.

We call $\delta_{f}:X\to X_{f}$ the _cyclic compactification_ of $X$ (induced
by $f$), [5, 29, 33].

######  Definition 3.3.

[29, 33] We say that a compact dynamical $S$-system $X$ is _cyclic_ if there
exists $f\in C(X)$ such that $(S,X)$ is topologically $S$-isomorphic to the
Gelfand space $X_{f}$ of the $S$-invariant unital subalgebra
${\mathcal{A}}_{f}\subset C(X)$ generated by the orbit $fS$.

######  Lemma 3.4.

Let $\gamma:X\to Y$ be an $S$-compactification and $f\in C(X)$.

  1. (1)

$f$ comes from $\gamma$ (i.e., $f=\bar{f}\circ\gamma$ for some $\bar{f}\in
C(Y)$) if and only if there exists a continuous onto $S$-map $q:Y\to X_{f}$
such that $\bar{f}=\tilde{f}\circ q$ and the following diagram is commutative

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{\pi_{f}}$$\scriptstyle{\gamma}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\textstyle{\mathbb{R}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
X_{f}}$$\scriptstyle{\widetilde{f}}$

  2. (2)

$q:Y\to X_{f}$ in (1) is an isomorphism of $S$-compactifications if and only
if $\bar{f}S$ separates points of $Y$ (where, as before,
$\bar{f}=\tilde{f}\circ q$).

###### Proof.

Use Gelfand’s description of compactifications in terms of the corresponding
algebras and the Stone-Weierstrass Theorem. ∎

######  Remark 3.5.

Let $X$ be a (not necessarily compact) $S$-space and
$f\in\operatorname{RUC}(X)$. Then, as was shown in [33], there exist a cyclic
$S$-system $X_{f}$, a continuous $S$-compactification $\pi_{f}:X\to X_{f}$,
and a continuous function $\tilde{f}:X_{f}\to{\mathbb{R}}$ such that
$f=\tilde{f}\circ\pi_{f}$; that is, $f$ comes from the $S$-compactification
$\pi_{f}:X\to X_{f}$. The collection of functions $\tilde{f}S$ separates
points of $X_{f}$.

######  Theorem 3.6.

Let $X$ be a compact $S$-space and $f\in C(X)$. The following conditions are
equivalent:

  1. (1)

$f\in\operatorname{Tame}(X)$ (i.e. $f$ comes from a tame dynamical system).

  2. (2)

$fS$ is a tame family.

  3. (3)

${{\rm{cls\,}}}_{p}(fS)\subset{\mathcal{F}}(X)$.

  4. (4)

$fS$ is an eventually fragmented family.

  5. (5)

For every countable infinite subset $A\subset S$ there exists a countable
infinite subset $A^{\prime}\subset A$ such that the corresponding pseudometric

$\rho_{f,A^{\prime}}(x,y):=\sup\\{|f(gx)-f(gy)|:\ \ g\in A^{\prime}\\}$

on $X$ is separable.

  6. (6)

The cyclic $S$-space $X_{f}$ is Rosenthal representable (i.e., WRN).

###### Proof.

The equivalence of (1), (2), (3) and (4) follows from Theorem 2.4. For (4)
$\Leftrightarrow$ (5) use Theorem 2.6. For (4) $\Leftrightarrow$ (6) we refer
to [31, 35]. ∎

##  4\. A characterization of tame symbolic systems

###  4.1. Symbolic systems and coding functions

The binary _Bernoulli shift system_ is defined as the cascade
$(\Omega,\sigma)$, where $\Omega:=\\{0,1\\}^{{\mathbb{Z}}}$. We have the
natural ${\mathbb{Z}}$-action on the compact metric space $\Omega$ induced by
the $\sigma$-shift:

${\mathbb{Z}}\times\Omega\to\Omega,\ \ \
\sigma^{m}(\omega_{i})_{i\in{\mathbb{Z}}}=(\omega_{i+m})_{i\in{\mathbb{Z}}}\ \
\ \forall(\omega_{i})_{i\in{\mathbb{Z}}}\in\Omega,\ \ \forall
m\in{\mathbb{Z}}.$

More generally, for a discrete monoid $S$ and a finite alphabet
$A:=\\{0,1,\dots,n\\}$ the compact space $\Omega:=A^{S}$ is an $S$-space under
the action

$S\times\Omega\to\Omega,\ \ (s\omega)(t)=\omega(ts),\ \omega\in A^{S},\ \
s,t\in S.$

A closed $S$-invariant subset $X\subset A^{S}$ defines a subsystem $(S,X)$.
Such systems are called subshifts or symbolic dynamical systems.

######  Definition 4.1.

  1. (1)

Let $S\times X\to X$ be an action on a (not necessarily compact) space $X$,
$f:X\to{\mathbb{R}}$ a bounded (not necessarily continuous) function, and
$z\in X$. Define a _coding function_ as follows:

$\varphi:=m(f,z):S\to{\mathbb{R}},\ s\mapsto f(sz).$

  2. (2)

When $f(X)\subseteq\\{0,1,\dots,d\\}$ every such code generates a point
transitive subshift $S_{\varphi}$ of $A^{S}$, where $A=\\{0,1,\dots,d\\}$ and

$S_{\varphi}:={\rm{cls\,}}_{p}\\{g\varphi:g\in S\\}\subset A^{S}\ \ \ \
(\text{where}\ g\varphi(t)=\varphi(tg))$

is the pointwise closure of the left $S$-orbit $S\varphi$ in the space
$\\{0,1,\cdots,d\\}^{S}$.

When $S={\mathbb{Z}}^{k}$ we say that $f$ is a _$(k,d)$ -code_. In the
particular case of the characteristic function $\chi_{D}:X\to\\{0,1\\}$ for a
subset $D\subset X$ and $S={\mathbb{Z}}$ we get a $(1,1)$-code, i.e. a binary
function $m(D,z):{\mathbb{Z}}\to\\{0,1\\}$ which generates a
${\mathbb{Z}}$-subshift of the Bernoulli shift on $\\{0,1\\}^{{\mathbb{Z}}}$.

Regarding some dynamical and combinatorial aspects of coding functions see
[18, 6].

Among others we will study the following question

######  Question 4.2.

When is a coding $\varphi$ function tame? Equivalently, when is the associated
transitive subshift system $S_{\varphi}\subset\\{0,1\\}^{{\mathbb{Z}}}$ with
$\varphi=m(D,z)$ tame?

Some restrictions on $D$ are really necessary because _every_ binary
bisequence $\varphi:{\mathbb{Z}}\to\\{0,1\\}$ can be encoded as
$\varphi=m(D,z)$.

It follows from results in [31] that a coding bisequence
$c:{\mathbb{Z}}\to{\mathbb{R}}$ is tame iff it can be represented as a
generalized matrix coefficient of a Rosenthal Banach space representation.
That is, iff there exist: a Rosenthal Banach space $V$, a linear isometry
$\sigma\in\operatorname{Iso}(V)$ and two vectors $v\in V$, $\varphi\in V^{*}$
such that

$c_{n}=\langle\sigma^{n}(v),\varphi\rangle=\varphi(\sigma^{n}(v))\ \ \ \
\forall n\in{\mathbb{Z}}.$

Let, as above, $A^{S}$ be the full symbolic shift $S$-system. For a nonempty
$L\subseteq S$ define the natural projection

$\pi_{L}:A^{S}\to A^{L}.$

The compact zero-dimensional space $A^{S}$ is metrizable iff $S$ is countable
(and, in this case, $A^{S}$ is homeomorphic to the Cantor set).

It is easy to see that the full shift system $\Omega=A^{S}$ (hence also every
subshift) is _uniformly expansive_. This means that there exists an entourage
${\varepsilon}_{0}\in\mu$ in the natural uniform structure of $A^{S}$ such
that for every distinct $\omega_{1}\neq\omega_{2}$ in $\Omega$ one can find
$s\in S$ with $(s\omega_{1},s\omega_{2})\notin{\varepsilon}_{0}$. Indeed, take

${\varepsilon}_{0}:=\\{(u,v)\in\Omega\times\Omega:\ u(e)=v(e)\\},$

where $e$, as usual, is the neutral element of $S$.

######  Lemma 4.3.

Every symbolic dynamical $S$-system $X\subset\Omega=A^{S}$ is cyclic
(Definition 3.3).

###### Proof.

It suffices to find $f\in C(X)$ such that the orbit $fS$ separates the points
of $X$ since then, by the Stone-Weierstrass theorem, $(S,X)$ is isomorphic to
its cyclic $S$-factor $(S,X_{f})$. The family

$\\{\pi_{s}:X\to A=\\{0,1,\dots,n\\}\subset{\mathbb{R}}\\}_{s\in S}$

of basic projections clearly separates points on $X$ and we let
$f:=\pi_{e}:X\to{\mathbb{R}}$. Now observe that $fS=\\{\pi_{s}\\}_{s\in S}$. ∎

A topological space $(X,\tau)$ is scattered (i.e., every nonempty subspace has
an isolated point) iff $X$ is $(\tau,\xi)$-fragmented, for arbitrary uniform
structure $\xi$ on the _set_ $X$.

######  Proposition 4.4.

[57, Prop. 7.15] Every scattered compact jointly continuous $S$-space $X$ is
RN (that is, Asplund representable).

###### Proof.

A compactum $X$ is scattered iff $C(X)$ is Asplund, [62]. Now use the
canonical $S$-representation of $(S,X)$ on the Asplund space $V:=C(X)$. ∎

The following result recovers and extends [29, Sect. 10] and [57, Sect. 7].

######  Theorem 4.5.

For a discrete monoid $S$ and a finite alphabet $A$ let $X\subset A^{S}$ be a
subshift. The following conditions are equivalent:

  1. (1)

$(S,X)$ is Asplund representable (that is, RN).

  2. (2)

$(S,X)$ is HNS.

  3. (3)

$X$ is scattered.

If, in addition, $X$ is metrizable (e.g., if $S$ is countable) then each of
the conditions above is equivalent also to:

  4. (4)

$X$ is countable.

###### Proof.

(1) $\Rightarrow$ (2): It was proved in [29, Lemma 9.8].

(2) $\Rightarrow$ (3): Let $\mu$ be the natural uniformity on $X$ and
$\mu_{S}$ the (finer) uniformity of uniform convergence on $X\subset X^{S}$
(we can treat $X$ as a subset of $X^{S}$ under the assignment
$x\mapsto\hat{x}$, where $\hat{x}(s)=sx$). If $X$ is HNS then the family
$\tilde{S}$ is fragmented. This means that $X$ is $\mu_{S}$-fragmented. As we
already mentioned, every subshift $X$ is uniformly $S$-expansive. Therefore,
$\mu_{S}$ coincides with the discrete uniformity $\mu_{\Delta}$ on $X$ (the
largest possible uniformity on the _set_ $X$). Hence, $X$ is also
$\mu_{\Delta}$-fragmented. This means that $X$ is a scattered compactum.

(3) $\Rightarrow$ (1): Use Proposition 4.4.

If $X$ is metrizable then

(4) $\Leftrightarrow$ (3): A scattered compactum is metrizable iff it is
countable. ∎

Every zero-dimensional compact ${\mathbb{Z}}$-system $X$ can be embedded into
a product $\prod X_{f}$ of (cyclic) subshifts $X_{f}$ (where, one may consider
only continuous functions $f:X\to\\{0,1\\}$) of the Bernoulli system
$\\{0,1\\}^{{\mathbb{Z}}}$.

For more information about countable (HNS and WAP) subshifts see [70, 10, 2].

######  Problem 4.6.

Find a nice characterization for WAP (necessarily, countable)
${\mathbb{Z}}$-subshifts.

Next we consider tame subshifts.

######  Theorem 4.7.

Let $X$ be a subshift of $\Omega=A^{S}$. The following conditions are
equivalent:

  1. (1)

$(S,X)$ is a tame system.

  2. (2)

For every infinite subset $L\subseteq S$ there exists an infinite subset
$K\subseteq L$ and a countable subset $Y\subseteq X$ such that

$\pi_{K}(X)=\pi_{K}(Y).$

That is,

$\forall x=(x_{s})_{s\in S}\in X,\ \exists y=(y_{s})_{s\in S}\in
Y\quad{\text{with}}\quad x_{k}=y_{k}\ \ \forall k\in K.$

  3. (3)

For every infinite subset $L\subseteq S$ there exists an infinite subset
$K\subseteq L$ such that $\pi_{K}(X)$ is a countable subset of $A^{K}$.

  4. (4)

$(S,X)$ is Rosenthal representable (that is, WRN).

###### Proof.

(1) $\Leftrightarrow$ (2): As in the proof of Lemma 4.3 define $f:=\pi_{e}\in
C(X)$. Then $X$ is isomorphic to the cyclic $S$-space $X_{f}$. $(S,X)$ is a
tame system iff $C(X)=\operatorname{Tame}(X)$. By Lemma 4.3,
$C(X)={\mathcal{A}}_{f}$, so we have only to show that
$f\in\operatorname{Tame}(X)$.

By Theorem 3.6, $f:=\pi_{e}:X\to{\mathbb{R}}$ is a tame function iff for every
infinite subset $L\subset S$ there exists a countable infinite subset
$K\subset L$ such that the corresponding pseudometric

$\rho_{f,K}(x,y):=\sup_{k\in K}\\{|(\pi_{e})(kx)-(\pi_{e})(ky)|\\}=\sup_{k\in
K}\\{|x_{k}-y_{k}|\\}$

on $X$ is separable. The latter assertion means that there exists a countable
subset $Y$ which is $\rho_{f,K}$-dense in $X$. Thus for every $x\in X$ there
is a point $y\in Y$ with $\rho_{f,K}(x,y)<1/2$. As the values of the function
$f=\pi_{0}$ are in the set $A$, we conclude that $\pi_{K}(x)=\pi_{K}(y)$,
whence

$\pi_{K}(X)=\pi_{K}(Y).$

The equivalence of (2) and (3) is obvious.

(1) $\Rightarrow$ (4): $(S,X)$ is Rosenthal-approximable (Theorem 3.6.1). On
the other hand, $(S,X)$ is cyclic (Lemma 4.3). By Theorem 3.6.7 we can
conclude that $(S,X)$ is WRN.

