#  on ergodicity of mostly expanding semi-group actions

A.Ehsani, F.H.Ghane and M.Zaj  Department of Mathematics, Ferdowsi University
of Mashhad, Mashhad, Iran<EMAIL_ADDRESS>Department of
Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran
<EMAIL_ADDRESS>Department of Mathematics, Ferdowsi University of Mashhad,
Mashhad, Iran<EMAIL_ADDRESS>
###### Abstract.

In this work, we address ergodicity of smooth actions of finitely generated
semigroups on an $m$-dimensional closed manifold $M$. We provide sufficient
conditions for such an action to be ergodic with respect to the Lebesgue
measure. Our results improve the main result in [8], where the ergodicity for
one dimensional fiber was proved. We will introduce Markov partition for
finitely generated semi-group actions and then we establish ergodicity for a
large class of finitely generated semi-groups of
$C^{1+\alpha}$-diffeomorphisms that admit a Markov partition.

Moreover, we present some transitivity criteria for semi-group actions and
provide a weaker form of dynamical irreducibility that suffices to ergodicity
in our setting.

###### Key words and phrases:

finitely generated semi-group action of diffeomorphisms, minimal semi-group
action, transitive semi-group action, ergodicity of semi-group actions.

##  1\. introduction

One of the main goals in dynamical systems is to describe the typical behavior
of orbits and find that how they evolve in time. An interesting approach in
this direction is given by ergodic theory, which describes the typical
behavior of orbits from a measurable point of view. In this context ergodic
measures play a key role in understanding the ergodic behavior of a dynamical
system.

In this paper, we discuss ergodic properties of random iterations of a finite
family of $C^{1+\alpha}$-diffeomorphisms defined on an $m$-dimensional compact
manifold.

Some knowledge of ergodic properties of semi-group or group action of
diffeomorphisms is already available, through works of several authors. For
instance, see [12, 8] for the case that the ambient space is $S^{1}$ and [6]
for general case.
Motivated by these results, we consider semi-groups or groups of
$C^{1+\alpha}$-diffeomorphisms on a closed manifold $M$ exhibiting mostly
expanding behavior and establish their ergodicity with respect to Lebesgue
measure.

Write $\mathcal{G}^{+}$ (or $\mathcal{G}$) for the semigroup (or group)
generated by a finite family of diffeomorphisms on a compact manifold $M$.

A subset $B\subset M$ is _forward invariant_ for $\mathcal{G}^{+}$ , if
$f(B)\subset B$ for all $f\in\mathcal{G}^{+}$. For the group action case, we
can replace it by $f(B)=B$ for all $f\in\mathcal{G}$.

An invariant probability measure $\mu$ for a semigroup (or group) action is
_ergodic_ if for every measurable forward $\mathcal{G}^{+}$-invariant (or
$\mathcal{G}$-invariant) subset $A\subset M$ either $\mu(A)=0$ or
$\mu(M\setminus A)=0$. This definition can be extended to a measure $\mu$,
which is _quasi-invariant_ , i.e. $f_{*}\mu$ is absolutely continuous with
respect to $\mu$ for each element $f$ in the acting semigroup or group. Note
that for any $C^{1}$-map, the Lebesgue measure is quasi-invariant [8].

In this context, the following question is natural [8]:

######  Question 1.1.

Under what conditions a smooth action of a semigroup or group on a compact
manifold is Lebesgue ergodic?

The main results of [12] and [8], solve the question in the affirmative for
the one dimensional case under some additional assumptions. Indeed, for
finitely generated groups of $C^{1+\alpha}$ diffeomorphisms on $S^{1}$, if
$\mathcal{G}$ acts minimally and its Lyapunov expansion exponent is positive
then the action is ergodic.
We recall that the action of a semigroup (or group) is _minimal_ if every
closed invariant subset is either empty or coincides with the whole space,
equivalently total orbit of each point is dense in the ambient space.

A point $x\in M$ is said to be non-expandable provided that for every
$g\in\mathcal{G}^{+}$ (or $g\in\mathcal{G}$) one has $m(Dg(x))\leq 1$,
otherwise is called an expandable point.

In [8], the authors proved that if $\mathcal{G}$ is a finitely generated
subgroup of $Diff^{2}(S^{1})$ such that it acts minimally and possesses a
finite number of non-expandable points with some aditional assumption, then
the action of $\mathcal{G}$ is ergodic with respect to the Lebesgue measure.

Several authors also worked on ergodicity of semi-group actions on compact
$m$-dimensional manifolds. As shown in [6], ergodicity happens whenever the
semi-group action is expanding minimal $C^{1+\alpha}$-conformal.
The concept of expanding action in [6] has different means: a semi-group
action is _expanding_ provided that its inverse semigroup behaves locally
expandable.

In general, minimality [9] does not imply ergodicity. In the opposite
direction ergodicity does not imply minimality. Indeed, one can easily
construct examples of ergodic group actions having a global fixed points.
Moreover, the main result of this paper illustrates an example of an ergodic
semi-group action for which minimality is not a necessary condition.

Here, we improve the results of [12, 8] for the case that the ambient space is
$m$-dimensional, $m\geqslant 2$. In our setting, we allow that the non-
expandable points exist and they are contained in an open region $V\subset M$.

In fact, we contribute sufficient conditions for ergodicity that applicable to
a large class of finitely generated semi-groups of
$C^{1+\alpha}$-diffeomorphisms.
Our hypothesis that formulated in section 2 are conditions of the type that
Oliveira and Viana [16] introduced for deterministic non-uniformly expanding
maps. We extend the approach used by them to semi-groups of
$C^{1+\alpha}$-diffeomorphisms on a compact $m$-dimensional manifold. However,
this extension involves some difficulties. For instance, Markov partitions are
not known to exist for semi-group actions.

Let us note that non-uniformly expanding maps with some random noise was
addressed in several works. In particular, the stochastical stability of
general classes of non-uniformly expanding maps were established in [3], [4],
[18] and [7]. Also in [6], the authors introduced a strong form of non-uniform
expanding property for random maps. This concept in our setting has slightly
different means. Indeed, we will establish the ergodicity of semi-group
actions which equip with a weak form of non-uniform expanding property. This
property is referred to as _orbital non-uniform expanding property_.

In section 6, some criteria for dynamical irreducibility is obtained. It
provides a weak form of irreducibility in comparing with minimality. Also, the
concept of weak cycle property will be introduced. This concept is equivalent
to density of backward orbits for a subset of points with full Lebesgue
measure and suffices to establish ergodicity from local ergodicity. Finally,
we construct some examples that fit our assumptions in the last section.

##  2\. Preliminaries and results

Suppose that $X$ is a compact metric space and
$\mathcal{F}=\\{f_{1},\cdots,f_{k}\\}$ is a finite family of homeomorphisms of
$X$. Let us consider the _symbol space_ $\Sigma_{k}^{+}$ which is the set of
one sided infinite words over the alphabet $\\{1,\ldots,k\\}$. For any
sequence
$\omega=(\omega_{0}\omega_{1}\cdots\omega_{n}\cdots)\in\Sigma_{k}^{+}$, we
write $f^{n}_{\omega}(x)=f_{\omega_{n-1}}\circ f_{\omega}^{n-1}(x)$;
$n\in\mathbb{N}$ and $f_{\omega}^{0}=Id$. A fiber orbit corresponding to a
one-way infinite word $\omega=(\omega_{0}\omega_{1}\cdots)\in\Sigma_{k}^{+}$
at a point $x$ is defined by
$\mathcal{O}^{+}(x,\omega)=\\{f^{n}_{\omega}(x)\\}_{n=0}^{\infty}$.

######  Definition 2.1.

We say that a finite family $\mathcal{R}=\\{R_{1},\cdots,R_{k}\\}$ of subsets
of $X$ is a topological partition if it satisfies the following:
(1) each $R_{i}$ is open in $X$;
(2) $R_{i}\cap R_{j}=\emptyset$, for each $i,j=1,\cdots,k$, with $i\neq j$;
(3) $X=\overline{R_{1}}\cup\cdots\cup\overline{R_{k}}$.

In below, we generalize the concept of Markov partition to finitely semi-group
actions.

Take a topological partition $\mathcal{R}=\\{R_{1},\cdots,R_{k}\\}$ of a
compact metric space $X$ with together a finite family
$\mathcal{F}=\\{f_{1},\cdots,f_{k}\\}$ of homeomorphisms of $X$. Let us
consider the restriction $f_{i}|_{R_{i}}$, for each $i=1,\cdots,k$.
We say that $(\mathcal{R},\mathcal{F})$ satisfies the $(n,\omega)$-fold
intersection property for a positive integer $n\geq 3$ and
$\omega=(\omega_{0}\omega_{1}\omega_{2}\cdots)\in\Sigma_{k}^{+}$ if
$R_{\omega_{j}}\cap f^{-1}_{\omega_{j}}(R_{\omega_{j+1}})\neq\emptyset$,
$0\leq j\leq n-1$, implies that
$\cap_{j=0}^{n-1}(f_{\omega}^{j})^{-1}(R_{\omega_{j+1}})\neq\emptyset$.
A topological partition $\mathcal{R}=\\{R_{1},\cdots,R_{k}\\}$ is _Markov_ for
semigroup generated by $\mathcal{F}=\\{f_{1},\cdots,f_{k}\\}\subset home(X)$,
if $(\mathcal{R},\mathcal{F})$ satisfies the $(n,\omega)$-fold intersection
property for all $n\geq 3$ and $\omega\in\Sigma_{k}^{+}$.

It is not hard to see that $(\mathcal{R},\mathcal{F})$ is Markov provided that
it satisfies the following condition: if $f_{i}(R_{i})\cap
R_{j}\neq\emptyset$, then $f_{i}(R_{i})\supset R_{j}$, for each
$i,j=1,\cdots,k$.

###  2.1. Statement of the main result

In the rest of the paper we take M to be a compact Riemannian manifold, $m$
denotes the normalized Lebesgue measure and
$\\{f_{1},\ldots,f_{p},f_{p+1},\ldots,f_{p+q}\\}$ a finite family of
$C^{1+\alpha}$-diffeomorphisms defined on $M$ satisfying the following
conditions:

$(A_{0})$ There exists a topological partition
$\mathcal{R}=\\{R_{1},\cdots,R_{p},R_{p+1},\ldots,R_{p+q}\\}$ such that for
each $i=1,\cdots,p+q$, the clouser $\overline{R_{i}}$ has finite inner
diameter, i.e. any two points in $\overline{R_{i}}$ may be joined by a curve
contained in $\overline{R_{i}}$ whose length is bounded by a constant $L$.
Moreover, $m(\partial R_{i})=0$, for each $i=1,\cdots,p+q$.

$(A_{1})$ If $f_{i}(R_{i})\cap R_{j}\neq\emptyset$ then $f_{i}(R_{i})\supset
R_{j}$ which implies that $f_{i}(\overline{R_{i}})\supset\overline{R_{j}}$.

$(A_{2})$ There exists $\sigma_{1},\sigma_{2}>1$ such that

  * •

$\|Df_{i}(x)^{-1}\|^{-1}\geq\sigma_{1}$ for every $x\in R_{i}$, $1\leq i\leq
p$,

  * •

$\|Df_{p+j}(x)^{-1}\|^{-1}\geq\sigma_{2}^{-1}$ for every $x\in R_{p+j}$,
$1\leq j\leq q$

and $\sigma_{2}$ is close enough to 1.

$(A_{3})$ $|det(Df_{p+j}(x))|>q$ for every $x\in R_{p+j}$, $1\leq j\leq q$.

In fact, the acting of semi-group $\mathcal{G}^{+}$ generated by
$\\{f_{1},\ldots,f_{p+q}\\}$ is expanding on $R_{1}\cup\ldots\cup R_{p}$ and
is never very contacting. According to condition $(A_{1})$, the topological
partition $\mathcal{R}$ is Markov for $\mathcal{F}$. Also, by $(A_{0})$,
$m(\partial\mathcal{R})=0$ and therefore,
$m(\mathcal{O}_{\mathcal{G}}^{+}(\partial\mathcal{R}))=0$, where
$\mathcal{O}_{\mathcal{G}}^{+}(\partial\mathcal{R})$ is the total orbit of
$\partial\mathcal{R}$, i.e.

$\mathcal{O}_{\mathcal{G}}^{+}(\partial\mathcal{R})=\bigcup_{g\in\mathcal{G}^{+}}g(\partial\mathcal{R}).$

Moreover, we take $N=M\backslash\mathcal{O}_{\mathcal{G}}^{+}(\partial R)$.
Then $m(N)=1$.

######  Definition 2.2.

We say that a semi-group $\mathcal{G}^{+}$ satisfies the weak cycle property
if for each open set $B\subset M$, there exists a sequence
$\\{h_{i}\\}\subset\mathcal{G}^{+}$ such that
$M\circeq\bigcup_{i=1}^{\infty}h_{i}(B)$, this means that
$m(M\setminus\bigcup_{i=1}^{\infty}h_{i}(B)=0)$.

Now, we state the main result of the paper that illustrates the Lebesgue
ergodicity for a large class of finitely generated semigroups of
$C^{1+\alpha}$-diffeomorphisms. Here the concept of ergodicity is referred to
ergodicity of quasi-invariant measures and our focus is on the Lebesgue
measure since it is quasi invariant for $C^{1}$ maps.

######  Theorem A.

Suppose that $\mathcal{G}^{+}$ is a finitely generated semi-group of
$Diff^{1+\alpha}(M)$ for which the following holds:

$(1)$ $\mathcal{G}^{+}$ satisfies the weak-cycle property;

$(2)$ there exists a finite family
$\\{f_{1},\ldots,f_{p+q}\\}\subset\mathcal{G}^{+}$ satisfies conditions
$(A0)$, $(A1)$, $(A2)$ and $(A3)$ above.

Then $\mathcal{G}^{+}$ is ergodic with respect to the Lebesgue measure.

##  3\. Non-uniform expanding semigroups

Non-uniformly expanding maps introduced in [1, 17]. Then general conclusions
for systems exhibiting non-uniform expanding behavior provided in [2]. Let us
recall that a local diffeomorphism $f:M\to M$ is _non-uniformly expanding_ if
there exists $c>0$ such that for Lebesgue almost every point $x\in M$ one has:

$\limsup_{n\to\infty}\frac{1}{n}\sum_{j=0}^{n-1}\log\parallel
Df(f^{j}(x))^{-1}\parallel<-c.$

This approach has been most effective in studying ergodic aspects of systems
and extended to the random setting [5]: the authors considered random
perturbations at each iterate a map, that is close to a non-uniformly
expanding map, chosen independently according to some probabilistic law
$\theta_{\varepsilon}$, where $\varepsilon>0$ is the noise level.

The notion of non-uniformly expanding on random orbits was addressed in [6]
with slightly different means.

To be more precise, consider a measure preserving system
$(\Omega,\sigma,\mathbb{P})$, where $\mathbb{P}$ is a Borel measure,
$\sigma:\omega\to\omega$ is $\mathbb{P}$-invariant ergodic transformation and
$\Omega$ is a compact separable metric space with a random continuous map
$F:\Omega\rightarrow C^{r}(M,M)$. We denote $f_{\omega}:=F(\omega)$,
$f_{\omega}^{n}:=f_{\sigma^{n-1}\omega}\circ\dots\circ f_{\omega}$ and
$f_{\omega}^{0}:=id$.
Now, given $x\in M$ and $\omega\in\Omega^{\mathbb{Z}}$, the sequence
$(f^{n}_{\omega}(x))_{n\geq 1}$ is called a _random orbit_ of $x$.

We say that $F$ is a _non-uniformly expanding on random orbits_ if there
exists $c>0$ such that for $(\mathbb{P}\times m)$-almost
$(\omega,x)\in\Omega\times m$ it holds:

$\limsup_{n\to\infty}\frac{1}{n}\sum_{i=0}^{\infty}\parallel
Df_{\sigma^{i}\omega}(f^{j}_{\omega}(x)^{-1})\parallel<-c.$

They proved that [Proposition 4.4, [6]] there are no non-uniform expanding
finitely generated semi-group of diffeomorphisms which acts on random orbits.

In this paper, we deal with random product of finitely many maps and semi-
group action generated by these maps and then introduce a weak form of non-
uniform expanding property that we addressed here.