(4) $\Rightarrow$ (1): Follows directly by Theorem 3.6.1. ∎

######  Remark 4.8.

From Theorem 4.7 we can deduce the following peculiar fact. If $X$ is a tame
subshift of $\Omega=\\{0,1\\}^{{\mathbb{Z}}}$ and $L\subset{\mathbb{Z}}$ an
infinite set, then there exist an infinite subset $K\subset L$, $k\geq 1$, and
$a\in\\{0,1\\}^{2k+1}$ such that $X\cap[a]\not=\emptyset$ and $\forall
x,x^{\prime}\in X\cap[a]$ we have $x|_{K}=x^{\prime}|_{K}$. Here
$[a]=\\{z\in\\{0,1\\}^{\mathbb{Z}}:z(j)=a(j),\ \forall|j|\leq k\\}$. In fact,
since $\pi_{K}(X)$ is a countable closed set it contains an isolated point,
say $w$, and then the open set $\pi_{K}^{-1}(w)$ contains a subset $[a]\cap X$
as required.

###  4.2. Tame and HNS subsets of ${\mathbb{Z}}$

We say that a subset $D\subset{\mathbb{Z}}$ is _tame_ if the characteristic
function $\chi_{D}:{\mathbb{Z}}\to{\mathbb{R}}$ is a tame function on the
group ${\mathbb{Z}}$. That is, when this function _comes_ from a pointed
compact tame ${\mathbb{Z}}$-system $(X,x_{0})$. Analogously, we say that $D$
is HNS (or _Asplund_), _WAP_ , or _Hilbert_ if
$\chi_{D}:{\mathbb{Z}}\to{\mathbb{R}}$ is an Asplund, WAP or Hilbert function
on ${\mathbb{Z}}$, respectively. By basic properties of the cyclic system
$X_{D}:={\rm{cls\,}}\\{\chi_{D}\circ
T^{n}:n\in{\mathbb{Z}}\\}\subset\\{0,1\\}^{\mathbb{Z}}$ (see Remark 3.5), the
subset $D\subset{\mathbb{Z}}$ is tame (Asplund, WAP) iff the associated
subshift $X_{D}$ is tame (Asplund, WAP).

Surprisingly it is not known whether $X_{f}:={\rm{cls\,}}\\{f\circ
T^{n}:n\in{\mathbb{Z}}\\}\subset{\mathbb{R}}^{\mathbb{Z}}$ is a Hilbert system
when $f:{\mathbb{Z}}\to{\mathbb{R}}$ is a Hilbert function (see [39]). The
following closely related question from [58] is also open: Is it true that
Hilbert representable compact metric ${\mathbb{Z}}$-spaces are closed under
factors?

######  Remark 4.9.

The definition of WAP sets was introduced by Ruppert [69]. He has the
following characterisation ([69, Theorem 4]):

$D\subset{\mathbb{Z}}$ is a WAP subset if and only if every infinite subset
$B\subset{\mathbb{Z}}$ contains a finite subset $F\subset B$ such that the set

$\bigcap_{b\in F}(b+D)\setminus\bigcap_{b\in B\setminus F}(b+D)$

is finite. See also [27].

######  Theorem 4.10.

Let $D$ be a subset of ${\mathbb{Z}}$. The following conditions are
equivalent:

  1. (1)

$D$ is a tame subset (i.e., the associated subshift
$X_{D}\subset\\{0,1\\}^{{\mathbb{Z}}}$ is tame).

  2. (2)

For every infinite subset $L\subseteq{\mathbb{Z}}$ there exists an infinite
subset $K\subseteq L$ and a countable subset $Y\subseteq\beta{\mathbb{Z}}$
such that for every $x\in\beta{\mathbb{Z}}$ there exists $y\in Y$ such that

$n+D\in x\Longleftrightarrow n+D\in y\ \ \ \forall n\in K$

(treating $x$ and $y$ as ultrafilters on the set ${\mathbb{Z}}$).

###### Proof.

By the universality of the greatest ambit $({\mathbb{Z}},\beta{\mathbb{Z}})$
it suffices to check when the function

$f=\chi_{\overline{D}}:\beta{\mathbb{Z}}\to\\{0,1\\},\ f(x)=1\Leftrightarrow
x\in\overline{D},$

the natural extension function of $\chi_{D}:{\mathbb{Z}}\to\\{0,1\\}$, is tame
(in the usual sense, as a function on the compact cascade
$\beta{\mathbb{Z}}$), where we denote by $\overline{D}$ the closure of $D$ in
$\beta{\mathbb{Z}}$ (a clopen subset). Applying Theorem 3.6 to $f$ we see that
the following condition is both necessary and sufficient: For every infinite
subset $L\subseteq{\mathbb{Z}}$ there exists an infinite subset $K\subseteq L$
and a countable subset $Y\subseteq\beta{\mathbb{Z}}$ which is dense in the
pseudometric space $(\beta{\mathbb{Z}},\rho_{f,K})$. Now saying that $Y$ is
dense is the same as the requiring that $Y$ be ${\varepsilon}$-dense for every
$0<{\varepsilon}<1$. However, as $f$ has values in $\\{0,1\\}$ and
$0<{\varepsilon}<1$ we conclude that for every $x\in\beta{\mathbb{Z}}$ there
is $y\in Y$ with

$x\in n+\overline{D}\Longleftrightarrow y\in n+\overline{D}\ \ \ \forall n\in
K,$

and the latter is equivalent to

$n+D\in x\Longleftrightarrow n+D\in y\ \ \ \forall n\in K.$

∎

######  Theorem 4.11.

Let $D$ be a subset of ${\mathbb{Z}}$. The following conditions are
equivalent:

  1. (1)

$D$ is an Asplund subset (i.e., the associated subshift
$X_{D}\subset\\{0,1\\}^{{\mathbb{Z}}}$ is Asplund).

  2. (2)

There exists a _countable_ subset $Y\subseteq\beta{\mathbb{Z}}$ such that for
every $x\in\beta{\mathbb{Z}}$ there exists $y\in Y$ such that

$n+D\in x\Longleftrightarrow n+D\in y\ \ \ \forall n\in{\mathbb{Z}}.$

###### Proof.

(Sketch) One can modify the proof of Theorem 4.10. Namely, if in assertion (4)
of Theorem 3.6 eventual fragmentability of $F$ is replaced by fragmentability
then this characterization of Asplund functions, [29] follows. ∎

######  Example 4.12.

${\mathbb{N}}$ is an Asplund subset of ${\mathbb{Z}}$ which is not a WAP
subset. In fact, let $X_{{\mathbb{N}}}$ be the corresponding subshift. Clearly
$X_{{\mathbb{N}}}$ is homeomorphic to the two-point compactification of
${\mathbb{Z}}$, with $\\{\bf{0}\\}$ and $\\{\bf{1}\\}$ as minimal subsets.
Since a transitive WAP system admits a unique minimal set, we conclude that
$X_{{\mathbb{N}}}$ is not WAP (see e.g. [23]). On the other hand, since
$X_{{\mathbb{N}}}$ is countable we can apply Theorem 4.5 to show that it is
HNS. Alternatively, using Theorem 4.11, we can take $Y$ to be
${\mathbb{Z}}\cup\\{p,q\\}$, where we choose $p$ and $q$ to be any two non-
principal ultrafilters such that $p$ contains ${\mathbb{N}}$ and $q$ contains
$-{\mathbb{N}}$.

##  5\. Entropy and null systems

We begin by recalling the basic definitions of topological (sequence) entropy.
Let $(X,T)$ be a cascade, i.e., a ${\mathbb{Z}}$-dynamical system, and
$A=\\{a_{0}<a_{1}<\ldots\\}$ a sequence of integers. Given an open cover
$\mathcal{U}$ define

$h^{A}_{top}(T,\mathcal{U})=\limsup_{n\to\infty}\frac{1}{n}N(\bigvee_{i=0}^{n-1}T^{-a_{i}}(\mathcal{U}))$

The topological entropy along the sequence $A$ is then defined by

$h^{A}_{top}(T)=\sup\\{h^{A}_{top}(T,\mathcal{U}):\mathcal{U}\ \text{an open
cover of $X$}\\}.$

When the phase space $X$ is zero-dimensional, one can replace open covers by
clopen partitions. We recall that a dynamical system $(T,X)$ is called null if
$h^{A}_{top}(T)=0$ for every infinite $A\subset{\mathbb{Z}}$. With
$A={\mathbb{N}}$ one retrieves the usual definition of topological entropy.
Finally when $Y\subset\\{0,1\\}^{{\mathbb{Z}}}$, and $A\subset{\mathbb{Z}}$ is
a given subset of ${\mathbb{Z}}$, we say that $Y$ is free on $A$ or that $A$
is an interpolation set for $Y$, if $\\{y|_{A}:y\in Y\\}=\\{0,1\\}^{A}$.

By theorems of Kerr and Li [51], [52] every null ${\mathbb{Z}}$-system is
tame, and every tame system has zero topological entropy. From results of
Glasner-Weiss [38] (for (1)) and Kerr-Li [52] (for (2) and (3)), the following
results can be easily deduced. (See Propositions 3.9.2, 6.4.2 and 5.4.2 of
[52] for the positive topological entropy, the untame, and the nonnull claims,
respectively.)

######  Theorem 5.1.

  1. (1)

A subshift $X\subset\\{0,1\\}^{{\mathbb{Z}}}$ has positive topological entropy
iff there is a subset $A\subset{\mathbb{Z}}$ of positive density such that $X$
is free on $A$.

  2. (2)

A subshift $X\subset\\{0,1\\}^{{\mathbb{Z}}}$ is not tame iff there is an
infinite subset $A\subset{\mathbb{Z}}$ such that $X$ is free on $A$.

  3. (3)

A subshift $X\subset\\{0,1\\}^{{\mathbb{Z}}}$ is not null iff for every
$n\in{\mathbb{N}}$ there is a finite subset $A_{n}\subset{\mathbb{Z}}$ with
$|A_{n}|\geq n$ such that $X$ is free on $A_{n}$.

###### Proof.

We consider the second claim; the other claims are similar.

Certainly if there is an infinite $A\subset{\mathbb{Z}}$ on which $X$ is free
then $X$ is not tame (e.g. use Theorem 4.7). Conversely, if $X$ is not tame
then, by Propositions 6.4.2 of [52], there exists a non diagonal IT pair
$(x,y)$. As $x$ and $y$ are distinct there is an $n$ with, say,
$x(n)=0,y(n)=1$. Since $T^{n}(x,y)$ is also an IT pair we can assume that
$n=0$. Thus $x\in U_{0}$ and $y\in U_{1}$, where these are the cylinder sets
$U_{i}=\\{z\in X:z(0)=i\\},i=0,1$. Now by the definition of an IT pair there
is an infinite set $A\subset{\mathbb{Z}}$ such that the pair $(U_{0},U_{1})$
has $A$ as an independence set. This is exactly the claim that $X$ is free on
$A$. ∎

The following theorem was proved (independently) by Huang [43], Kerr and Li
[52], and Glasner [25]. See [28] for a recent generalization of this result.

######  Theorem 5.2.

_(A structure theorem for minimal tame dynamical systems)_ Let $(G,X)$ be a
tame minimal metrizable dynamical system with $G$ an abelian group. Then:

  1. (1)

$(G,X)$ is an almost one to one extension $\pi:X\to Y$ of a minimal
equicontinuous system $(G,Y)$.

  2. (2)

$(G,X)$ is uniquely ergodic and the factor map $\pi$ is, measure
theoretically, an isomorphism of the corresponding measure preserving system
on $X$ with the Haar measure on the equicontinuous factor $Y$.

######  Examples 5.3.

  1. (1)

According to Theorem 5.2 the Morse minimal system, which is uniquely ergodic
and has zero entropy, is nevertheless not tame as it fails to be an almost 1-1
extension of its adding machine factor. We can therefore deduce that, a
fortiori, it is not null.

  2. (2)

Let $L=IP\\{10^{t}\\}_{t=1}^{\infty}\subset{\mathbb{N}}$ be the IP-sequence
generated by the powers of ten, i.e.

$L=\\{10^{a_{1}}+10^{a_{2}}+\cdots+10^{a_{k}}:1\leq
a_{1}<a_{2}<\cdots<a_{k}\\}.$

Let $f=1_{L}$ and let
$X=\bar{\mathcal{O}}_{\sigma}(f)\subset\\{0,1\\}^{\mathbb{Z}}$, where $\sigma$
is the shift on $\Omega=\\{0,1\\}^{\mathbb{Z}}$. The subshift $(\sigma,X)$ is
not tame. In fact it can be shown that $L$ is an interpolation set for $X$.

  3. (3)

Take $u_{n}$ to be the concatenation of the words $a_{n,i}0^{n}$, where
$a_{n,i},\ i=1,2,3,\dots,2^{n}$ runs over $\\{0,1\\}^{n}$. Let
$v_{n}=0^{|u_{n}|}$, $w_{n}=u_{n}v_{n}$ and $w_{\infty}$ the infinite
concatenation $\\{0,1\\}^{\mathbb{N}}\ni w_{\infty}=w_{1}w_{2}w_{3}\cdots$.
Finally define $w\in\\{0,1\\}^{\mathbb{Z}}$ by $w(n)=0$ for $n\leq 0$ and
$w(n)=w_{\infty}(n)$. Then
$X=\bar{\mathcal{O}}_{\sigma}(w)\subset\\{0,1\\}^{\mathbb{Z}}$ is a countable
subshift, hence HNS and a fortiori tame, but for an appropriately chosen
sequence the sequence entropy of $X$ is $\log 2$. Hence, $X$ is not null.
Another example of a countable nonnull subshift can be found in [43, Example
5.12].

  4. (4)

In [52, Section 11] Kerr and Li construct a Toeplitz subshift ( = a minimal
almost one-to-one extension of an adding machine) which is tame but not null.

  5. (5)

In [42, Theorem 13.9] the authors show that for interval maps being tame is
the same as being null.

######  Remark 5.4.

Let $T:[0,1]\to[0,1]$ be a continuous self-map on the closed interval. In an
unpublished paper [54] the authors show that the enveloping semigroup $E(X)$
of the cascade (an ${\mathbb{N}}\cup\\{0\\}$-system) $X=[0,1]$ is either
metrizable or it contains a topological copy of $\beta{\mathbb{N}}$. The
metrizable enveloping semigroup case occurs exactly when the system is HNS.
This was proved in [37] for group actions but it remains true for semigroup
actions, [33]. The other case occurs iff $\sigma$ is Li-Yorke chaotic.
Combining this result with Example 5.3.5 one gets: HNS = null = tame, for any
cascade $(T,[0,1])$.

##  6\. Some examples of tame functions and systems

In this section we give some methods for constructing tame systems and
functions. It is closely related to the question whether given family of real
(not necessarily, continuous) functions is tame.

Recall (see for example [6]) that a bisequence ${\mathbb{Z}}\to\\{0,1\\}$ is
_Sturmian_ if it is recurrent and has the minimal complexity $p(n)=n+1$.

######  Example 6.1.

  1. (1)

(See [29]) Consider an irrational rotation $(R_{\alpha},{\mathbb{T}})$. Choose
$x_{0}\in{\mathbb{T}}$ and split each point of the orbit $x_{n}=x_{0}+n\alpha$
into two points $x_{n}^{\pm}$. This procedure results is a dynamical system
$(\sigma,X)$ which is a minimal almost 1-1 extension of
$(R_{\alpha},{\mathbb{T}})$. Then
$E(X,\sigma)\setminus\\{\sigma^{n}\\}_{n\in{\mathbb{Z}}}$ is homeomorphic to
the two arrows space, a basic example of a non-metrizable Rosenthal compactum.
It follows that $E(\sigma,X)$ is also a Rosenthal compactum. Hence,
$(\sigma,X)$ is tame but not HNS.

  2. (2)

Let $P_{0}$ be the set $[0,c)$ and $P_{1}$ the set $[c,1)$; let $z$ be a point
in $[0,1)$ (identified with ${\mathbb{T}}$) via the rotation $R_{\alpha}$ we
get the binary bisequence $u_{n}$, $n\in{\mathbb{Z}}$ defined by $u_{n}=0$
when $R_{\alpha}^{n}(z)\in P_{0},u_{n}=1$ otherwise. These are called
_Sturmian like codings_. With $c=1-\alpha$ we get the classical Sturmian
bisequences. For example, when $\alpha:=\frac{\sqrt{5}-1}{2}$ the
corresponding sequence, computed at $z=0$, is called the _Fibonacci
bisequence_.

######  Example 6.2.

  1. (1)

In his paper [15] Ellis, following Furstenberg’s classical work [19],
investigates the projective action of $GL(n,{\mathbb{R}})$ on the projective
space $\mathbb{P}^{n-1}$. It follows from his results that the corresponding
enveloping semigroup is not first countable. However, in a later work [1],
Akin studies the action of $G=GL(n,{\mathbb{R}})$ on the sphere
$\mathbb{S}^{n-1}$ and shows that here the enveloping semigroup is first
countable (but not metrizable). It follows that the dynamical systems
$D_{1}=(G,\mathbb{P}^{n-1})$ and $D_{2}=(G,\mathbb{S}^{n-1})$ are tame but not
HNS. Note that $E(D_{1})$ is Fréchet, being a quotient of a first countable
compact space, namely $E(D_{2})$.

  2. (2)

(Huang [43]) An almost 1-1 extension $\pi:X\to Y$ of an equicontinuous metric
${\mathbb{Z}}$-system $Y$ with $X\setminus X_{0}$ countable, where
$X_{0}=\\{x\in X:|\pi^{-1}\pi(x)|=1\\}$, is tame.

We will see that many coding functions are tame, including some
multidimensional analogues of Sturmian sequences. The latter are defined on
the groups ${\mathbb{Z}}^{k}$ and instead of the characteristic function
$f:=\chi_{D}$ (with $D=[0,c)$) one may consider coloring of the space leading
to shifts with finite alphabet. We give a precise definition which (at least
in some partial cases) was examined in several papers. Regarding some
dynamical and combinatorial aspects of coding functions see for example [7,
18, 65], and the survey paper [6].

######  Definition 6.3.

Consider an arbitrary finite partition

${\mathbb{T}}=\cup_{i=0}^{d}[c_{i},c_{i+1})$

of ${\mathbb{T}}$ by the ordered $d$-tuple of points
$c_{0}=0,c_{1},\dots,c_{d},c_{d+1}=1$ and any coloring map

$f:{\mathbb{T}}\to A:=\\{0,\dots,d\\}.$

Now for a given $k$-tuple $(\alpha_{1},\dots,\alpha_{k})\in{\mathbb{T}}^{k}$
and a given point $z\in{\mathbb{T}}$ consider the corresponding coding
function

$m(f,z):{\mathbb{Z}}^{k}\to\\{0,\dots,d\\}\ \ \ (n_{1},\dots,n_{k})\mapsto
f(z+n_{1}\alpha_{1}+\cdots+n_{k}\alpha_{k}).$

We call such a sequence a _multidimensional $(k,d)$-Sturmian like sequence_.

######  Lemma 6.4.

  1. (1)

Let $q:X_{1}\to X_{2}$ be a map between sets and
$\\{f_{n}:X_{2}\to{\mathbb{R}}\\}_{n\in{\mathbb{N}}}$ a bounded sequence of
functions _(with no continuity assumptions on $q$ and $f_{n}$)_. If
$\\{f_{n}\circ q\\}$ is an independent sequence on $X_{1}$ then $\\{f_{n}\\}$
is an independent sequence on $X_{2}$.

  2. (2)

If $q$ is onto then the converse is also true. That is $\\{f_{n}\circ q\\}$ is
independent if and only if $\\{f_{n}\\}$ is independent.

  3. (3)

Let $\\{f_{n}\\}$ be a bounded sequence of continuous functions on a
topological space $X$. Let $Y$ be a _dense_ subset of $X$. Then $\\{f_{n}\\}$
is an independent sequence on $X$ if and only if the sequence of restrictions
$\\{f_{n}|_{Y}\\}$ is an independent sequence on $Y$.

###### Proof.

Claims (1) and (2) are straightforward.

(3) Since $\\{f_{n}\\}$ is an independent sequence for every pair of finite
disjoint sets $P,M\subset{\mathbb{N}}$, the set

$\bigcap_{n\in P}f_{n}^{-1}(-\infty,a)\cap\bigcap_{n\in
M}f_{n}^{-1}(b,\infty)$

is non-empty. This set is open because every $f_{n}$ is continuous. Hence,
each of them meets the dense set $Y$. As $f_{n}^{-1}(-\infty,a)\cap
Y=f_{n}|_{Y}^{-1}(-\infty,a)$ and $f_{n}^{-1}(b,\infty)\cap
Y=f_{n}|_{Y}^{-1}(b,\infty)$, this implies that $\\{f_{n}|_{Y}\\}$ is an
independent sequence on $Y$.

Conversely if $\\{f_{n}|_{Y}\\}$ is an independent sequence on a subset
$Y\subset X$ then by (1) (where $q$ is the embedding $Y\hookrightarrow X$),
$\\{f_{n}\\}$ is an independent sequence on $X$. ∎

Below we will sometimes deal with (not necessarily continuous) functions
$f:X\to{\mathbb{R}}$ such that the orbit $fS$ of $f$ in ${\mathbb{R}}^{X}$ is
a tame family (Definition 2.3). An example of such Baire 1 function (which is
not tame, being discontinuous), is the characteristic function $\chi_{D}$ of
an arc $D=[a,a+s)\subset{\mathbb{T}}$ defined on the system
$(R_{\alpha},{\mathbb{T}})$, where $R_{\alpha}$ is an irrational rotation of
the circle ${\mathbb{T}}$. See Theorem t:tame-typeOften.

######  Lemma 6.5.

Let $S$ be a semigroup, $X$ a (not necessarily compact) $S$-space and
$f:X\to{\mathbb{R}}$ a bounded (not necessarily continuous) function.

  1. (1)

Let $f\in\operatorname{RUC}(X)$; then $f\in\operatorname{Tame}(X)$ if and only
if $fS$ is a tame family. Moreover, there exists an $S$-compactification
$\nu:X\to Y$ where the action $S\times Y\to Y$ is continuous, $Y$ is a tame
system and $f=\tilde{f}\circ\nu$ for some $\tilde{f}\in C(Y)$.

  2. (2)

Let $G$ be a topological group and $f\in\operatorname{RUC}(G)$. Then
$f\in\operatorname{Tame}(G)$ if and only if $fG$ is a tame family.

  3. (3)

Let $L$ be a discrete semigroup and $f:L\to{\mathbb{R}}$ a bounded function.
Then $f\in\operatorname{Tame}(L)$ if and only if $fL$ is a tame family.

  4. (4)

Let $h:L\to S$ be a homomorphism of semigroups, $S\times Y\to Y$ be an action
(without any continuity assumptions) on a set $Y$ and $f:Y\to{\mathbb{R}}$ be
a bounded function such that $fL$ is a tame family. Then for every point $y\in
Y$ the corresponding coding function $m(f,y):L\to{\mathbb{R}}$ is tame on the
discrete semigroup $(L,\tau_{discr})$.

###### Proof.

For (1) consider the cyclic $S$-compactification $\nu:X\to Y=X_{f}$ (see
Definition 3.3). Since $f\in\operatorname{RUC}(X)$ the action $S\times
X_{f}\to X_{f}$ is jointly continuous (Remark 3.5). By the basic property of
the cyclic compactification there exists a continuous function
$\tilde{f}:X_{f}\to{\mathbb{R}}$ such that $f=\tilde{f}\circ\nu$. The family
$fS$ has no independent sequence. By Lemma 6.4.3 we conclude that also
$\tilde{f}S$ has no independent sequence. This means, by Theorem 3.6, that
$\tilde{f}$ is tame. Hence (by Definition) so is $f$. The converse follows
from Lemma 6.4.1.

(2) and (3) follow easily from (1) (with $X=G=L$) taking into account that on
a discrete semigroup $L$ every bounded function $L\to{\mathbb{R}}$ is in
$\operatorname{RUC}(L)$.

(4) By (3) it is enough to show for the coding function $f_{0}:=m(f,y)$ that
the family $f_{0}L$ has no independent subsequence. Define $q:L\to Y,s\mapsto
h(s)y$. Then $f_{0}t=(ft)\circ q$ for every $t\in L$. If $f_{0}t_{n}$ is an
independent sequence for some sequence $t_{n}\in L$ then Lemma 6.4.1 implies
that the sequence of functions $ft_{n}$ on $Y$ is independent. This
contradicts the assumption that $fL$ has no independent subsequence. ∎

Let $f:X\to Y$ be a function between topological spaces. We denote by
$cont(f)$ and $disc(f)$ the points of continuity and discontinuity for $f$
respectively.

######  Definition 6.6.

Let $F$ be a family of functions on $X$. We say that $F$ is:

  1. (1)

_Strongly almost continuous_ if for every $x\in X$ we have $x\in cont(f)$ for
almost all $f\in F$ (i.e. with the exception of at most a finite set of
elements which may depend on $x$).

  2. (2)

_Almost continuous_ if for every infinite (countable) subset $F_{1}\subset F$
there exists an infinite subset $F_{2}\subset F_{1}$ such that $F_{2}$ is
strongly almost continuous on $X$.

######  Example 6.7.

  1. (1)

Let $G\times X\to X$ be a group action, $G_{0}\leq G$ a subgroup and
$f:X\to{\mathbb{R}}$ a function such that

$G_{0}x\cap disc(f)\ \text{and}\ St(x)\cap G_{0}\ \ \text{are \ finite}\
\forall x\in X,$

where $St(x)\leq G$ is the stabilizer subgroup of $x$. Then the family
$fG_{0}$ is strongly almost continuous (indeed, use the following equality
$g^{-1}cont(f)=cont(fg),\ g\in G$).

  2. (2)

A coarse sufficient condition for (1) is: $disc(f)$ is finite and $St(x)\cap
G_{0}$ is finite $\forall x\in X$.

  3. (3)

As a particular case of (2) we have the following example. For every compact
group $G$ and a function $f:G\to{\mathbb{R}}$ with finitely many
discontinuities, $fG_{0}$ is strongly almost continuous on $X=G$ for every
subgroup $G_{0}$ of $G$.

######  Theorem 6.8.

Let $X$ be a compact metric space and $F$ a bounded family of real valued
functions on $X$ such that $F$ is almost continuous. Further assume that:

  * (*)

for every sequence $\\{f_{n}\\}_{n\in{\mathbb{N}}}$ in $F$ there exists a
subsequence $\\{f_{n_{m}}\\}_{m\in{\mathbb{N}}}$ and a countable subset
$C\subset X$ such that $\\{f_{n_{m}}\\}_{m\in{\mathbb{N}}}$ pointwise
converges on $X\setminus C$ to a function $\phi:X\setminus C\to{\mathbb{R}}$
where $\phi\in{\mathcal{B}}_{1}(X\setminus C)$.

Then $F$ is a tame family.

###### Proof.

Assuming the contrary let $\\{f_{n}\\}$ be an independent sequence in $F$.
Then, by assumption, there exists a countable subset $C\subset X$ and a
subsequence $\\{f_{n_{m}}\\}$ such that $\\{f_{n_{m}}:X\setminus
C\to{\mathbb{R}}\\}$ pointwise converges on $X\setminus C$ to a function
$\phi:X\setminus C\to{\mathbb{R}}$ such that
$\phi\in{\mathcal{B}}_{1}(X\setminus C)$.

Independence is preserved by subsequences so this subsequence
$\\{f_{n_{m}}\\}$ remains independent. For simplicity of notation assume that
$\\{f_{n}\\}$ itself has the properties of $\\{f_{n_{m}}\\}$. Moreover we can
suppose in addition, by Definition 6.6, that $\\{f_{n}\\}$ is strongly almost
continuous.