######  Definition 3.1.

Consider a finitely generated semigroup $\mathcal{G}^{+}$ (or group
$\mathcal{G}$) with generators $\\{f_{0},\ldots,f_{k-1}\\}$. We say that
$\mathcal{G}^{+}$ (or $\mathcal{G}$) is orbital non-uniformly expanding if
there exists $c>0$ such that for Lebesgue almost every $x\in M$, there is a
sequence $\omega\in\Sigma_{+}^{k}$ (or $\omega\in\Sigma^{k}$) satisfying

(1) $\limsup_{n\to\infty}\dfrac{1}{n}\sum_{i=0}^{n-1}\log\parallel
Df_{\omega_{i}}(f_{\omega}^{i}(x))^{-1}\|\leq-c.$

In the rest of this section, we show that a large class of finitely generated
semi-groups of $C^{1+\alpha}$ diffeomorphisms on a compact manifold $M$
exhibit orbital non-uniformly expanding property.

######  Definition 3.2.

Let $x\in N=M-\partial\mathcal{R}$ be given. Suppose that $x\in
R_{\omega_{0}}$ and the indices
$\omega_{0},\omega_{1},\cdots,\omega_{j-1}\in\\{1,\cdots,p+q\\}$ are chosen
such that for each $k=0,...,j-2$,

$f_{\omega_{k}}\circ\cdots\circ f_{\omega_{0}}(x)\in R_{\omega_{k+1}}.$

Now, we choose an index $\omega_{j}\in\\{1,\cdots,p+q\\}$ satisfying

$f_{\omega_{j-1}}\circ\cdots\circ f_{\omega_{0}}\in R_{\omega_{j}}.$

In this way, inductively we may associate a sequence $\omega=\omega(x)$, to
each $x\in N$, by taking $\omega=(\omega_{0},\omega_{1},\cdots)$ which is
called the _itinerary_ of $x$. Let $\pi_{n}$ denote the projection that maps
the sequence $\omega=(\omega_{0},\omega_{1},\cdots)$ to the finite word
$\omega(x,n)=(\omega_{0},\omega_{1},\cdots,\omega_{n-1})$ which we refer to it
as $n$-_itinerary_ of $x$.

######  Proposition 3.3.

Consider semi group $\mathcal{G}^{+}\subset Diff^{1+\alpha}(M)$ for which
there exist a finite family
$\\{f_{1},\ldots,f_{p},f_{p+1},\ldots,f_{p+q}\\}\subset\mathcal{G}^{+}$ and a
topological partition
$\mathcal{R}=\\{R_{1},\cdots,R_{p},R_{p+1},\ldots,R_{p+q}\\}$ satisfying
conditions $(A0)$, $(A1)$, $(A2)$ and $(A3)$. Then $\mathcal{G}^{+}$ is
orbital non-uniformly expanding.

###### Proof.

Consider the subset $\\{f_{1},\ldots,f_{p},\ldots,f_{p+q}\\}$ of
$\mathcal{G}^{+}$ and a family $\mathcal{R}$ of open subsets of $M$ satisfying
the assumptions. Also, for each $x\in N$, take a sequence $\omega=\omega(x)$
which is the itinerary of $x$. Since $N$ has full Lebesgue measure and
condition $(A3)$ ensures that the approach applied by Alves, Bonatti and Viana
[Lemma A1, [2]] can be extended to our setting, hence the following claim
holds.

######  Claim 3.4.

For Lebesgue almost every point $x\in M$, the $\omega$-orbital branch of $x$
spends a fraction $\epsilon_{0}>0$ of the time in $R_{1}\cup\cdots\cup R_{p}$,
for some $\epsilon_{0}>0$; that is

$\\#\\{0\leq k<n:f_{\omega}^{k}(x)\in R_{1}\cup\cdots\cup
R_{p}\\}\geq\epsilon_{0}n$

for large enough $n$, where $\omega$ is the itinerary of $x$.

Now, we use the claim to show that $\mathcal{G}^{+}$ is orbital non-uniformly
expanding, that is the equality (1) holds for Lebesgue almost every point
$x\in M$. Take $\epsilon_{0}>0$ as introduced in the claim and $\sigma_{2}$
close enough to 1 so that $\sigma_{1}^{-\epsilon_{0}}\sigma_{2}\leq e^{-c}$,
for some $c>0$. Let $x$ be any point satisfies the conclusion of the claim and
$\omega=\omega(x)$ is the itinerary of $x$. Then

$\displaystyle\prod_{j=0}^{n-1}\|Df_{\omega_{j}}(f_{\omega}^{j}(x))^{-1}\|\leq\sigma_{1}^{-\epsilon_{0}n}\sigma_{2}^{(1-\epsilon_{0})n}\leq
e^{-cn}$

for every large enough $n$. This means that $x$ satisfies the conclusion of
the proposition. ∎

##  4\. Hyperbolic times and hyperbolic cyliders

Throughout this section, we take
$\\{f_{1},\ldots,f_{p},f_{p+1},\ldots,f_{p+q}\\}\subset\mathcal{G}^{+}$ a
finite family of $C^{1+\alpha}$-diffeomorphisms and a topological partition
$\mathcal{R}=\\{R_{1},\cdots,R_{p},R_{p+1},\ldots,R_{p+q}\\}$ satisfying the
conditions $(A0)$, $(A1)$, $(A2)$, and $(A3)$.

First, we present the concept of _hyperbolic time_ of a point $x\in M$ for
semigroup action $\mathcal{G}^{+}$. This concept was introduced in [2] for
differentiable deterministic maps.

######  Definition 4.1.

Given $0<c<1$, we say that $n\in\mathbb{N}$ is a hyperbolic time for $x\in M$
if for every $1\leq k\leq n$, one has

(2) $\prod_{j=n-k}^{n-1}\|Df_{\omega_{j}}(f_{\omega}^{j}(x))^{-1}\|\leq
e^{-ck},$

where $\omega=\omega(x)=(\omega_{0},\omega_{1},\ldots,\omega_{j},\ldots)$ is
the itinerary of $x$.
Any nonempty set of the form

$C^{n}=C^{n}[\omega_{0},\ldots,\omega_{n-1}]=\\{x\in M:x\in
R_{\omega_{0}},f_{\omega_{0}}(x)\in R_{\omega_{1}},\ldots,f_{\omega}^{n}(x)\in
R_{\omega_{n-1}}\\}$

is called a _cylinder_ of length $n$. We say that $C^{n}$ is a _hyperbolic
cylinder_ provided that $n$ is a hyperbolic time for each point $x\in C^{n}$
and its itinerary $\omega$.

Let $\mathcal{C}^{n}$ be the family of all cylinders of length $n$ and
$\mathcal{C}^{n}_{h}$ the subset of hyperbolic cylinders.

######  Remark 4.2.

The closure of $\mathcal{C}^{n}[\omega_{0},\dots,\omega_{n-1}]$ is defined as

$\bar{C}^{n}[\omega_{0},\dots,\omega_{n-1}]=\\{y\in
M:y\in\bar{R}_{\omega_{0}},f_{\omega_{0}}(y)\in\bar{R}_{\omega_{1}},\dots,f_{\omega}^{n}(y)\in\bar{R}_{\omega_{n-1}}\\}.$

Hence, $f_{\omega_{j-1}}\circ\dots\circ
f_{\omega_{0}}(\bar{C}^{n}[\omega_{0},\dots,\omega_{n-1}])=\bar{C}^{n-j}[\omega_{j},\dots,\omega_{n-1}]$,
for any $1\leq j<n$.
So, $f_{\omega_{n-1}}\circ\dots\circ
f_{\omega_{0}}(\bar{C}^{n}[\omega_{0},\dots,\omega_{n-1}])=f_{\omega_{n-1}}(\bar{R}_{\omega_{n-1}})$
and thus its inner diameter is bounded by the constant

$K_{2}=K_{1}max_{1\leq i\leq p+q}\\{\parallel Df_{i}(x)\parallel:x\in
R_{i}\\},$

where $K_{1}$ is the maximum inner diameter of $\bar{R}_{i}$ over all
$i=1,\dots,p+q$.

######  Definition 4.3.

Let $x\in M$ with itinerary $\omega$. We say that the frequency of hyperbolic
times for $x$ is greater than $\epsilon_{0}>0$ if for large $n\in\mathbb{N}$,
there are $l\geq\epsilon_{0}n$ and integers $1\leq n_{1}<\cdots<n_{l}\leq n$
which are hyperbolic times for $x$.

######  Remark 4.4.

Let us note that if $\omega$ is the itinerary of $x$ then $\sigma^{k}\omega$
is the itinerary of $f_{\omega}^{k}(x)$, where $\sigma$ is the Bernoulli shift
transformation.
If $n$ is a hyperbolic time for $x$ with itinerary $\omega$, then $n-s$ is a
hyperbolic time for $f^{s}_{\omega}(x)$, for any $1\leq s<n$.
It is not hard to see that the converse is also true. Indeed, if $k<n$ is a
hyperbolic time for $x$ and there exists $1\leq s\leq k$ such that $n-s$ is a
hyperbolic time for $f^{s}_{\omega}(x)$ then $n$ is a hyperbolic time for $x$.