By the definition of independence, there exist $a<b$ such that for every pair
of disjoint finite sets $P,M\subset{\mathbb{N}}$ we have

$\bigcap_{n\in P}A_{n}\cap\bigcap_{n\in M}B_{n}\neq\emptyset,$

where $A_{n}:=f_{n}^{-1}(-\infty,a)$ and $B_{n}:=f_{n}^{-1}(b,\infty)$. Now
define a tree of nested sets as follows:

$\Omega_{1}:=X$

$\Omega_{2}:=\Omega_{1}\cap A_{1}=A_{1}$ $\Omega_{3}:=\Omega_{1}\cap
B_{1}=B_{1}$

$\Omega_{4}:=\Omega_{2}\cap A_{2}\hskip 34.14322pt\Omega_{5}:=\Omega_{2}\cap
B_{2}\hskip 34.14322pt\Omega_{6}:=\Omega_{3}\cap A_{2}\hskip
28.45274pt\Omega_{7}:=\Omega_{3}\cap B_{2}$,

and so on. In general,

$\Omega_{2^{n+1}+2k}:=\Omega_{2^{n}+k}\cap A_{n+1},\hskip
8.5359pt\Omega_{2^{n+1}+2k+1}:=\Omega_{2^{n}+k}\cap B_{n+1}$

for every $0\leq k<2^{n}$ and every $n\in{\mathbb{N}}$.

We obtain a system $\\{\Omega_{n}\\}_{n\in{\mathbb{N}}}$ which satisfies:

$\Omega_{2n}\cup\Omega_{2n+1}\subset\Omega_{n}\ {\text{ and}}\
\Omega_{2n}\cap\Omega_{2n+1}=\emptyset\ {\text{for each}}\ n\in{\mathbb{N}}.$

Since $\\{(A_{n},B_{n})\\}_{n\in{\mathbb{N}}}$ is independent (in the sense of
[68]), every $\Omega_{n}$ is nonempty.

For every binary sequence $u=(u_{1},u_{2},\dots)\in\\{0,1\\}^{{\mathbb{N}}}$
we have the corresponding uniquely defined _branch_

$\alpha_{u}:=\Omega_{1}\supset\Omega_{n_{1}}\supset\Omega_{n_{2}}\supset\cdots$

where for each $i\in{\mathbb{N}}$ with $2^{i-1}\leq n_{i}<2^{i}$ we have

$n_{i+1}=2n_{i}\ \text{iff}\ u_{i}=0\ \text{and}\ n_{i+1}=2n_{i}+1\
\text{iff}\ u_{i}=1.$

Let us say that $u,v\in\\{0,1\\}^{{\mathbb{N}}}$ are _essentially distinct_ if
they have infinitely many different coordinates. Equivalently, if $u$ and $v$
are in different cosets of the Cantor group $\\{0,1\\}^{{\mathbb{N}}}$ with
respect to the subgroup $H$ consisting of the binary sequences with finite
support. Since $H$ is countable there are uncountably many pairwise
essentially distinct elements in the Cantor group. We choose a subset
$T\subset\\{0,1\\}^{{\mathbb{N}}}$ which intersects each coset in exactly one
point. Clearly, $card(T)=2^{\omega}$. Now for every branch $\alpha_{u}$ where
$u\in T$ choose one element

$x_{u}\in\bigcap_{i\in{\mathbb{N}}}cl(\Omega_{n_{i}}).$

Here we use the compactness of $X$ which guarantees that
$\bigcap_{i\in{\mathbb{N}}}cl(\Omega_{n_{i}})\neq\emptyset$. We obtain a set
$X_{T}:=\\{x_{u}:\ u\in T\\}\subset X$ and an onto function $T\to X_{T},\
u\mapsto x_{u}$.

Claim:

  1. (1)

The function $T\to X_{T},\ u\mapsto x_{u}$ is injective. In particular,
$X_{T}\setminus C$ is uncountable.

  2. (2)

$|\phi(x_{u})-\phi(x_{v})|\geq{\varepsilon}:=b-a$ for every distinct
$x_{u},x_{v}\in X_{T}\setminus C$.

Proof of the Claim: (1) Let $u=(u_{i})$ and $v=(v_{i})$ are distinct elements
in $T$. Denote by $\alpha_{u}:=\\{\Omega_{n_{i}}\\}_{i\in{\mathbb{N}}}$ and
$\alpha_{v}:=\\{\Omega_{m_{i}}\\}_{i\in{\mathbb{N}}}$ the corresponding
branches. Then, by the definition of $X_{T}$, we have the uniquely defined
points $x_{u}\in\cap_{i\in{\mathbb{N}}}cl(\Omega_{n_{i}})$ and
$x_{v}\in\cap_{i\in{\mathbb{N}}}cl(\Omega_{m_{i}})$ in $X_{T}$.

Since $u,v\in T$ are essentially distinct they have infinitely many different
indices.

As $\\{f_{n}\\}$ is strongly almost continuous there exists a sufficiently
large $t_{0}\in{\mathbb{N}}$ such that the points $x_{u}$ and $x_{v}$ are both
points of continuity of $f_{n}$ for every $n\geq t_{0}$.

Now note that if $u_{i}\neq v_{i}$ then the sets $\Omega_{n_{i}+1}$ and
$\Omega_{m_{i}+1}$ are contained (respectively) in the pair of disjoint sets
$A_{k}:=f_{k}^{-1}(-\infty,a)$ and $B_{k}:=f_{k}^{-1}(b,\infty)$. Since $u$
and $v$ are essentially distinct we can assume that $i$ is sufficiently large
in order to ensure that $k\geq t_{0}$. That is, we necessarily have exactly
one of the cases:

$(a)\ \ \ \Omega_{n_{i}+1}\subset A_{k},\quad\Omega_{m_{i}+1}\subset B_{k}$

or

$(b)\ \ \ \Omega_{n_{i}+1}\subset B_{k},\quad\Omega_{m_{i}+1}\subset A_{k}.$

For simplicity we only check the first case (a). For (a) we have
$x_{u}\in{\rm{cls\,}}(\Omega_{n_{i}+1})\subset{\rm{cls\,}}(f_{k}^{-1}(-\infty,a))$
and
$x_{v}\in{\rm{cls\,}}(\Omega_{n_{i}+1})\subset{\rm{cls\,}}(f_{k}^{-1}(b,\infty))$.
Since $\\{x_{u},x_{v}\\}\subset cont(f_{n})$ are continuity points for every
$n\geq t_{0}$ and since $k\geq t_{0}$ by our choice, we obtain
$f_{k}(x_{u})\leq a$ and $f_{k}(x_{v})\geq b$. So, we can conclude that
$|f_{k}(x_{u})-f_{k}(x_{v})|\geq{\varepsilon}:=b-a$ for every $k\geq t_{0}$.
In particular, $x_{u}$ and $x_{v}$ are distinct. This proves (1).

(2) Furthermore, if our distinct $x_{u},x_{v}\in X_{T}$ are in addition from
$X_{T}\setminus C$ then by (*) we have $\lim f_{k}(x_{u})=\phi(x_{u})$ and
$\lim f_{k}(x_{v})=\phi(x_{v})$. It follows that
$|\phi(x_{u})-\phi(x_{v})|\geq{\varepsilon}$ and the condition (2) of our
claim is also proved.

Since $X_{T}\setminus C$ is an uncountable subset of a Polish space $X$ there
exists an uncountable subset $Y\subset X_{T}\setminus C$ such that any point
of $y$ is a condensation point in $X_{T}\setminus C$. This follows from the
proof of Cantor-Bendixson theorem, [48]). More precisely, define

$Q:=\\{x\in X_{T}\setminus C:\ \text{there exists a countable open nbd}\
O_{x}\ \text{of}\ x\ \text{in the space}\ X_{T}\setminus C\\}.$

Observe that $Q=\bigcup\\{O_{x}:\ x\in Q\\}$. Since $Q$ is second countable,
by Lindelof property there exists a countable subcover. Hence, $Q$ is at most
a countable subset of $X_{T}\setminus C$ and any point $y\in
Y:=(X_{T}\setminus C)\setminus Q$ is a condensation point.

Now, it follows by assertion (2) of the Claim that for every open subset $U$
in $X$ with $U\cap Y\neq\emptyset$ we have ${\mathrm{diam}}(\phi(U\cap
Y))\geq{\varepsilon}$. This means that $\phi:X\setminus C\to{\mathbb{R}}$ is
not fragmented. Since $C$ is countable and $X$ is compact metrizable the
subset $X\setminus C$ is Polish. On Polish spaces fragmentability and Baire 1
property are the same for real valued functions (Lemma 2.2.2). So, we obtain
that $\phi:X\setminus C\to{\mathbb{R}}$ is not Baire 1. This contradicts the
assumption that $\phi\in{\mathcal{B}}_{1}(X\setminus C)$.

∎

######  Theorem 6.9.

Let $X$ be a compact metric space, and $F$ a bounded family of real valued
functions on $X$ such that $F$ is almost continuous. Assume that
${\rm{cls\,}}_{p}(F)\subset{\mathcal{B}}_{1}(X)$. Then $F$ is a tame family.

###### Proof.

Assuming the contrary let $F$ has an independent sequence
$F_{1}:=\\{f_{n}\\}$. Since ${\rm{cls\,}}_{p}(F)\subset{\mathcal{B}}_{1}(X)$
we have ${\rm{cls\,}}_{p}(F_{1})\subset{\mathcal{B}}_{1}(X)$. By the BFT
theorem [8, Theorem 3F] the compactum ${\rm{cls\,}}_{p}(F_{1})$ is a Fréchet
topological space. Every (countably) compact Fréchet space is sequentially
compact, [16, Theorem 3.10.31], hence ${\rm{cls\,}}_{p}(F_{1})$ is
sequentially compact. Therefore the sequence $\\{f_{n}\\}$ contains a
pointwise convergent subsequence, say $f_{n}\to\phi\in{\mathcal{B}}_{1}(X)$.
Now apply Theorem 6.8 to get a contradiction, taking into account that the
properties almost continuity and independence are both inherited by
subsequences. ∎

######  Theorem 6.10.

  1. (1)

Let $X$ be a compact metric $S$-space, $S_{0}$ a subsemigroup of $S$. Let
$f:X\to{\mathbb{R}}$ be a bounded function such that
${\rm{cls\,}}_{p}(fS_{0})\subset{\mathcal{B}}_{1}(X)$ and with $fS_{0}$ almost
continuous. Then $fS_{0}$ is a tame family.

  2. (2)

Let $X$ be a compact metric $G$-space and $G_{0}\leq G$ a subgroup of $G$ such
that (i) $(G_{0},X)$ is tame, (ii) for every $p\in E(G,X)$ and every $x\in X$
the preimage $p^{-1}(x)$ is countable and (iii) $G_{0}\cap St(x)$ is finite
for every $x\in X$. Suppose further that $f:X\to{\mathbb{R}}$ is a bounded
function with only finitely many points of discontinuity. Then $fG_{0}$ is a
tame family.

  3. (3)

In particular, (2) holds in the following useful situation: $X=G$ is a compact
metric group, $h:G_{0}\to X$ is a homomorphism of groups and
$f:X\to{\mathbb{R}}$ has finitely many discontinuities. Then $fG_{0}$ is a
tame family.

  4. (4)

In all the cases above (1), (2), (3), a coding function
$m(f,z):S_{0}\to{\mathbb{R}}$ is a tame function on the discrete semigroup
$S_{0}$ for every $z\in X$ and every homomorphism $h:S_{0}\to S$ (or,
$G_{0}\to G$). Also the corresponding subshift $(S_{0},X_{f})$ is tame.

###### Proof.

(1) Apply Theorem 6.9.

(2) First note that $fG_{0}$ is strongly almost continuous (Example 6.7.2).
Now assuming the contrary $fG_{0}$ has an independent subsequence
$\\{fg_{n}\\}$. Since $(G_{0},X)$ is a tame metric system we can assume, with
no loss in generality, that the sequence $\\{g_{n}\\}$ converges to an element
$p\in E(G_{0},X)$. Then $f(g_{n}(x))$ converges to $f(px)$ for every $x\in
X\setminus C$, where $C:=p^{-1}(disc(f))$ is a countable set. Since $X$ is a
tame system, $p:X\to X$ is a fragmented function. Then also the restricted
function $p_{0}:X\setminus C\to X$ is fragmented. Since $f:X\to{\mathbb{R}}$
is uniformly continuous we obtain that the composition $f\circ
p_{0}:X\setminus C\to{\mathbb{R}}$ is fragmented. Since $C$ is countable,
$X\setminus C$ is Polish. Therefore, by Lemma 2.2.2, $f\circ p_{0}:X\setminus
C\to{\mathbb{R}}$ is Baire 1. This however is in contradiction with Theorem
6.8.

(4) Follows from (1) and Lemma 6.5. ∎

Theorem 6.10.4 directly implies the following:

######  Example 6.11.

For every irrational rotation $\alpha$ of the circle ${\mathbb{T}}$ and an arc
$D:=[a,b)\subset{\mathbb{T}}$ the function

$\varphi_{D}:={\mathbb{Z}}\to{\mathbb{R}},\quad n\mapsto\chi_{D}(n\alpha)$

is a tame function on the group ${\mathbb{Z}}$. In particular, for
$D:=[-\frac{1}{4},\frac{1}{4})$ we get that
$\varphi_{D}(n)={\rm{sgn}}\cos(2\pi n\alpha)$ is a tame function on
${\mathbb{Z}}$.

######  Theorem 6.12.

The multidimensional Sturmian $(k,d)$-sequences
${\mathbb{Z}}^{k}\to\\{0,1,\dots,d\\}$ (Definition 6.3) are tame.

###### Proof.

In terms of Definition 6.3 consider the homomorphism

$h:{\mathbb{Z}}^{k}\to{\mathbb{T}},\ \ \ (n_{1},\dots,n_{k})\mapsto
n_{1}\alpha_{1}+\dots+n_{k}\alpha_{k}.$

The function $f$ induced by a given partition
${\mathbb{T}}=\cup_{i=0}^{d}[c_{i},c_{i+1})$

$f:{\mathbb{T}}\to A:=\\{0,\dots,d\\},\ \ \ f(t)=i\ \text{iff}\
t\in[c_{i},c_{i+1}).$

has only finitely many discontinuities. Now Theorem 6.10 (items (3) and (4))
guarantees that the corresponding $(k,d)$-coding function
$m(f,z):{\mathbb{Z}}^{k}\to A\subset{\mathbb{R}}$ is tame for every
$z\in{\mathbb{T}}$. ∎

Note that the tameness of the functions on ${\mathbb{Z}}^{k}$ from Theorem
6.12 follows also by results from [36]. In fact, such functions come from
circularly ordered metric dynamical ${\mathbb{Z}}^{k}$-systems. See also
Theorem 7.7 and Remark 7.8 below. At least for $(1,d)$-codes, Theorem 6.12,
can be derived also from results of Pikula [65], and of Aujogue [3].