The next lemma asserts that for points satisfy the orbital non-uniformly
expanding property (1), there are infinitely many hyperbolic times. Moreover,
the set of hyperbolic times has positive density at infinity, and its proof is
based on a lemma due to Pliss (see e.g. [14]).

######  Lemma 4.5.

Suppose that $x\in M$, $\omega$ is its itinerary and $n\geq 1$ is such that

(3)
$\frac{1}{n}\sum_{j=0}^{n-1}\log\|Df_{\omega_{j}}(f_{\omega}^{j}(x))^{-1}\|\leq-c<0.$

Then, there is $\epsilon_{0}>0$, depending only on $\mathcal{G}^{+}$ and $c$,
and a sequence of hyperbolic times $1\leq n_{1}<\cdots<n_{t}\leq n$ for $x$,
with $t\geq\epsilon_{0}n$; that is the frequency of hyperbolic times for $x$
is larger than $\epsilon_{0}$.

###### Proof.

Similar to Corollary 3.2 of [2]. See also Proposition 4.4 of [5]. ∎

Suppose that $x$ satisfies the conclusion of Proposition 3.3, so for large
enough $n>0$

$\displaystyle\sum_{j=0}^{n-1}\log\|Df_{\omega_{j}}(f_{\omega}^{j}(x))^{-1}\|<-nc,$

thus

$\displaystyle\prod_{j=0}^{n-1}\|Df_{\omega_{j}}(f_{\omega}^{j}(x))^{-1}\|<e^{-nc}.$

Then $n$ is a hyperbolic time for $x$.
Let $\mathcal{H}$ be the set of all points $x\in M$ with itinerary $\omega$
satisfying the orbital non-uniformly expanding property (1). Clearly
$\mathcal{H}$ has full Lebesgue measure.

Here, the ball of radius $r>0$ is meant with respect to the Riemannian
distance $dist(x,y)$ on $M$. If $D\subset M$ is any path connected domain, we
define the distance $dist_{D}(x,y)$ between two points $x$ and $y$ in $D$ to
be the infimum the lengths of all curves joining $x$ to $y$ inside $D$. In
particular, $dist_{D}(x,y)\geq dist(x,y)$ for every $x$ and $y$ in $D$.

######  Lemma 4.6.

There is $r>0$ such that if $n$ is a hyperbolic time for $x\in\mathcal{H}$
with itinerary $\omega=(\omega_{0},\omega_{1},\ldots)$, then

(4) $\|Df_{\omega_{0}}(y)^{-1}\|\leq
e^{\frac{c}{2}}\|Df_{\omega_{0}}(x)^{-1}\|,$

for any point $y$ in the ball $B(x,re^{\frac{-nc}{2}})$.

###### Proof.

By continuity we can choose $r>0$ small enough so that the uniform bounds
$\sigma_{1}$ and $\sigma_{2}$ in conditions $(A1)$ and $(A2)$ hold for the
mappings $f_{i}$ in $r$-neighborhoods of $R_{i}$, $i=1,\ldots,p+q$. Moreover,
we take $r$ small enough so that the inverse of the exponential map $exp_{x}$
is defined on the $r$-neighborhood of every point $x\in M$ and it is isometry
on $B(x,r)$.

Now, by continuity and compactness of $M$ we can take $r>0$ small enough such
that if $n$ is a hyperbolic time for $x$ with itinerary
$\omega=(\omega_{0},\omega_{1},\ldots)$ then

$\|Df_{\omega_{0}}(y)^{-1}\|\leq e^{\frac{c}{2}}\|Df_{\omega_{0}}(x)^{-1}\|,$

whenever $y\in B(x,re^{\frac{-c}{2}}).$ Since $B(x,re^{\frac{-nc}{2}})\subset
B(x,re^{\frac{-c}{2}})$, the statement of the lemma follows by the above
inequality. ∎

A _dynamical ball_ of center $x$, itinerary $\omega\in\Sigma^{k}_{+}$, radius
$r$, and length $n\geq 1$ is defined by

$B(\omega,x,n,r)=\\{y\in M:dist(f_{\omega}^{j}(x),f_{\omega}^{j}(y))\leq
r,\forall\ 0\leq j\leq n\\}.$

######  Proposition 4.7.

There exists $r>0$ such that if $n$ is a hyperbolic time for
$x\in\mathcal{H}$, then there exists a neighborhood $V(x,n)$ of $x$ for which
the following hold:
1) $f_{\omega}^{n}$ maps $V(x,n)$ diffeomorphically onto
$B(f_{\omega}^{n}(x),r)$, where $B(f_{\omega}^{n}(x),r)$ is a ball of radius
$r$ and with center $f^{n}_{\omega}(x)$;
2) for every $y\in V(x,n)$ and $1\leq j\leq n$, we have

$\|D(f^{j}_{\sigma_{\omega}^{n-j}})^{-1}(z)\|\leq e^{\frac{-jc}{2}},$

where $z$ belongs to $B(f_{\omega}^{n}(x),r)$.

###### Proof.

Analogous to the proof of Proposition 4.9 of [5] and according to Lemma 4.6. ∎

It is not hard to see that the neighborhood $V(x,n)$ is a dynamical ball about
$x$.

######  Corollary 4.8.

For every $1\leq j\leq n$ and $x,y$ in the closure of any
$C^{n}=C^{n}[\omega_{0},\dots,\omega_{n-1}]\in\mathcal{C}^{n}_{h}$ we have:

$d_{f^{n-j}_{\omega}(\bar{C}^{n})}(f^{n-j}_{\omega}(x),f^{n-j}_{\omega}(y))\leq
e^{\frac{-jc}{2}}d_{f_{\omega}^{n}(\bar{C}^{n})}(f^{n}_{\omega}(x),f^{n}_{\omega}(y))\leq
K_{2}e^{\frac{-jc}{2}},$

where $f_{\omega}^{j}=f_{\omega_{j-1}}\circ\ldots\circ f_{\omega_{0}}$.

###### Proof.

Any curve joining $f^{n}_{\omega}(x)$ to $f^{n}_{\omega}(y)$ inside
$f^{n}_{\omega}(\bar{C}^{n})$ lifts to a unique curve joining
$f^{n-j}_{\omega}(x)$ to $f^{n-j}_{\omega}(y)$ inside
$f^{n-j}_{\omega}(\bar{C}^{n})$. Now we conclude the result by the above
proposition. ∎

######  Lemma 4.9.

There is $L_{1}>0$ such that for any $x,y$ in closure of any
$C^{n}=C^{n}[\omega_{0},\ldots,\omega_{n-1}]\in\mathcal{C}^{n}_{h}$

$L^{-1}_{1}\leq\frac{\mid detDf^{n}_{\omega}(x)\mid}{\mid
detDf^{n}_{\omega}(y)\mid}\leq L_{1}.$

###### Proof.

Note that by assumption $g_{j}=log\mid detDf_{j{\mid\bar{R}_{j}}}\mid$, for
$j=1,\dots,p+q$, is $\alpha$-Hölder and thus for each $x,y\in R_{j}$, it holds
that

$\mid g_{j}(x)-g_{j}(y)\mid\leq C_{0}d(x,y)^{\alpha},$

for some constants $C_{0}>0$ and $\alpha>0$. Now, suppose that
$C^{n}=C^{n}[\omega_{0},\ldots,\omega_{n-1}]$ is a hyperbolic cylinder and
$x,y\in\overline{C}^{n}$. This means that $x,y\in R_{\omega_{0}},\
f_{\omega_{0}}(x),f_{\omega_{0}}(y)\in R_{\omega_{1}}$ and for each $1\leq
i\leq n$, $f_{\omega}^{i}(x),f_{\omega}^{i}(y)\in R_{\omega_{i}}$. According
to Corollary 4.8, for each $1\leq j\leq n$ and $x,y\in\overline{C}^{n}$ it
holds that