###  6.1. Strong almost 1-1 equivalence and tameness

Recall that a $G$-factor $\pi:X\to Y$ is said to be an _almost one-to-one
extension_ if

$X_{0}:=\\{x\in X:\ \ |\pi^{-1}(\pi(x))|=1\\}$

is a residual subset of $X$. We will say that $\pi:X\to Y$ is a _strongly
almost 1-1 extension_ if $X\setminus X_{0}$ is at most countable.

We say that compact dynamical $G$-systems $X,Y$ are _strongly almost 1-1
equivalent_ if there exist a continuous $G$-map $\pi:X\to Y$ and two countable
subsets $X_{1}\subset X,Y_{1}\subset Y$ such that the restriction
$\pi:X\setminus X_{1}\to Y\setminus Y_{1}$ is bijective. One may show that a
surjective strongly almost 1-1 equivalence $\pi:X\to Y$ is exactly a strongly
almost 1-1 extension.

######  Remark 6.13.

In [46, p. 30] Jolivet calls a strong almost 1-1 equivalence “semi-
conjugation”. However, the name semi-conjugation is often used as a synonym to
factor map; so we use “strong almost 1-1 equivalence” instead.

The next lemma is well known; for completeness we provide the short proof.

######  Lemma 6.14.

Let $\pi:X\to Y$ be a continuous onto $G$-map of compact metric $G$-systems.
Set

$X_{0}:=\\{x\in X:\ \ |\pi^{-1}(\pi(x))|=1\\}.$

Then the restriction map $\pi:X_{0}\to Y_{0}$ is a topological homeomorphism
of $G$-subspaces, where $Y_{0}:=\pi(X_{0})$.

###### Proof.

Since $G$ is a group $X_{0}$ and $Y_{0}$ are $G$-invariant. The map
$\pi:X_{0}\to Y_{0}$ is a continuous bijection. For every converging sequence
$y_{n}\to y$, where $y_{n},y\in Y_{0}$ the preimage
$\pi^{-1}(\\{y\\}\cup\\{y_{n}\\}_{n\in{\mathbb{N}}})$ is a compact subset of
$X$. On the other hand, $\pi^{-1}(\\{y\\}\cup\\{y_{n}\\})\subset X_{0}$ by the
definition of $X_{0}$. It follows that the restriction of $\pi$ to
$\pi^{-1}(\\{y\\}\cup\\{y_{n}\\})$ is a homeomorphism and $\pi^{-1}(y_{n})$
converges to $\pi^{-1}(y)$. ∎

From Theorem 6.8 one can derive the following result which generalizes the
above mentioned result of Huang [43] from Example 6.2.2.

######  Theorem 6.15.

[35] Let $\pi:X\to Y$ be a strongly almost 1-1 extension of compact metric
$G$-systems. Assume that the dynamical system $(G,Y)$ is tame and that the set
$p^{-1}(y)$ is (at most) countable for every $p\in E(Y)$ and $y\in Y$,
222E.g., this latter condition is always satisfied when $Y$ is distal. Another
example of such (non-distal) system is the Sturmian system. then $(G,X)$ is
also tame.

###### Proof.

We have to show that every $f\in C(X)$ is tame. Assuming the contrary, suppose
$fG$ contains an independent sequence $fs_{n}$. Since $Y$ is metrizable and
tame, one can assume (by Theorem 3.2) that the sequence $s_{n}$ converges
pointwise to some element $p$ of $E(G,Y)$. Consider the set $Y_{0}\cap
p^{-1}Y_{0}$, where $Y_{0}=\pi(X_{0})$. Since $p^{-1}(y)$ is countable for
every $y\in Y\setminus Y_{0}$ it follows that $Y\setminus(Y_{0}\cap
p^{-1}Y_{0})$ is countable. Therefore, by the definition of $X_{0}$ and the
countability of $X\setminus X_{0}$, we see that $X\setminus\pi^{-1}(Y_{0}\cap
p^{-1}Y_{0})$ is also countable. Now observe that the sequence $(fs_{n})(x)$
converges for every $x\in\pi^{-1}(Y_{0}\cap p^{-1}Y_{0})$. Indeed if we denote
$y=\pi(x)$ then $s_{n}y$ converges to $py$ in $Y$. In fact we have $py\in
Y_{0}$ (by the choice of $x$) and $s_{n}y\in Y_{0}$. By Lemma 6.14,
$\pi:X_{0}\to Y_{0}$ is a $G$-homeomorphism. So we obtain that $s_{n}x$
converges to $\pi^{-1}(py)$ in $X_{0}$. Since $f:X\to{\mathbb{R}}$ is
continuous, $(fs_{n})(x)$ converges to $f(\pi^{-1}(py))$ in ${\mathbb{R}}$.
Each $fs_{n}$ is a continuous function, hence so is also its restriction to
$\pi^{-1}(Y_{0}\cap p^{-1}Y_{0})$. Therefore the limit function
$\phi:\pi^{-1}(Y_{0}\cap p^{-1}Y_{0})\to{\mathbb{R}}$ is Baire 1. Since
C:=$X\setminus\pi^{-1}(Y_{0}\cap p^{-1}Y_{0})$ is countable and $fs_{n}$ is an
independent sequence and Theorem 6.8 provides the sought-after contradiction.
∎

######  Corollary 6.16.

(Huang [43] for cascades) Let $\pi:X\to Y$ be a strong almost 1-1 equivalence
of compact metric $G$-systems, where $Y$ is equicontinuous. Then $X$ is tame.

###### Proof.

Consider the induced factor $X\to f(X)\subset Y$ and apply Theorem 6.15. ∎

Recall the following:

######  Problem 6.17.

(A version of Pisot conjecture) [46, page 31] Is it true that every
(unimodular) irreducible Pisot substitution dynamical system is semi-conjugate
to a toral translation ?

By Corollary 6.16 a related question is:

######  Problem 6.18.

(A weaker form of Pisot Conjecture) Is it true that every substitutional
symbolic dynamical system with Pisot conditions above is always tame ?

######  Remark 6.19.

T. Jolivet [46, Theorem 3.1.1], in the context of Pisot conjecture, discusses
some (substitution) dynamical systems which semi-conjugate to a translation on
the two-dimensional torus. In particular:

(a) (Rouzy) Tribonacci 3-letter substitution.

(b) Arnoux-Rouzy substitutions.

(c) Brun substitution.

(d) Jacobi-Perron substitution. By Corollary 6.16 all these systems are tame.

###  6.2. A special class of generalized Sturmian systems

Let $R_{\alpha}$ be an irrational rotation of the torus ${\mathbb{T}}^{d}$. In
many cases a reasonably chosen subset $D\subset{\mathbb{T}}^{d}$ will yield a
generalized Sturmian system.

######  Example 6.20.

Let $\alpha=(\alpha_{1},\dots,\alpha_{d})$ be a vector in ${\mathbb{R}}^{d},\
d\geq 2$ with $1,\alpha_{1},\dots,\alpha_{d}$ independent over ${\mathbb{Q}}$.
Consider the minimal equicontinuous dynamical system $(R_{\alpha},Y)$, where
$Y={\mathbb{T}}^{d}={\mathbb{R}}^{d}/{\mathbb{Z}}^{d}$ (the $d$-torus) and
$R_{\alpha}y=y+\alpha$. Let $D$ be a small closed $d$-dimensional ball in
${\mathbb{T}}^{d}$ and let $C=\partial D$ be its boundary, a $(d-1)$-sphere.
Fix $y_{0}\in{\rm{int}}D$ and let $X=X(D,y_{0})$ be the symbolic system
generated by the function

$x_{0}\in\\{0,1\\}^{\mathbb{Z}}\ {\text{defined by}}\
x_{0}(n)=\chi_{D}(R_{\alpha}^{n}y_{0}),\qquad
X=\overline{\mathcal{O}_{\sigma}x_{0}}\subset\\{0,1\\}^{\mathbb{Z}},$

where $\sigma$ denotes the shift transformation. This is a well known
construction and it is not hard to check that the system $(\sigma,X)$ is
minimal and admits $(R_{\alpha},Y)$ as an almost 1-1 factor:

$\pi:(\sigma,X)\to(R_{\alpha},Y).$

######  Theorem 6.21.

There exists a ball $D\subset{\mathbb{T}}^{d}$ as above such that the
corresponding symbolic dynamical system $(\sigma,X)$ is tame. For such $D$ we
then have a precise description of $E(\sigma,X)\setminus{\mathbb{Z}}$ as the
product set ${\mathbb{T}}^{d}\times\mathcal{F}$, where $\mathcal{F}$ is the
collection of ordered orthonormal bases for ${\mathbb{R}}^{d}$.

###### Proof.

1. First we show that a sphere $C\subset[0,1)^{d}\cong{\mathbb{T}}^{d}$ can be chosen so that for every $y\in{\mathbb{T}}^{d}$ the set $(y+\\{n\alpha:n\in Z\\})\cap C$ is finite. We thank Benjamin Weiss for providing the following proof of this fact.

  1. (1)

For the case $d=2$ the argument is easy. If $A$ is any countable subset of the
square $[0,1)\times[0,1)$ there are only a countable number of circles that
contain three points of $A$. These circles have some countable collection of
radii. Take any circle with a radius which is different from all of them and
no translate of it will contain more than two points from the set $A$. Taking
$A=\\{n\alpha:n\in{\mathbb{Z}}\\}$ we obtain the required circle.

  2. (2)

We next consider the case $d=3$, which easily generalizes to the general case
$d\geq 3$. What we have to show is that there can not be infinitely many
points in

$A=\\{(n\alpha_{1}-[n\alpha_{1}],\alpha_{2}-[n\alpha_{2}],\alpha_{3}-[n\alpha_{3}]):n\in{\mathbb{Z}}\\}$

that lie on a plane. For if that is the case, we consider all $4$-tuples of
elements from the set $A$ that do not lie on a plane to get a countable set of
radii for the spheres that they determine. Then taking a sphere with radius
different from that collection we obtain our required sphere. In fact, if a
sphere contains infinitely many points of $A$ and no $4$-tuple from $A$
determines it then they all lie on a single plane.

So suppose that there are infinitely many points in $A$ whose inner product
with a vector $v=(z,x,y)$ is always equal to $1$. This means that there are
infinitely many equations of the form:

($*$)
$z\alpha_{1}+x\alpha_{2}+y\alpha_{3}=1/n+z[n\alpha_{1}]/n+x[n\alpha_{2}]/n+y[n\alpha_{3}]/n.$

Subtract two such equations with the second using $m$ much bigger than $n$ so
that the coefficient of $y$ cannot vanish. We can express $y=rz+sx+t$ with
$r,s$ and $t$ rational. This means that we can replace ($*$) by

($**$)
$z\alpha_{1}+x\alpha_{2}+y\alpha_{3}=1/n+t[n\alpha_{3}]/n+z([n\alpha_{1}]/n+r[n\alpha_{3}]/n)+x([n\alpha_{2}]/n+s[n\alpha_{3}]/n).$

Now $r,s$ and $t$ have some fixed denominators and (having infinitely many
choices) we can take another equation like ($**$) where $n$ (and the
corresponding $r,s,t$) is replaced by some much bigger $k$, then subtract
again to obtain an equation of the form $x=pz+q$ with $p$ and $q$ rational.
Finally one more step will show that $z$ itself is rational. However, in view
of ($*$), this contradicts the independence of
$1,\alpha_{1},\alpha_{2},\alpha_{3}$ over ${\mathbb{Q}}$ and our proof is
complete.

2. Next we show that for $C$ as above

> for every converging sequence $n_{i}\alpha$, say
> $n_{i}\alpha\to\beta\in{\mathbb{T}}^{d}\cong
> E(R_{\alpha},{\mathbb{T}}^{d})$, there exists a subsequence
> $\\{n_{i_{j}}\\}$ such that for every $y\in{\mathbb{T}}^{d}$,
> $y+n_{i_{j}}\alpha$ is either eventually in the interior of $D$ or
> eventually in its exterior.

Clearly we only need to consider points $y\in C-\beta$. Renaming we can now
assume that $n_{i}\alpha\to 0$ and that $y\in C$. Passing to a subsequence if
necessary we can further assume that the sequence of unit vectors
$\frac{n_{i}\alpha}{\|n_{i}\alpha\|}$ converges,

$\frac{n_{i}\alpha}{\|n_{i}\alpha\|}\to v_{0}\in\mathbb{S}^{d-1}.$

In order to simplify the notation we now assume that $C$ is centered at the
origin. For every point $y\in C$ where $\langle y,v_{0}\rangle\neq 0$ we have
that $y+n_{i}\alpha$ is either eventually in the interior of $D$ or eventually
in its exterior. On the other hand, for the points $y\in C$ with $\langle
y,v_{0}\rangle=0$ this is not necessarily the case. In order to deal with
these points we need a more detailed information on the convergence of
$n_{i}\alpha$ to $\beta$. At this stage we consider the sequence of orthogonal
projections of the vectors $n_{i}\alpha$ onto the subspace
$V_{1}=\\{u\in{\mathbb{R}}^{d}:\langle u,v_{0}\rangle=0\\}$, say
$u_{i}={\rm{proj\,}}_{v_{0}}(n_{i}\alpha)\to u={\rm{proj\,}}_{v_{0}}(\beta)$.
If it happens that eventually $u_{i}=0$, this means that all but a finite
number of the $n_{i}\alpha$’s are on the line defined by $v_{0}$ and our
required property is certainly satisfied 333Actually this possibility can not
occur, as is shown in the first step of the proof.. Otherwise we choose a
subsequence (again using the same index) so that

$\frac{u_{i}}{\|u_{i}\|}\to v_{1}\in\mathbb{S}^{d-2}.$

Again (as we will soon explain) it is not hard to see that for points $y\in
C\cap V_{1}$ with $\langle y,v_{1}\rangle\neq 0$ we have that $y+n_{i}\alpha$
is either eventually in the interior of $D$ or eventually in its exterior. For
points $y\in C\cap V_{1}$ with $\langle y,v\rangle=0$ we have to repeat this
procedure. Considering the subspace $V_{2}=\\{u\in V_{1}:\langle
u,v_{1}\rangle=0\\}$, we define the sequence of projections
$u^{\prime}_{i}={\rm{proj\,}}_{v_{1}}(u_{i})\in V_{2}$ and pass to a further
subsequence which converges to a vector $v_{2}$

$\frac{u^{\prime}_{i}}{\|u^{\prime}_{i}\|}\to v_{2}\in\mathbb{S}^{d-3}.$

Inductively this procedure will produce an ordered orthonormal basis
$\\{v_{0},v_{1},\dots,v_{d-1}\\}$ for ${\mathbb{R}}^{d}$ and a final
subsequence (which for simplicity we still denote as $n_{i}$) such that

> for each $y\in{\mathbb{T}}^{d}$, $y+n_{i}\alpha$ is either eventually in the
> interior of $D$ or it is eventually in its exterior.

This is clear for points $y\in{\mathbb{T}}^{d}$ such that $y+\beta\not\in C$.
Now suppose we are given a point $y$ with $y+\beta\in C$. We let $k$ be the
first index with $\langle y+\beta,v_{k}\rangle\neq 0$. As
$\\{v_{0},v_{1},v_{2},\dots,v_{d-1}\\}$ is a basis for ${\mathbb{R}}^{d}$ such
$k$ exists. We claim that the sequence $y+n_{i}\alpha$ is either eventually in
the interior of $D$ or it is eventually in its exterior. To see this consider
the affine hyperplane which is tangent to $C$ at $y+\beta$ (which contains the
vectors $\\{v_{0},\dots,v_{k-1}\\}$). Our assumption implies that the sequence
$y+n_{i}\alpha$ is either eventually on the opposite side of this hyperplane
from the sphere, in which case it certainly lies in the exterior of $D$, or it
eventually lies on the same side as the sphere. However in this latter case it
can not be squeezed in between the sphere and the tangent hyperplane, as this
would imply $\langle y+\beta,v_{k}\rangle=0$, contradicting our assumption.
Thus it follows that in this case the sequence $y+n_{i}\alpha$ is eventually
in the interior of $D$.

3. Let now $p$ be an element of $E(\sigma,X)$. We choose a net $\\{n_{\nu}\\}\subset{\mathbb{Z}}$ with $\sigma^{n_{\nu}}\to p$. It defines uniquely an element $\beta\in E(Y)\cong{\mathbb{T}}^{d}$ so that $\pi(px)=\pi(x)+\beta$ for every $x\in X$. Taking a subnet if necessary we can assume that the net $\frac{\beta-n_{\nu}\alpha}{\|\beta-n_{\nu}\alpha\|}$ converges to some $v_{0}\in S^{d-1}$. And, as above, proceeding by induction we assume likewise that all the corresponding limits $\\{v_{0},\dots,v_{k-1}\\}$ exist.

Next we choose a sequence $\\{n_{i}\\}$ such that $n_{i}\alpha\to\beta$,
$\frac{\beta-n_{i}\alpha}{\|\beta-n_{i}\alpha\|}\to v_{0}$ etc. We conclude
that $\sigma^{n_{i}}\to p$. Thus every element of $E(\sigma,X)$ is obtained as
a limit of a sequence in ${\mathbb{Z}}$ and is therefore of Baire class 1.

4. From the proof we see that the elements of $E(\sigma,X)\setminus{\mathbb{Z}}$ can be parametrized by the set ${\mathbb{T}}^{d}\times\mathcal{F}$, where $\mathcal{F}$ is the collection of ordered orthonormal bases for ${\mathbb{R}}^{d}$, $\ p\mapsto(\beta,\\{v_{0},\dots,v_{d-1}\\})$. ∎

##  7\. Remarks about order preserving systems

###  7.1. Order preserving action on the unit interval

Recall that for the group $G={\mathrm{H}}_{+}[0,1]$ comprising the orientation
preserving self-homeomorphisms of the unit interval, the $G$-system $X=[0,1]$
with the obvious $G$-action is tame [33]. One way to see this is to observe
that the enveloping semigroup of this dynamical system naturally embeds into
the Helly compact space (and hence is a Rosenthal compact space). By Theorem
3.2, $(G,X)$ is tame. We list here some other properties of
${\mathrm{H}}_{+}[0,1]$.

######  Remark 7.1.

Let $G:={\mathrm{H}}_{+}[0,1]$. Then

  1. (1)

_(Pestov[63])_ $G$ is extremely amenable.

  2. (2)

[30]
$\operatorname{WAP}(G)=\operatorname{Asp}(G)=\operatorname{SUC}(G)=\\{constants\\}$
and every Asplund representation of $G$ is trivial.

  3. (3)

[33] $G$ is representable on a (separable) Rosenthal space.

  4. (4)

_(Uspenskij[74, Example 4.4])_ $G$ is Roelcke precompact.

  5. (5)

$\operatorname{UC}(G)\subset\operatorname{Tame}(G)$, that is, the Roelcke
compactification of $G$ is tame.

  6. (6)

$\operatorname{Tame}(G)\neq\operatorname{UC}(G)$.

  7. (7)

$\operatorname{Tame}(G)\neq\operatorname{RUC}(G)$, that is, $G$ admits a
transitive dynamical system which is not tame.

  8. (8)

[60] ${\mathrm{H}}_{+}[0,1]$ and ${\mathrm{H}}_{+}({\mathbb{T}})$ are minimal
topological groups.

In properties (5) and (6) we answer two questions of T. Ibarlucia which are
related to [47]. For the details see [35].

######  Theorem 7.2.

The Polish group $G={\mathrm{H}}_{+}({\mathbb{T}})$ is Roelcke precompact.

###### Proof.

First a general fact: if a topological group $G$ can be represented as $G=KH$,
where $K$ is a compact subset and $H$ a Roelcke-precompact subgroup, then $G$
is also Roelcke-precompact. This is easy to verify either directly or by
applying [66, Prop. 9.17]. As was mentioned in Theorem 7.1.4,
${\mathrm{H}}_{+}[0,1]$ is Roelcke precompact. Now, observe that in our case
$G=KH$, where $H:=St(1)\cong{\mathrm{H}}_{+}[0,1]$ is the stability group of
$1\in{\mathbb{T}}$ and $K\cong{\mathbb{T}}$ is the subgroup of $G$ consisting
of the rotations of the circle. Indeed, the coset space $G/H$ is homeomorphic
to ${\mathbb{T}}$ and there exists a natural continuous section
$s:{\mathbb{T}}\to K\subset G$. ∎

###  7.2. Circularly ordered systems

In [36] we introduce the class of circularly ordered (c-ordered) dynamical
systems which naturally generalizes the class of linearly ordered systems. A
compact $S$-system $X$ is said to be c-ordered (notation $(S,X)\in$ CODS) if
the topological space $X$ is c-ordered and every $s$-translation $X\to X$ is
c-order preserving.

######  Example 7.3.

  1. (1)

With every c-ordered compact space there is the associated topological group
${\mathrm{H}}_{+}(X)$ of c-order preserving homeomorphisms. Certainly, $X$ is
a c-ordered ${\mathrm{H}}_{+}(X)$-system. Every linearly ordered $G$-system is
c-ordered.

  2. (2)

The Sturmian like ${\mathbb{Z}}^{k}$-subshifts (see Theorem 6.12) admit a
circular order. Moreover, their enveloping semigroups also are c-ordered
systems, [36].

  3. (3)

Every element $g$ of the projective group ${\mathrm{PGL}}(2,{\mathbb{R}})$
defines a homeomorphism on the circle ${\mathbb{T}}\to{\mathbb{T}}$ which is
either c-order preserving or $c$-order reversing.

######  Theorem 7.4.

[36]

  1. (1)

Every c-ordered compact, not necessarily metrizable, $S$-space $X$ is
Rosenthal representable (that is, $\mathrm{WRN}$), hence, in particular, tame.
So, $\mathrm{CODS}\subset\mathrm{WRN}\subset\mathrm{Tame}$.

  2. (2)

The topological group ${\mathrm{H}}_{+}(X)$ (with compact open topology) is
Rosenthal representable for every c-ordered compact space $X$. For example,
this is the case for ${\mathrm{H}}_{+}({\mathbb{T}})$.

The Ellis compactification $j:G\to E(G,{\mathbb{T}})$ of the group
$G={\mathrm{H}}_{+}({\mathbb{T}})$ is a topological embedding. In fact,
observe that the compact open topology on $j(G)\subset
C_{+}({\mathbb{T}},{\mathbb{T}})$ coincides with the pointwise topology. This
observation implies, by [34, Remark 4.14], that $\operatorname{Tame}(G)$
separates points and closed subsets. For any group $G$ having sufficiently
many tame functions the universal tame semigroup compactification $G\to
G^{\operatorname{Tame}}$ is a topological embedding.

######  Remarks 7.5.

  1. (1)

Regarding Theorem 7.1.2 we note that recently Ben-Yaacov and Tsankov [4] found
some other Polish groups $G$ for which $\operatorname{WAP}(G)=\\{constants\\}$
(and which are therefore also reflexively trivial).

  2. (2)

Although $G={\mathrm{H}}_{+}({\mathbb{T}})$ is representable on a (separable)
Rosenthal Banach space, the group ${\mathrm{H}}_{+}({\mathbb{T}})$ is Asplund-
trivial. Indeed, it is algebraically simple [20, Theorem 4.3] and contains a
copy of ${\mathrm{H}}_{+}[0,1]=St(z)$ (a stabilizer group of some point
$z\in{\mathbb{T}}$) which is Asplund-trivial [30]. Now, as in [30, Lemma
10.2], use an observation of Pestov, which implies that any continuous Asplund
representation of ${\mathrm{H}}_{+}({\mathbb{T}})$ is trivial.

######  Question 7.6.

Is it true that the universal tame compactification $u_{t}:G\to
G^{\operatorname{Tame}}$ is an embedding for every Polish group $G$ ?

The universal Polish group $G=H[0,1]^{{\mathbb{N}}}$ is a natural candidate
for a counterexample.

###  7.3. Noncommutative Sturmian like symbolic systems

The following construction yields many tame coding functions for subgroups of
${\mathrm{H}}_{+}({\mathbb{T}})$, and via any abstract homomorphism
$h:G\to{\mathrm{H}}_{+}({\mathbb{T}})$, we obtain coding functions on $G$.

######  Theorem 7.7.

Let $h:G\to{\mathrm{H}}_{+}({\mathbb{T}})$ be a group homomorphism and let

$f:{\mathbb{T}}\to A:=\\{0,\dots,d\\}$

be a finite coloring map induced by a finite partition of the circle
${\mathbb{T}}$ comprising disjoint arcs.

Then, for any given point $z\in{\mathbb{T}}$ we have:

  1. (1)

the coding function $\varphi=m(f,z):G\to\\{0,\dots,d\\}$ is tame on the
discrete copy of $G$.

  2. (2)

the corresponding symbolic $G$-system
$G_{\varphi}\subset\\{0,1,\cdots,d\\}^{G}$ is tame.

  3. (3)

if the action of $G$ on ${\mathbb{T}}$ is minimal then, in many cases, the
$G$-system $G_{\varphi}$ is minimal and circularly ordered.

###### Proof.

(A sketch) The coloring map $f:{\mathbb{T}}\to A:=\\{0,\dots,d\\}$ has bounded
variation. Every circle homeomorphism $g\in{\mathrm{H}}_{+}({\mathbb{T}})$ is
circular order preserving. This implies that the orbit $fG=\\{fg:g\in G\\}$,
as a bounded family of real (discontinuous) functions on ${\mathbb{T}}$, has
bounded total variation. As we know by [59, 36] any such family on
${\mathbb{T}}$ (or, on any other circularly ordered set) is tame. From Lemma
6.5.4 we conclude that $\varphi=m(f,z)$ is a tame function. This yields (1)
and (2). For (3) we use some results from [36]. ∎

######  Remark 7.8.

Some particular cases of this construction (for a suitable $h$) are as
follows:

  1. (1)

Sturmian and Sturmian like multidimensional symbolic
${\mathbb{Z}}^{k}$-systems.

  2. (2)

Consider a subgroup $G$ of ${\mathrm{PSL}}_{2}({\mathbb{R}})$, isomorphic to
$F_{2}$, which is generated by two Möbius transformations as in [32], say an
irrational rotation and a parabolic transformation. When $d=1$ (two colors) we
get the corresponding minimal tame subshift $X\subset\\{0,1\\}^{F_{2}}$.

  3. (3)

One can consider coding functions on any Fuchsian group $G$. E.g., for the
noncommutative modular group
$G={\mathrm{PSL}}_{2}({\mathbb{Z}})\simeq{\mathbb{Z}}_{2}\ast{\mathbb{Z}}_{3}$.

  4. (4)

More generally, at least in the assertions (1) and (2) of Theorem 7.7, one may
replace the circle ${\mathbb{T}}$ by any circularly ordered set $X$ with
circular order preserving $f:X\to A$.

##  8\. Tame minimal systems and topological groups

Recall that for every topological group $G$ there exists a unique universal
minimal $G$-system $M(G)$. Frequently $M(G)$ is nonmetrizable. For example,
this is the case for every locally compact noncompact $G$. On the other hand,
many interesting massive Polish groups are extremely amenable that is, having
trivial $M(G)$. See for example [63, 64, 74, 76]. The first example of a
nontrivial yet metrizable $M(G)$ was found by Pestov. In [63] he shows that
for $G:={\mathrm{H}}_{+}({\mathbb{T}})$ the universal minimal system $M(G)$
can be identified with the natural action of $G$ on the circle ${\mathbb{T}}$.
Glasner and Weiss [40, 41] gave an explicite description of $M(G)$ for the
symmetric group $S_{\infty}$ and for ${\mathrm{H}}(C)$ (the Polish group of
homeomorphisms of the Cantor set $C$). Using model theory Kechris, Pestov and
Todorc̆ević gave in [49] many new examples of various subgroups of
$S_{\infty}$ with metrizable (and computable) $M(G)$.