$d_{f^{n-j}_{\omega}(\bar{C}^{n})}(f^{n-j}_{\omega}(x),f^{n-j}_{\omega}(y))\leq
e^{\frac{-jc}{2}}d_{f_{\omega}^{n}(\bar{C}^{n})}(f^{n}_{\omega}(x),f^{n}_{\omega}(y))\leq
K_{2}e^{\frac{-jc}{2}},$

and therefore,

$log\frac{\mid detDf^{n}_{\omega}(x)\mid}{\mid
detDf^{n}_{\omega}(y)\mid}=\sum^{n-1}_{i=0}\mid
g_{\omega_{i}}(f^{i}_{\omega}(x))-g_{\omega_{i}}(f^{i}_{\omega}(y))\mid\leq\sum^{n-1}_{i=0}C_{0}d_{f_{\omega}^{j}(\overline{C}^{n})}(f^{i}_{\omega}(x),f^{i}_{\omega}(y))^{\alpha}$
$\leq C_{0}K_{2}^{\alpha}\sum^{n-1}_{i=0}e^{\frac{-(n-i)c\alpha}{2}}\leq
C_{0}K_{2}^{\alpha}\sum^{\infty}_{i=0}e^{\frac{-ic\alpha}{2}}.$

Take
$L_{1}=\exp(C_{0}K_{2}^{\alpha}\sum^{\infty}_{i=0}e^{\frac{-ic\alpha}{2}}).$ ∎

######  Corollary 4.10.

There exists a constant $L_{2}>0$ such that if
$C^{n}=C^{n}[\omega_{0},\ldots,\omega_{n-1}]$ is a hyperbolic cylinder and
$A_{1},A_{2}$ are two subsets of $\overline{C}^{n}$ then

(5) $\displaystyle
L_{2}^{-1}\dfrac{m(A_{1})}{m(A_{2})}\leq\dfrac{m(f_{\omega}^{n}(A_{1}))}{m(f_{\omega}^{n}(A_{2}))}\leq
L_{2}\dfrac{m(A_{1})}{m(A_{2})}$

###### Proof.

We apply the change of variable formula for $f_{\omega}^{n}$ and analogous to
Corollaries 4.10, 4.11 and 4.12 of [5], we conclude the result. ∎

##  5\. Proof of Theorem A

Let us take a finite family $\\{f_{1},\ldots,f_{p+q}\\}\subset\mathcal{G}^{+}$
of $C^{1+\alpha}$ diffeomorphisms on $M$ and a Markov partition
$\mathcal{R}=\\{R_{1},\ldots,R_{p+q}\\}$ for which the conditions $(A0)$,
$(A1)$, $(A2)$ and $(A3)$ hold. Moreover, suppose that $\mathcal{G}^{+}$ has
weak-cycle property. We will prove that the semi-group action
$\mathcal{G}^{+}$ is ergodic with respect to the Lebesgue measure. We take
$\mathcal{A}$ the set of points $x\in M$ that belong to the closure of any
hyperbolic cylinder $C^{n}$ for infinitely many values
$n_{1}(x)<n_{2}(x)<\ldots<n_{k}(x)<\ldots$ of $n$. Clearly
$\mathcal{H}\subseteq\mathcal{A}$. In particular, each $x\in\mathcal{A}$ has
infinitely many hyperbolic times and hence $m(\mathcal{A})=1$.

Now, for each $n\geq 1$, we consider a covering $\mathcal{A}_{n}$ of
$\mathcal{A}$ by pairwise disjoint measurable sets such that
$A_{n}\in\mathcal{A}_{n}$ satisfying $C^{m}\subseteq
A_{n}\subseteq\overline{C}^{m}$ for some $C^{m}\in\mathcal{C}_{h}^{m}$ and
$m\geq n$. Indeed, we can take the family $\mathcal{A}_{n}$ which consists of
cylinders $C^{m}\in\mathcal{C}_{h}^{m}$ with $m\geq n$ that are not contained
in any $C^{k}\in\mathcal{C}_{h}^{k}$ with $m>k\geq n$.

######  Proposition 5.1.

Suppose that $B\subset M$ is a measurable $\mathcal{G}^{+}$-forward invariant
subset with positive Lebesgue measure. Then there exists an element $R_{i}$ of
the topological partition $\mathcal{R}$ such that $m(R_{i}\cap B^{c})=0$.

###### Proof.

Let $B\subset M$ be a measurable $\mathcal{G}^{+}$-forward invariant subset
with positive Lebesgue measure. The next claim is needed.

######  Claim 5.2.

Given $\delta>0$, there exist $n\geq 1$ and a subset $\\{A_{n,i}:i\in I\\}$ of
$\mathcal{A}_{n}$ such that

$m(B\bigtriangleup\bigcup_{i\in I}A_{n,i})\leq\delta.$

Indeed, it is enough to apply the technique of [Lemma 3.11, [16]] to our
setting. Take compact subsets $K\subset B$ and $K^{\prime}\subset B^{c}$ such
that

$m(B\bigtriangleup K)\leq\frac{\delta}{3},\ and\ m(B^{c}\bigtriangleup
K^{\prime})\leq\frac{\delta}{3}.$

We set $\rho=dist(K,K^{\prime})$. By Corollary 4.8, one has
$diam(\mathcal{A}_{n})\leq K_{2}e^{\frac{-nc}{2}}<\rho$ provided that $n$ is
large enough. Since $\mathcal{A}$ is covered by $\mathcal{A}_{n}$, so there
exists a family $\\{A_{n,i}:i\in I\\}\subset\mathcal{A}_{n}$ so that
$m(K\setminus\bigcup_{i\in I}A_{n,i})\leq\frac{\delta}{3}.$ Since all
$A_{i,n},\ i\in I$, intersect $K$, they are disjoint from $K^{\prime}$.
Therefore,

$m(B\bigtriangleup\bigcup_{i\in I}A_{n,i})\leq m(B\setminus
K)+m(K\setminus\bigcup_{i\in I}A_{n,i})+m(B^{c}\setminus
K^{\prime})\leq\delta.$

Now, according to the above claim and the approach of [Corollary 3.12, [16]]
the following claim holds.

######  Claim 5.3.

For each $\epsilon>0$, there exist $n\geq 1$ and $A_{n}\in\mathcal{A}_{n}$
such that

$\frac{m(B\cap A_{n})}{m(A_{n})}>1-\epsilon.$

By Claim 5.3, there exists a sequence $A_{n}$ of measurable sets for which the
following holds: $C^{m_{n}}\subset A_{n}\subset\bar{C}^{m_{n}}$, for some
hyperbolic cylinder $C^{m_{n}}$ with $m_{n}\geq n$ and

$\frac{m(A_{n}\cap B^{c})}{m(A_{n})}\to 0,\ n\to\infty.$

Then, according to the distortion Lemma 4.9, Corollary 4.10 and the assumption
that $B$ is forward invariant, one has that

(6) $\frac{m(f^{m_{n}}_{\omega({m_{n}})}(A_{n})\cap
B^{c})}{m(f^{m_{n}}_{\omega({m_{n}})}(A_{n}))}\to 0,\ when\ n\to\infty,$

where $\omega({m_{n}})$ is the itinerary of the cylinder
$C^{m_{n}}=C^{m_{n}}[\omega_{0},\ldots,\omega_{m_{n}-1}]$ and
$f^{m_{n}}_{\omega({m_{n}})}=f_{\omega_{m_{n}-1}}\circ\ldots\circ
f_{\omega_{0}}$.
By Remark 4.6,

$R_{\omega_{m_{n}-1}}\subset f^{m_{n}}_{\omega(m_{n})}(C^{m_{n}})\subset
f^{m_{n}}_{\omega(m_{n})}(A_{n}),$

for some $\omega_{m_{n}-1}\in\\{1,\dots,p+q\\}$. Fix any $j$ such that
$\omega_{m_{n}-1}=j$ for infinitely many values of $n$. We know that $R_{j}$
is an open subset and therefore has positive $m$-measure. Then $(6)$ implies
that $m(R_{j}\cap B^{c})=0$. ∎

Now, we will finish the proof of Theorem A. As you have seen in the previous
proposition, for forward invariant set $B$ with $m(B)>0$, there exists a
cylinder $R_{j}$ satisfying $m(R_{j}\cap B^{c})=0$. It is enough to show that
$m(B)=1$.

Let us fix the cylinder $R_{j}$. According to weak-cycle property of
$\mathcal{G}^{+}$, there exists a sequence $\\{g_{n}\\}\subset\mathcal{G}^{+}$
such that

$M\circeq\bigcup_{n\geq 1}g_{n}(R_{j}).$

Since $B$ is $\mathcal{G}^{+}$-forward invariant, $B^{c}$ is also
$\mathcal{G}^{+}$-forward invariant. Hence, one has that

$g_{n}(R_{j})\setminus B\subset g_{n}(R_{j}\setminus B),\ for\ all\ n\geq 1.$

The Lebesgue measure $m$ is quasi-invariant for $C^{1}$\- diffeomorphisms
which imply that $m(g_{n}(R_{j}\setminus B))=0$ and therefore,
$m(g_{n}(R_{j})\setminus B)=0$. Since $\\{g_{n}(R_{j})\\}$ is a countable
family, $m(B)=1$ which terminates the proof.