Note that the universal almost periodic factor $M_{AP}(G)$ of $M(G)$ is the
Bohr compactification $b(G)$ of $G$. When the induced homomorphism
$G\to{\mathrm{Homeo}}\,b(G)$ is injective (trivial) the topological group $G$
is called maximally (resp., minimally) almost periodic. Every topological
group $G$ has a universal minimal tame system $M_{t}(G)$ which is the largest
tame $G$-factor of $M(G)$. It is not necessarily AP (in contrast to the HNS
and WAP cases). There are (even discrete) minimally almost periodic groups
which however admit effective minimal tame systems, or in other words, groups
for which the corresponding homomorphism $G\to{\mathrm{Homeo}}\,(M_{t}(G))$ is
injective. For example, the countable group ${\mathrm{PSL}}_{2}({\mathbb{Q}})$
is minimally almost periodic (von Neumann and Wigner) even in its discrete
topology. It embeds densely into the group ${\mathrm{PSL}}_{2}({\mathbb{R}})$
which acts effectively and transitively on the circle. Thus, the circle
provides a topologically effective minimal action for every dense subgroup $G$
of ${\mathrm{PSL}}_{2}({\mathbb{R}})$. In particular, it is effective (though
not topologically effective) for the discrete copy of
${\mathrm{PSL}}_{2}({\mathbb{Q}})$).

######  Question 8.1.

Which Polish groups (e.g., discrete countable groups) $G$ have effective tame
minimal actions ? Equivalently, when is the homomorphism
$G\to{\mathrm{Homeo}}\,(M_{t}(G))$ injective?

Next we will discuss in more details the question “when is $M(G)$ tame ?”.

######  Definition 8.2.

We say that a topological group $G$ is _intrinsically tame_ if one of the
following equivalent conditions is satisfied:

  1. (1)

every continuous action of $G$ on a compact space $X$ admits a $G$-subsystem
$Y\subset X$ which is tame.

  2. (2)

any minimal compact $G$-system is tame.

  3. (3)

the universal minimal $G$-system $M(G)$ is tame.

  4. (4)

the natural projection $M(G)\to M_{t}(G)$ is an isomorphism.

The $G$-space $M_{t}(G)$ can also be described as a minimal left ideal in the
universal tame $G$-system $G^{\operatorname{Tame}}$. Recall that $G\to
G^{\operatorname{Tame}}$ is a semigroup $G$-compactification determined by the
algebra $\operatorname{Tame}(G)$. The latter is isomorphic to its own
enveloping semigroup and thus has a structure of a compact right topological
semigroup. Moreover, any two minimal left ideals there, are isomorphic as
dynamical systems.

In [29] we defined, for a topological group $G$ and a dynamical property $P$,
the notion of $P$-fpp ($P$ fixed point property). Namely $G$ has the $P$-fpp
if every $G$-system which has the property $P$ admits a $G$ fixed point.
Clearly this is the same as demanding that every minimal $G$-system with the
property $P$ be trivial. Thus for $P=\operatorname{Tame}$ a group $G$ has the
tame-fpp iff $M_{t}(G)$ is trivial.

We will need the following theorem which extends a result in [25].

######  Theorem 8.3.

Let $(G,X)$ be a metrizable minimal tame dynamical system and suppose it
admits an invariant probability measure. Then $(G,X)$ is point distal. If
moreover, with respect to $\mu$ the system $(G,\mu,X)$ is weakly mixing then
it is a trivial one point system.

###### Proof.

With notations as in [25] we observe that for any minimal idempotent $v\in
E(G,X)$ the set $C_{v}$ of continuity points of $v$ restricted to the set
$\overline{vX}$, is a dense $G_{\delta}$ subset of $\overline{vX}$ and
moreover $C_{v}\subset vX$ ([25, Lemma 4.2.(ii)]). Also, by [25, Proposition
4.3] we have $\mu(vX)=1$, and it follows that $\overline{vX}=X$. The proof of
the claim that $(G,X)$ is point distal is now finished as in [25, Proposition
4.4].

Finally, if the measure preserving system $(G,\mu,X)$ is weakly mixing it
follows that it is also topologically weakly mixing. By the Veech-Ellis
structure theorem for point distal systems [77, 14], if $(G,X)$ is nontrivial
it admits a nontrivial equicontinuous factor, say $(G,Y)$. However $(G,Y)$,
being a factor of $(G,X)$, is at the same time also topologically weakly
mixing which is a contradiction. ∎

######  Theorem 8.4.

  1. (1)

Every extremely amenable group is intrinsically tame.

  2. (2)

The Polish group ${\mathrm{H}}_{+}({\mathbb{T}})$ of orientation preserving
homeomorphisms of the circle is intrinsically tame.

  3. (3)

The Polish groups ${\mathrm{A}ut}\,(\mathbf{S}(2))$ and
${\mathrm{A}ut}\,(\mathbf{S}(3))$, of automorphisms of the circular directed
graphs $\mathbf{S}(2)$ and $\mathbf{S}(3)$, are intrinsically tame.

  4. (4)

A discrete group which is intrinsically tame is finite.

  5. (5)

For an abelian infinite countable discrete group $G$, its universal minimal
tame system $M_{t}(G)$ is a highly proximal extension of its Bohr
compactification $G^{AP}$ (see e.g. [25]).

  6. (6)

The Polish group ${\mathrm{H}}(C)$, of homeomorphisms of the Cantor set, is
not intrinsically tame.

  7. (7)

The Polish group $G=S_{\infty}$, of permutations of the natural numbers, is
not intrinsically tame. In fact $M_{t}(G)$ is trivial; i.e. $G$ has the tame-
fpp.

###### Proof.

(1) Is trivial.

(2) Follows from Pestov’s theorem [63], which identifies $(G,M(G))$ for
$G={\mathrm{H}}_{+}({\mathbb{T}})$ as the tautological action of $G$ on
${\mathbb{T}}$, and from Theorem 7.4 which asserts that this system is tame
(being c-ordered).

(3) The universal minimal $G$-systems for the groups
${\mathrm{A}ut}\,(\mathbf{S}(2))$ and ${\mathrm{A}ut}\,(\mathbf{S}(3))$ are
computed in [75]. In both cases it is easy to check that every element of the
enveloping semigroup $E(M(G))$ is an order preserving map. As there are only
$2^{\aleph_{0}}$ order preserving maps, it follows that the cardinality of
$E(M(G))$ is $2^{\aleph_{0}}$, whence, in both cases, the dynamical system
$(G,M(G))$ is tame.

In order to prove Claim (4) we assume, to the contrary, that $G$ is infinite
and apply a result of B. Weiss [78], to obtain a minimal model, say
$(G,X,\mu)$, of the Bernoulli probability measure preserving system
$(G,\\{0,1\\}^{G},(\frac{1}{2}(\delta_{0}+\delta_{1}))^{G})$. Now $(G,X,\mu)$
is metrizable, minimal and tame, and it carries a $G$-invariant probability
measure with respect to which the system is weakly mixing. Applying Theorem
8.3 we conclude that $X$ is trivial. This contradiction finishes the proof.
444Modulo an extension of Weiss’ theorem, which does not yet exist, a similar
idea would work for any locally compact group. The more general statement
would be: A locally compact group which is intrinsically tame is compact.

(5) In [43], [52] and [25] it is shown that a metric minimal tame $G$-system
is an almost one-to-one extension of an equicontinuous system. Now tameness is
preserved under sub-products, and because our group $G$ is countable, it
follows that $M_{t}(G)$ is a minimal sub-product of all the minimal tame
metrizable systems. In turn this implies that $M_{t}(G)$ is a (non-metrizable)
highly proximal extension of the Bohr compactification $G^{AP}$ of $G$.

(6) To see that $G={\mathrm{H}}(C)$ is not intrinsically tame it suffices to
show that the tautological action $(G,C)$, which is a factor of $M(G)$, is not
tame. To that end note that the shift transformation $\sigma$ on
$X=\\{0,1\\}^{\mathbb{Z}}$ is a homeomorphism of the Cantor set. Now the
enveloping semigroup $E(\sigma,X)$ of the cascade $(\sigma,X)$, a subset of
$E(G,X)$, is homeomorphic to $\beta{\mathbb{N}}$.

(7) To see that $G=S_{\infty}$ is not intrinsically tame we recall first that,
by [38], the universal minimal dynamical system for this group can be
identified with the natural action of $G$ on the compact metric space
$X=LO({\mathbb{N}})$ of linear orders on ${\mathbb{N}}$. Also, it follows from
the analysis of this dynamical system that for any minimal idempotent $u\in
E(G,X)$ the image of $u$ contains exactly two points, say
$uX=\\{x_{1},x_{2}\\}$. A final fact that we will need concerning the system
$(G,X)$ is that it carries a $G$-invariant probability measure $\mu$ of full
support [38]. Now to finish the proof, suppose that $(G,X)$ is tame. Then
there is a sequence $g_{n}\in G$ such that $g_{n}\to u$ in $E(G,X)$. If $f\in
C(X)$ is any continuos real valued function, then we have, for each $x\in X$,

$\lim_{n\to\infty}f(g_{n}x)=f(ux)\in\\{f(x_{1}),f(x_{2})\\}.$

But then, choosing a function $f\in C(X)$ which vanishes at the points $x_{1}$
and $x_{2}$ and with $\int f\,d\mu=1$, we get, by Lebesgue’s theorem,

$1=\int f\,d\mu=\lim_{n\to\infty}\int f(g_{n}x)\,d\mu=\int f(ux)\,d\mu=0.$

Finally, the property of supporting an invariant measure, as well as the fact
that the cardinality of the range of minimal idempotents is $\leq 2$, are
inherited by factors and thus the same argument shows that $M(G)$ admits no
nontrivial tame factor. Thus $M_{t}(G)$ is trivial. ∎

We will say that $G$ is _intrinsically c-ordered_ if the $G$-system $M(G)$ is
circularly ordered. Using this terminology Theorem 8.4 says that the Polish
groups $G={\mathrm{H}}_{+}({\mathbb{T}})$, ${\mathrm{A}ut}\,(\mathbf{S}(2))$
and ${\mathrm{A}ut}\,(\mathbf{S}(3))$ are intrinsically c-ordered. Note that
for $G={\mathrm{H}}_{+}({\mathbb{T}})$ every compact $G$-space $X$ contains a
copy of ${\mathbb{T}}$ as $G$-subspace or a $G$-fixed point.

The (nonamenable) group $G={\mathrm{H}}_{+}({\mathbb{T}})$ has one more
remarkable property. Besides $M(G)$, one can also effectively compute the
affine analogue of $M(G)$. Namely, the _universal irreducible affine system_
of $G$ (we denote it by $I\\!A(G)$) which was defined and studied in [21, 22].
It is uniquely determined up to affine isomorphisms. For any topological group
$G$ the corresponding affine compactification $G\to I\\!A(G)$ coincides with
the affine compactification $G\to P(M_{sp}(G))$, where, $M_{sp}(G)$ is the
_universal strongly proximal minimal system_ of $G$ and $P(M_{sp}(G))$ is the
space of probability measures on the compact space $M_{sp}(G)$. For more
information regarding affine compactifications of dynamical systems we refer
to [33].

######  Definition 8.5.

We say that $G$ is _convexly intrinsically tame_ if one of the following
equivalent conditions is satisfied:

  1. (1)

every compact affine dynamical system $(G,Q)$ admits an affine tame
$G$-subsystem.

  2. (2)

every compact affine dynamical system $(G,Q)$ admits a tame $G$-subsystem.

  3. (3)

every irreducible affine $G$-system is tame.

  4. (4)

the universal irreducible affine $G$-system $I\\!A(G)$ is tame.

Note that the $G$-system $P(X)$ is affinely universal for a $G$-system $X$;
also, $P(X)$ is tame whenever $X$ is [31, Theorem 6.11], [33]. In particular,
it follows that any intrinsically tame group is convexly intrinsically tame.

It is well known that a topological group $G$ is amenable iff $M_{sp}(G)$ is
trivial (see [22]). Thus $G$ is amenable iff $I\\!A(G)$ is trivial and it
follows that every amenable group is trivially convexly intrinsically tame.

Thus we have the following diagram which emphasizes the analogy between the
two pairs of properties:

extreme amenability intrinsically tame amenability convexly intrinsically tame

######  Remark 8.6.

Given a class $P$ of compact $G$-systems which is stable under subdirect
products, one can define the notions of an intrinsically $P$ group and a
convexly intrinsically $P$ group in a manner analogous to the one we adopted
for $P=\operatorname{Tame}$. We then note that in this terminology a group is
convexly intrinsically HNS (and, hence, also conv-int-WAP) iff it is amenable.
This follows easily from the fact that the algebra $\operatorname{Asp}(G)$ is
left amenable, [32]. This “collapsing effect” together with the special role
of tameness in the dynamical BFT dichotomy 3.2 suggest that the notion of
convex intrinsic tameness is a natural analogue of amenability.

At least for discrete groups, if $G$ is intrinsically HNS then it is finite.
In fact, for any group, an HNS minimal system is equicontinuous (see [29]), so
that for a group $G$ which is intrinsically HNS the universal minimal system
$M(G)$ coincides with its Bohr compactification $G^{AP}$. Now for a discrete
group, it is not hard to show that an infinite minimal equicontinuous system
admits a nontrivial almost one to one (hence proximal) extension which is
still minimal. Thus $M(G)$ must be finite. However, by a theorem of Ellis
[13], for discrete groups the group $G$ acts freely on $M(G)$, so that $G$
must be finite as claimed. Probably similar arguments will show that a locally
compact intrinsically HNS group is necessarily compact.

######  Theorem 8.7.

  1. (1)

$S_{\infty}$ is amenable (hence convexly intrinsically tame) but not
intrinsically tame.

  2. (2)

${\mathrm{H}}(C)$ is not convexly intrinsically tame.

  3. (3)

${\mathrm{H}}([0,1]^{{\mathbb{N}}})$ is not convexly intrinsically tame.

  4. (4)

${\mathrm{H}}_{+}({\mathbb{T}})$ is a (convexly) intrinsically tame
nonamenable topological group.

  5. (5)

$SL_{n}({\mathbb{R}})$, $n>1$ (more generally, any semisimple Lie group $G$
with finite center and no compact factors) is convexly intrinsically tame
nonamenable topological group.

###### Proof.

(1) $S_{\infty}$ is amenable, [12]. It is not intrinsically tame by Theorem
8.4.7.

(2) Natural action of ${\mathrm{H}}(C)$ on the Cantor set $C$ is minimal and
strongly proximal, but this action is not tame; it contains, as a subaction, a
copy of the full shift $({\mathbb{Z}},C)\cong(\sigma,\\{0,1\\}^{\mathbb{Z}})$.