##  6\. Transitivity criteria

In this section we obtain some results for transitivity of semi-group actions
of homeomorphisms defined on a compact manifold $M$. We introduce some kinds
of transitive property and finally we provide a strong form of transitivity
that suffices to conclude ergodicity in our setting. First, we need to
formalize some notions.

Consider a finite family $\\{f_{1},\ldots,f_{k}\\}$ of homeomorphisms defined
on a compact manifold $M$. Let us take $\mathcal{G}^{+}$ the semi-group
generated by these homeomorphisms. For given $x\in M$, the _forward total
orbit_ of $x$ is defined by

$\mathcal{O}_{\mathcal{G}}^{+}(x)=\\{h(x):h\in\mathcal{G}^{+}\\}.$

Analogously, the _backward total orbit_ of $x$ is defined by

$\mathcal{O}_{\mathcal{G}}^{-}(x)=\\{h^{-1}(x):h\in\mathcal{G}^{+}\\}.$

We say that $\mathcal{G}^{+}$ acts _minimally_ on $M$ if any point has a dense
forward total orbit.
For any sequence
$\omega=(\omega_{0},\omega_{1},\ldots,\omega_{n},\ldots)\in\Sigma_{k}^{+}$, we
write

$f_{\omega}^{n}(x)=f_{\omega_{n}}\circ f_{\omega}^{n-1}(x);\ n\in\mathbb{N}\
and\ f_{\omega}^{0}=Id.$

The set $\\{f_{\omega}^{n}(x):n\geq 0\\}$ is called $\omega$-_orbital branch_
of $x$. We say that $x$ possesses an _orbital dense orbit_ if there exists a
sequence $\omega\in\Sigma_{k}^{+}$ such that $\omega$-orbital branch of $x$ is
dense in $M$.

It is easy to see that if $\mathcal{G}^{+}$ acts minimally on $M$, then each
$x$ has orbital dense orbit.
In a finitely generated semi-group action, if minimality of action preserves
under small perturbation of generators then we say that $\mathcal{G}^{+}$ is
_robustly minimal_. Some examples of robustly minimal semi-group actions are
already available, for instance see [10] and [11]. Let us mention that
transitive semi-group actions have a weak form of dynamical irreducible
property with respect to minimal semi-groups.

Here, we introduce three kinds of transitive property. The action of
$\mathcal{G}^{+}$ is _weak transitive_ if it possesses a point with a dense
forward total orbit. We say that $\mathcal{G}^{+}$ is _transitive_ whenever
$U$ and $V$ are two open subsets of $M$, there exists $h\in\mathcal{G}^{+}$
such that $h(U)\cap V\neq\emptyset$. Finally, $\mathcal{G}^{+}$ is _strong
transitive_ if it admits a full measure subset $\widetilde{M}\subset M$ such
that every point $x\in\widetilde{M}$ has a dense forward orbit.

In bellow, we illustrate the relationship between these different concepts of
transitive property. Moreover, to obtain some results about the transitive
properties of semi-group actions, we adapt some techniques from deterministic
systems [13] to our setting.

A semi-group $\mathcal{G}^{+}$ is _contractive_ at point $x\in M$ whenever
there exist a sequence $\\{h_{n}\\}\subset\mathcal{G}^{+}$ and open set $B$
containing $x$ for which $diam(h_{n}(B))\to 0$, as $n\to\infty$.

######  Lemma 6.1.

Let $\\{f_{1},\dots,f_{k}\\}$ be a finite family of homeomorphisms on a
compact manifold $M$ and $\mathcal{G}^{+}$ be the semi-group generated by
them. Suppose that there exist $x\in M$ for which the total backward orbit and
total forward orbit are dense in $M$. If the semi-group $\mathcal{G}^{+}$ is
contractive at $x$, then the action of $\mathcal{G}^{+}$ is transitive, i.e.
for each two open subsets $U$ and $V$ of $M$ there exists
$h\in<\mathcal{G}>^{+}$ such that $h(U)\cup V\neq\emptyset$.
In particular, the point $x$ possesses an orbital branch which is dense in
$M$.

###### Proof.

Suppose that $U$ and $V$ are two open subsets of $M$. Since the total forward
orbit of $x$ is dense in $M$, there exist $k_{1},k_{2}\in\mathcal{G}^{+}$ such
that $k_{1}(x)\in U$ and $k_{2}(x)\in V$.
According to the density of $\mathcal{O}^{-}_{\mathcal{G}}(x)$, we can choose
two sequences $\\{z_{i}\\}\subset U$ and $\\{h_{i}\\}\subset\mathcal{G}^{+}$
that satisfying the following property: $h_{i}(z_{i})=x$ and $z_{i}$ converges
to $k_{1}(x)$ when $i$ goes to infinity and therefore, $h_{i}(k_{1}(x))$ is
contained in a neighborhood $B$ of $x$, for large enough $i$.
On the other hand, continuity of $k_{2}$ and this fact that $k_{2}(x)\in V$
imply that $k_{2}(B_{r}(x))\subset V$, for some small $r>0$.
Now, by contractibility of $\mathcal{G}^{+}$ at $x$, one can find
$g\in<\mathcal{G}>^{+}$ such that $g(B)\subset B_{r}(x)$, and therefore
$k_{2}(g(h_{i}(k_{1}(x))))\in V.$ Let us take $h:=k_{2}\circ g\circ h_{i}\circ
k_{1}$, then $h(U)\cap V\neq\emptyset$.
Finally, if we apply the above argument for a countable basis of $M$, the
second statement follows immediately. ∎

######  Remark 6.2.

Let us take $\mathcal{G}^{+}\subset Diff^{1+\alpha}(M)$ with generators
$g_{1},\ldots,g_{k}$ and there exists
$\\{f_{1},\ldots,f_{p+q}\\}\subset\mathcal{G}^{+}$ satisfies the conditions
proposed in subsection 2.1. Then Corollary 4.8 ensures that the inverse semi-
group $\mathcal{G}^{-}=<g_{1}^{-1},\ldots,g_{k}^{-1}>$ is contractive at each
point $x\in\mathcal{H}$. Hence, Lemma 6.2 implies that if $\mathcal{G}^{+}$
possesses a point $x\in\mathcal{H}$ with forward and backward dense orbit,
then both semi-groups $\mathcal{G}^{+}$ and $\mathcal{G}^{-}$ are transitive.

######  Lemma 6.3.

Suppose that $\\{f_{1},\ldots,f_{k}\\}$ is a finite family of homeomorphisms
on $M$ and $\mathcal{G}^{+}$ is the semi-group generated by these maps. If
$\mathcal{G}^{+}$ has any point with forward orbital dense orbit, then the set
of points with total forward dense orbit is a residual subset of $M$.

###### Proof.

Suppose that there exists a point $x\in M$ and $\omega\in\Sigma^{+}_{k}$ such
that the $\omega$-orbital orbit of $x$,
$\mathcal{O}^{+}_{\mathcal{G}}(\omega,x)=\\{f^{j}_{\omega}(x):j\in\mathbb{N}\\}$,
is dense in $M$. Let us take a countable basis
$\mathcal{V}=\\{V_{j}:j\in\mathbb{N}\\}$ of $M$ and we set

$\mathcal{A}_{n}=\\{z\in M:\mathcal{O}_{\mathcal{G}}^{+}(z)\ is\
\frac{1}{n}-dense\\}.$

By density of $\mathcal{O}^{+}_{\mathcal{G}}(\omega,x)$ there exist sequences
$\\{x_{j_{l}}\\}\subset\mathcal{O}^{+}_{\mathcal{G}}(\omega,x)$ and
$k_{n,l}\subset\mathbb{N}$, for each $n,l\in\mathbb{N}$, for which the
following holds:
$x_{j_{l}}\in V_{l}$ and the segment orbit
$\\{x_{j_{l}}=f^{j_{l}}_{\omega}(x),f^{j_{l}+1}_{\omega}(x),\ldots
f^{j_{l}+k_{n,l}}_{\omega}(x)\\}$ is $\frac{1}{2n}$-dense. By continuity,
there exists $\\{r_{n,l}>0\\}_{n,l\in\mathbb{N}}$ such that
$f^{i}_{\sigma^{j_{l}-1}\omega}(B_{r_{n,l}}(x_{j_{l}}))\subset
B_{\frac{1}{2n}}(f^{i}_{\sigma^{j_{l}-1}\omega}(x_{j_{l}}))$ for $0\leq i\leq
k_{n,l}$. This means that for each $y\in B_{r_{n,l}}(x_{j_{l}})$, the segment
orbit
$\\{y,f_{\sigma^{j_{l}-1}\omega}(y),\ldots,f^{k_{n,l}}_{\sigma^{j_{l}-1}\omega}(y)\\}$
is $\frac{1}{n}$-dense.
Therefore, $\cup_{l\in\mathbb{N}}B_{r_{n,l}}(x_{j_{l}})$ is an open and dense
subset. In particular, this proves that
$\cap_{n\in\mathbb{N}}\cup_{l\in\mathbb{N}}B_{r_{n,l}}(x_{j_{l}})$ is a
residual subset contained in $\cap_{n=1}^{\infty}\mathcal{A}_{n}$ which
consists of the points with dense forward orbit. This terminates the proof of
the lemma. ∎

######  Lemma 6.4.

Let $\mathcal{F}$ be a finite family of homeomorphism on $M$. If there exists
a point $x\in M$ with forward orbital dense orbit, then $\\{x\in
M:\overline{\mathcal{O}^{-}_{\mathcal{F}}(x)}=\overline{\mathcal{O}^{+}_{\mathcal{F}}(x)}=M\\}$
is a residual subset of $M$.

###### Proof.

From the previous lemma, $\\{x\in
M:\overline{\mathcal{O}^{+}_{\mathcal{F}}(x)}=M\\}$ is a residual subset of
$M$. Let us take $x\in M$ with forward orbital dense orbit, so there exists
$\omega\in\Sigma^{+}_{n}$ for which $\mathcal{O}^{+}_{\mathcal{F}}(x,\omega)$
is dense in $M$, i.e, the set
$\\{x,f_{\omega_{0}}(x),\dots,f^{j}_{\omega}(x),\dots\\}$ is dense in $M$. Let
us take $x_{l}=f^{l}_{\omega}(x)$ and $\mathcal{A}_{n}=\\{x\in
M:\mathcal{O}^{-}_{\mathcal{F}}(x)\ \ is\ \ \frac{1}{n}-dense\\}$. Since
$\mathcal{O}^{+}_{\mathcal{F}}(x,\omega)$ is dense, there exists $\\{k_{n}\\}$
such that $\\{x,\ldots,f^{k_{n}}_{\omega}(x)\\}$ is $\frac{1}{2n}$-dense. As
$\\{x,\ldots,f^{k_{n}}_{\omega}(x)\\}\subset\mathcal{O}^{-}(x_{j})$ for all
$j\geq k_{n}$, we get that $x_{j}\in\mathcal{A}_{n}$ for each $j\geq k_{n}$.
By continuity, one can find $r_{j}>0$ such that
$f^{l}_{\omega}(B_{r_{j}}(x))\subset B_{\frac{1}{2n}}(f^{l}_{\omega}(x))$, for
all $0\leq l\leq k_{j}$. This proves that $\mathcal{O}^{-}_{\mathcal{F}}(y)$
is $\frac{1}{n}$-dense for all $y\in f^{j}_{\omega}(B_{r_{j}}(x))$ and $j\geq
k_{n}$. As $f^{j}_{\omega}(B_{r_{j}}(x))$ is an open neighborhood of $x_{j}$
and $\\{x_{j}:j\geq k_{n}\\}$ is dense, then $\cup_{j\geq
k_{n}}f^{j}_{\omega}(B_{r_{j}}(x))\subset\mathcal{A}_{n}$ is an open and dense
subset $M$. Therefore $\cap_{n\geq 1}\cup_{j\geq
k_{n}}f^{j}_{\omega}(B_{r_{j}}(x))\subset\cap_{n\geq 1}\mathcal{A}_{n}$ is a
residual subset of $M$. Since $\cap_{n\geq 1}\mathcal{A}_{n}$ consists of
points with dense backward orbit, then $\\{x\in
M:\overline{\mathcal{O}^{-}_{\mathcal{F}}(x)}=M\\}\cap\\{x\in
M:\overline{\mathcal{O}^{+}_{\mathcal{F}}(x)}=M\\}$ is also residual. This
terminates the proof of the lemma.

∎

Let us recall that the a finitely generated semi-group $\mathcal{G}^{+}$
satisfies the _weak cycle_ property if for each open set $B\subset M$, there
exists a sequence $\\{h_{i}\\}\subset\mathcal{G}^{+}$ such that
$M\circeq\cup^{\infty}_{i=1}h_{i}(B)$, this means that $m(M\setminus\cup
h_{i}(B))=0$.

######  Lemma 6.5.

A finitely generated semi-group $\mathcal{G}^{+}$ of homeomorphisms on $M$ has
weak cycle property if and only if there exists a full measure subset
$\tilde{M}\subset M$ such that for each point $x\in\tilde{M}$, the total
backward orbit of $x$ is dense in $M$.

###### Proof.

First, suppose that $\mathcal{G}^{+}$ has weak cycle property. Let us take a
countable basis $\\{V_{j}:j\geq 1\\}$ of $M$. Weak cycle property implies that
there exists a sequence
$\\{h_{ij}\\}_{i,j\in\mathbb{N}}\subset\mathcal{G}^{+}$ so that
$M\circeq\cup^{\infty}_{i=1}h_{ij}(V_{j})$, for each $j\in\mathbb{N}$.
Suppose that $F_{j}=M\setminus\cup_{i}h_{ij}(V_{j})$, then $F_{j}$ has zero
Lebesgue measure, hence $F=\cup_{j}F_{j}$ has also zero Lebesgue measure. We
set $\tilde{M}=M\setminus F$ and we show that each $x\in\tilde{M}$ has a dense
backward orbit.
Indeed, if $x\in\tilde{M}$ then $x$ is not belong to $F$, which implies that
$x$ is not belong to $F_{j}$, for each $j$. This means that $x\in
h_{ij}(V_{j})$ for each $i$. Therefore $h^{-1}_{ij}(x)\in V_{j}$ and hence the
backward orbit of $x$ is dense in $M$.

Conversely, suppose that there exists a full measure subset $\tilde{M}\subset
M$ so that each $x\in\tilde{M}$ has dense backward orbit. This means that for
each open set $B$ there exists $h_{x}\in\mathcal{G}^{+}$ with
$h^{-1}_{x}(x)\in B$. Therefore $x\in h_{x}(B)$. Then, the family
$\\{h_{x}(B):x\in\tilde{M}\\}$ is an open covering of $\tilde{M}$ and hence
there exists a countable subcover $\\{h_{x_{i}}(B):i\in\mathbb{N}\\}$. If we
set $h_{i}:=h_{x_{i}}$, then we conclude that $M\circeq\cup_{i}h_{i}(B)$. ∎

##  7\. Examples

In this section, we exhibit some examples of finitely generated semigroups of
$Diff^{1+\alpha}(M)$ satisfying our hypothesis in subsection 2.1. To start the
construction, we recall some concepts of topology.

The _standard_ $m$-_simplex_ is the set

$\triangle_{m}=\\{x\in\mathbb{R}^{m+1}:x_{i}\geq 0\ and\
\sum_{i=0}^{m+1}x_{i}=1\\}.$

A _general_ $m$-_simplex_ is a subset of $M$ diffeomorphic to the standard
$m$-simplex $\triangle_{m}$ and a _general_ $n$-_face_ is a subset of $M$
diffeomorphic to the standard $n$-face.

A _triangulation_ of a compact $m$\- manifold $M$ is a finite collection
$\mathcal{T}$ of diffeomorphic images of $\triangle_{m}$ that cover $M$ and
satisfying the following condition: for any general $m$-simplices
$S_{i},S_{j}\in\mathcal{T}$, if $S_{i}\cap S_{j}\neq\emptyset$, their
intersection is a $(m-1)$-face of both $S_{i}$ and $S_{j}$.
The _barycentric subdivision_ of an $m$-dimensional simplex $S$ consists of
$(m+1)!$ simplices. Each piece, with vertices $v_{0},\ldots,v_{m}$ can be
associated with a permutation $p_{0},\ldots,p_{m}$ of the vertices of $S$, in
such a way that each vertex $v_{i}$ is the barycenter of the points
$p_{0},\ldots,p_{i}$. Barycentric subdivision is an important tool which is
used to obtain finer simplicial complexes.