(3) The group ${\mathrm{H}}([0,1]^{{\mathbb{N}}})$ is a universal Polish group
(see Uspenskij [73]). It is not convexly intrinsically tame. This can be
established by observing that the action of this group on the Hilbert cube is
minimal, strongly proximal and not tame. The strong proximality of this action
can be easily checked. The action is not tame because it is a _universal
action_ (see [55]) for all Polish groups on compact metrizable spaces.

(4) The (universal) minimal $G$-system ${\mathbb{T}}$ for
$G={\mathrm{H}}_{+}({\mathbb{T}})$ is strongly proximal. Hence, $I\\!A(G)$ in
this case is easily computable and it is exactly $P({\mathbb{T}})$ which, as a
$G$-system, is tame (by Theorem 8.4.4). Thus, ${\mathrm{H}}_{+}({\mathbb{T}})$
is a (convexly) intrinsically tame.

(5) By Furstenberg’s result [19] the universal minimal strongly proximal
system $M_{sp}(G)$ is the homogeneous space $X=G/P$, where $P$ is a minimal
parabolic subgroup (see [22]). Results of Ellis [15] and Akin [1] (Example
6.2.1) show that the enveloping semigroup $E(G,X)$ in this case is a Rosenthal
compact space, whence the system $(G,X)$ is tame by the dynamical BFT
dichotomy (Theorem 3.2). ∎

In particular, for $G=SL_{2}({\mathbb{R}})$ note that in any compact affine
$G$-space we can find either a 1-dimensional real projective $G$-space (a copy
of the circle) or a fixed point. For general $SL_{n}({\mathbb{R}})$, $n\geq 2$
– flag manifolds and their $G$-quotients.

## References

  * [1] E. Akin, Enveloping linear maps, in: Topological dynamics and applications, Contemporary Mathematics 215, a volume in honor of R. Ellis, 1998, pp. 121-131.
  * [2] E. Akin and E. Glasner, WAP Systems and Labeled Subshifts, arXiv:1410.4753, 2014, to appear in Mem. AMS.
  * [3] J. B. Aujogue, _Ellis enveloping semigroup for almost canonical model sets of an Euclidean space_ , Algebr. Geom. Topol. 15 (2015), no. 4, 2195-2237.
  * [4] I. Ben Yaacov and T. Tsankov, _Weakly almost periodic functions, model-theoretic stability, and minimality of topological groups_ , Trans. Amer. Math. Soc. 368 (2016), 8267-8294.
  * [5] J.F. Berglund, H.D. Junghenn and P. Milnes, Analysis on Semigroups, Wiley, New York, 1989.
  * [6] V. Berthe, S. Ferenczi, L.Q. Zamboni, _Interactions between Dynamics, Arithmetics and Combinatorics: the Good, the Bad, and the Ugly_ , Contemporary Math., 385 (2005), 333-364.
  * [7] V. Berthe, L. Vuillon, _Palindromes and two-dimensional Sturmian sequences,_ J. Automata, Languages and Combinatorics, 6 (2001), 121–138.
  * [8] J. Bourgain, D.H. Fremlin and M. Talagrand, Pointwise compact sets in Baire-measurable functions, Amer. J. of Math., 100:4 (1977), 845–886.
  * [9] B. Cascales, I. Namioka and J. Orihuela, _The Lindelof property in Banach spaces_ , Studia Math. 154 (2003), 165–192.
  * [10] D. Cenzer, A. Dashti, F. Toska and S. Wyman, Computability of countable subshifts in one dimension, Theory Comput. Syst. 51 (2012), no. 3, 352–371.
  * [11] A. Chernikov and P. Simon, Definably amenable NIP groups, arXiv:1502.04365v1, 2015.
  * [12] P. de la Harpe, Moyennabilité de quelques groupes topologiques de dimension infinie, C.R. Acad. Sci. Paris, Sér. A 277 (1973), 1037–1040.
  * [13] R. Ellis, Universal minimal sets, Proc. Amer. Math. Soc. 11 (1960), 540–543.
  * [14] R. Ellis, The Veech structure theorem, Trans. Amer. Math. Soc. 186 (1973), 203–218.
  * [15] R. Ellis, The enveloping semigroup of projective flows, Ergod. Th. Dynam. Sys. 13 (1993), 635–660.
  * [16] R. Engelking, General topology, revised and completed edition, Heldermann Verlag, Berlin, 1989.
  * [17] M. Fabian, Gateaux differentiability of convex functions and topology. Weak Asplund spaces, Canadian Math. Soc. Series of Monographs and Advanced Texts, New York, 1997.
  * [18] T. Fernique, _Multi-dimensional Sturmian sequences and generalized substitutions_ , Int. J. Found. Comput. Sci., 17 (2006), pp. 575–600.
  * [19] H. Furstenberg, A Poisson formula for semi-simple Lie groups, Ann. of Math. 77 (1963), 335–386.
  * [20] E. Ghys, _Groups acting on the circle_ , Enseign. Math. (2) 47 (2001), 329–407.
  * [21] S. Glasner, Compressibility properties in topological dynamics, Amer. J. Math., 97 (1975), 148–171.
  * [22] E. Glasner, _Proximal flows,_ Lect. Notes, 517, Springer, 1976.
  * [23] E. Glasner, Ergodic Theory via joinings, Math. Surveys and Monographs, AMS, 101, 2003.
  * [24] E. Glasner, On tame dynamical systems, Colloq. Math. 105 (2006), 283–295.
  * [25] E. Glasner, The structure of tame minimal dynamical systems, Ergod. Th. and Dynam. Sys. 27 (2007), 1819–1837.
  * [26] E. Glasner, Enveloping semigroups in topological dynamics, Topology Appl. 154, (2007), 2344–2363.
  * [27] E. Glasner, Translation-finite sets, ArXiv: 1111.0510, 2011.
  * [28] E. Glasner, The structure of tame minimal systems for general groups, Invent. math. DOI 10.1007/s00222-017-0747-z.
  * [29] E. Glasner and M. Megrelishvili, _Linear representations of hereditarily non-sensitive dynamical systems_ , Colloq. Math., 104 (2006), no. 2, 223–283.
  * [30] E. Glasner and M. Megrelishvili, New algebras of functions on topological groups arising from $G$-spaces, Fundamenta Math., 201 (2008), 1–51.
  * [31] E. Glasner and M. Megrelishvili, Representations of dynamical systems on Banach spaces not containing $l_{1}$, Trans. Amer. Math. Soc., 364 (2012), 6395–6424. ArXiv e-print: 0803.2320.
  * [32] E. Glasner and M. Megrelishvili, On fixed point theorems and nonsensitivity, Israel J. of Math., 190 (2012), 289–305. ArXiv e-print: 1007.5303.
  * [33] E. Glasner and M. Megrelishvili, _Banach representations and affine compactifications of dynamical systems_ , in: Fields institute proceedings dedicated to the 2010 thematic program on asymptotic geometric analysis, M. Ludwig, V.D. Milman, V. Pestov, N. Tomczak-Jaegermann (Editors), Springer, New-York, 2013. ArXiv version: 1204.0432.
  * [34] E. Glasner and M. Megrelishvili, _Representations of dynamical systems on Banach spaces,_ in: Recent Progress in General Topology III, (Eds.: K.P. Hart, J. van Mill, P. Simon), Springer-Verlag, Atlantis Press, 2014, 399–470.
  * [35] E. Glasner and M. Megrelishvili, _Eventual nonsensitivity and tame dynamical systems_ , arXiv:1405.2588, 2014.
  * [36] E. Glasner and M. Megrelishvili, _Circularly ordered dynamical systems,_ Monats. Math. 185 (2018), 415–441, arXiv:1608.05091, 2016.
  * [37] E. Glasner, M. Megrelishvili and V.V. Uspenskij, _On metrizable enveloping semigroups_ , Israel J. of Math. 164 (2008), 317–332.
  * [38] E. Glasner and B. Weiss, Quasifactors of zero-entropy systems, J. of Amer. Math. Soc. 8 (1995), 665–686.
  * [39] E. Glasner and B. Weiss, On Hilbert dynamical systems, Ergodic Theory Dynam. Systems, 32 (2012), no. 2, 629–642.
  * [40] E. Glasner and B. Weiss, Minimal actions of the group $S({\mathbb{Z}})$ of permutations of the integers, Geom. Funct. Anal., 12 (2002), 964–988.
  * [41] E. Glasner and B. Weiss, The universal minimal system for the group of homeomorphisms of the Cantor set, Fund. Math., 176 (2003), 277–289.
  * [42] E. Glasner and X. Ye, Local entropy theory, Ergodic Theory Dynam. Systems 29 (2009), 321–356.
  * [43] W. Huang, Tame systems and scrambled pairs under an abelian group action, Ergod. Th. Dynam. Sys. 26 (2006), 1549–1567.
  * [44] J.E. Jayne, J. Orihuela, A.J. Pallares and G. Vera, $\sigma$-fragmentability of multivalued maps and selection theorems, J. Funct. Anal. 117 (1993), no. 2, 243–273.
  * [45] J.E. Jayne and C.A. Rogers, Borel selectors for upper semicontinuous set-valued maps, Acta Math. 155 (1985), 41–79.
  * [46] T. Jolivet, _Combinatorics of Pisot Substitutions_ , TUCS Dissertations No 164, 2013.
  * [47] T. Ibarlucia, _The dynamical hierachy for Roelcke precompact Polish groups_ , Israel J. Math., 215 (2016), 965–1009.
  * [48] A.S. Kechris, Classical descriptive set theory, Graduate texts in mathematics, 156, 1991, Springer-Verlag.
  * [49] A.S. Kechris, V.G. Pestov, and S. Todorc̆ević, Fraïssé limits, Ramsey theory, and topological dynamics of automorphism groups, Geom. Funct. Anal. 15 (2005), no. 1, 106–189.
  * [50] P.S. Kenderov and W.B. Moors, _Fragmentability of groups and metric-valued function spaces_ , Top. Appl., 159 (2012), 183–193.
  * [51] D. Kerr and H. Li, Dynamical entropy in Banach spaces, Invent. Math. 162 (2005), 649–686.
  * [52] D. Kerr and H. Li, Independence in topological and $C^{*}$-dynamics, Math. Ann. 338 (2007), 869–926.
  * [53] A. Köhler, Enveloping semigrops for flows, Proc. of the Royal Irish Academy, 95A (1995), 179–191.
  * [54] A. Komisarski, H. Michalewski, P. Milewski, _Bourgain-Fremlin-Talagrand dychotomy and dynamical systems,_ preprint, 2004.
  * [55] M. Megrelishvili, Free topological $G$-groups, New Zealand J. of Math., 25 (1996), 59–72.
  * [56] M. Megrelishvili, Fragmentability and continuity of semigroup actions, Semigroup Forum, 57 (1998), 101–126.
  * [57] M. Megrelishvili, Fragmentability and representations of flows, Topology Proceedings, 27:2 (2003), 497–544. See also: www.math.biu.ac.il/ ${\tilde{}}$ megereli.
  * [58] M. Megrelishvili, _Topological transformation groups: selected topics,_ in: Open Problems In Topology II (Elliott Pearl, ed.), Elsevier Science, 2007, pp. 423–438.
  * [59] M. Megrelishvili, _A note on tameness of families having bounded variation_ , Topology and Appl. 217 (2017), 20–30.
  * [60] M. Megrelishvili and L. Polev, _Order and minimality of some topological groups_ , Topology and Appl., 201 (2016), 131–144.
  * [61] I. Namioka, Radon-Nikodým compact spaces and fragmentability, Mathematika 34 (1987), 258–281.
  * [62] I. Namioka and R.R. Phelps, Banach spaces which are Asplund spaces, Duke Math. J., 42 (1975), 735–750.
  * [63] V.G. Pestov, On free actions, minimal flows, and a problem by Ellis, Trans. Amer. Math. Soc., 350 (1998), 4149-4165.
  * [64] V.G. Pestov, _Dynamics of infinite-dimensional groups. The Ramsey-Dvoretzky-Milman phenomenon._ University Lecture Series, 40. American Mathematical Society, Providence, RI, 2006.
  * [65] R. Pikula, _Enveloping semigroups of affine skew products and sturmian-like systems_ , Dissertation, The Ohio State University, 2009.
  * [66] W. Roelcke and S. Dierolf, Uniform structures on topological groups and their quotients, McGraw-Hill, 1981.
  * [67] A.V. Romanov, _Ergodic properties of discrete dynamical systems and enveloping semigroups_ , Ergod. Th. & Dynam. Sys. 36 (2016), 198–214.
  * [68] H.P. Rosenthal, _A characterization of Banach spaces containing $l_{1}$_, Proc. Nat. Acad. Sci. U.S.A., 71 (1974), 2411–2413.
  * [69] W. Ruppert, _On weakly almost periodic sets,_ Semigroup Forum, 32 (1985) 267–281.
  * [70] S.A. Shapovalov, A new solution of one Birkhoff problem, J. Dynam. Control Systems, 6 (2000), no. 3, 331–339.
  * [71] M. Talagrand, _Pettis integral and measure theory,_ Mem. AMS No. 51, 1984.
  * [72] S. Todorc̆ević, Topics in topology, Lecture Notes in Mathematics, 1652, Springer-Verlag, 1997.
  * [73] V.V. Uspenskij, A universal topological group with countable base, Funct. Anal. Appl., 20 (1986), 160–161.
  * [74] V.V. Uspenskij, Compactifications of topological groups, Proceedings of the Ninth Prague Topological Symposium (Prague, August 19–25, 2001). Edited by P. Simon. Published April 2002 by Topology Atlas (electronic publication). Pp. 331–346, ArXiv:math.GN/0204144.
  * [75] L. Nguyen van Thé, More on the Kechris-Pestov-Todorcevic correspondence: precompact expansions, Fund. Math. 222 (2013), no. 1, 19–47.
  * [76] L. Nguyen van Thé, _Fixed points in compactifications and combinatorial counterparts,_ arXiv:1701.04257, 2017.
  * [77] W.A. Veech, Point-distal flows, Amer. J. Math., 92 (1970), 205–242.
  * [78] B. Weiss, Minimal models for free actions, Dynamical systems and group actions, 249-264, Contemp. Math., 567, Amer. Math. Soc., Providence, RI, 2012.