Now, we start by considering any smooth Riemannian manifold $M$ that supports
orbital expanding or orbital non-uniformly expanding finitely generated
semigroups of $Diff^{1+\alpha}(M)$. We are going to introduce a finite family
of $Diff^{1+\alpha}(M)$ that admits a Markov partition satisfying conditions
$(A_{0}),(A_{1}),(A_{2})$ and $(A_{3})$.

It is a well-known fact that $M$ possesses a triangulation
$\mathcal{T}=\\{S_{1},\ldots,S_{k}\\}$ , where $S_{i},\ i=1,\ldots,k$, are
diffeomorphic images of $\triangle_{m}$ and cover $M$. Suppose that $g_{i},\
i=1,\ldots,k$, are diffemorphisms which map $\triangle_{m}$ to $S_{i}$.
We divide $\triangle_{m}$ by barycentric subdivision to smaller $m$\-
dimensional $m$-simplices $\triangle_{m,1},\ldots,\triangle_{m,l}$. Take
affine maps $h_{j},j=1,\ldots,l$, which map $\triangle_{m,j}$ to
$\triangle_{m}$.
Clearly, each $h_{j}$ is an expanding map. Now, we set $f_{i,j}:=g_{i}\circ
h_{j}\circ g_{i}^{-1}$ and extend it to a $C^{1+\alpha}$-diffeomorphism on
$M$. Also, take $S_{i.j}:=g_{i}(\triangle_{m,j})$, then the interiors of
$S_{i,j}$, $i=1,\ldots,k$ , $j=1,\ldots,l$ exhibit a Markove partition, that
is if we take $R_{i,j}:=int(S_{i,j})$, then $(R_{i,j},f_{i,j})$ satisfying
conditions $(A0),\ldots,(A3)$, with $p=kl$ and $q=0$. Clearly, the semigroup
generated by $\\{f_{i.j}\\}$ is an orbital expanding finitely generated
$C^{1+\alpha}$ semi-group.

Now, we will give some changes in the subdivision corresponding to one of
generalized $m$-simplices that allows us to exhibit a finitely generated semi-
group of $C^{1+\alpha}$-diffeomorphisms on any smooth manifold that is orbital
non-uniformly expanding.

Again, we start with a triangulation $\mathcal{T}$ on a smooth compact
Riemannian manifold $M$ and choose a generalized $m$-simplex
$S^{*}\in\mathcal{T}$. Let $\\{\triangle_{m,1},\ldots,\triangle_{m,l}\\}$ be a
subdivision of standard $m$-simplex $\triangle_{m}$ in the following sense:
there exists a sub-simplex $\triangle_{m,j}$ which admits the entire of an
$(m-1)$-face of $\triangle_{m}$ as a face but other faces are strictly smaller
than the faces of $\triangle_{m}$. Moreover, other sub-simplicies are smaller
than $\triangle_{m}$ in all direction.

We take $h_{*}$ an affine transformation which maps $\triangle_{m,j}$ to
$\triangle_{m}$. Also, suppose that $h_{i}$, $i=1,\ldots,l$, $i\neq j$, are
affine transformations which map $\triangle_{m,i}$ to $\triangle_{m}$.
Let $g$ be a diffeomorphism which takes $\triangle_{m}$ diffeomorphically to
$S^{*}$.
We set $S_{j}^{*}:=g(\triangle_{m,j})$, $R_{j}^{*}:=int(S_{j}^{*})$ and
$f_{j}^{*}:=g\circ h^{*}\circ g^{-1}\mid_{S^{*}_{j}}$. Also, we take
$S_{i}^{*}:=g(\triangle_{m,i})$, $R_{i}^{*}:=int(S_{i}^{*})$ and
$f_{i}^{*}:=g\circ h_{i}\circ g^{-1}\mid_{S_{i}^{*}}$, for $i=1,\ldots,l$,
with $i\neq j$.
Clearly, this subdivision is not barycentric. Other elements of $\mathcal{T}$
admit barycentric subdivisions with the affine transformations which map them
to the whole of standard $m$-simplex, analogous to the previous example. It is
easy to see that this partition is also Markov and satisfies the conditions
$(A0)$, $(A1)$, $(A2)$ and $(A3)$ with $q=1$.

## Acknowledgments

We are grateful to M. Nassiri, A. Fakhari and M.Saleh for many useful
conversations and valuable suggestions.

## References

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UIshort = UI, long = user interface GUIshort = GUI, long = graphical user
interface TLXshort = NASA TLX, long = NASA-Task Load Index RTLXshort = NASA
Raw-TLX, long = NASA Raw-Task Load Index ERshort = ER, long = error rate
TCTshort = TCT, long = task completion time HCIshort = HCI, long = human-
computer interaction UXshort = UX, long = user experience RMSEshort = RMSE,
long = root mean squared error HMDshort = HMD, long = Head-Mounted Display
CNNshort = CNN, long = Convolutional Neural Network FOVshort = FoV, long =
field of view HRCshort = HRC, long = Human-Robot Collaboration HRIshort = HRI,
long = Human-Robot Interaction ANOVAshort = ANOVA, long = analysis of variance
RMANOVAshort = RM-ANOVA, long = Repeated Measures Analysis of Variance
JNDshort = JND, long =just-noticeable difference SUSshort = SUS, long =system
usability scale CSCWshort = CSCW, long = computer-supported cooperative work
CADshort = CAD, long = computer-aided design MRshort = MR, long = Mixed
Reality ARshort = AR, long = Augmented Reality AVshort = AV, long = Augmented
Virtuality VRshort = VR, long = Virtual Reality SARshort = SAR, long = Spatial
Augmented Reality ADLsshort = ADLs, long = Activities of Daily Living LEDshort
= LED, long = Light-Emitting Diode DoFshort = DoF, long = Degree-of-Freedom,
long-plural-form = Degrees-of-Freedom HHCshort = HHC, long = Human-Human
Collaboration AIshort = AI, long = Artifical Intelligence QUEADshort = QUEAD,
long = Questionnaire for the Evaluation of Physical Assistive Devices TiAshort
= TiA, long = Trust in Automation Questionnaire ADMCshort = ADMC, long =
Adaptive DoF Mapping Control, long-plural-form = Adaptive DoF Mapping Controls

# In Time and Space: Towards Usable Adaptive Control for Assistive Robotic
Arms

Max Pascher1,2 and Kirill Kronhardt1 and Felix Ferdinand Goldau3 and Udo
Frese3 and Jens Gerken1 1Max Pascher, Kirill Kronhardt, and Jens Gerken are
with the Westphalian University of Applied Sciences, Human-Computer
Interaction, 45897 Gelsenkirchen, Germany<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>Pascher is also with the
University of Duisburg-Essen, Human-Computer Interaction, 45127 Essen, Germany
<EMAIL_ADDRESS>Ferdinand Goldau and Udo Frese are with the
University of Bremen, Mathematics & Computer Science, 28359 Bremen, Germany
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Robotic solutions, in particular robotic arms, are becoming more frequently
deployed for close collaboration with humans, for example in manufacturing or
domestic care environments. These robotic arms require the user to control
several DoF to perform tasks, primarily involving grasping and manipulating
objects. Standard input devices predominantly have two DoFs, requiring time-
consuming and cognitively demanding mode switches to select individual DoFs.
Contemporary ADMC have shown to decrease the necessary number of mode switches
but were up to now not able to significantly reduce the perceived workload.
Users still bear the mental workload of incorporating abstract mode switching
into their workflow. We address this by providing feed-forward multimodal
feedback using updated recommendations of ADMC, allowing users to visually
compare the current and the suggested mapping in real-time. We contrast the
effectiveness of two new approaches that a) _continuously_ recommend updated
DoF combinations or b) use discrete _thresholds_ between current robot
movements and new recommendations. Both are compared in a VR in-person study
against a _classic_ control method. Significant results for lowered task
completion time, fewer mode switches, and reduced perceived workload
conclusively establish that in combination with feedforward, ADMC methods can
indeed outperform classic mode switching. A lack of apparent quantitative
differences between _Continuous_ and _Threshold_ reveals the importance of
user-centered customization options. Including these implications in the
development process will improve usability, which is essential for
successfully implementing robotic technologies with high user acceptance.

##  I Introduction

While robotic devices have long been put behind fences for safety reasons,
advances in the development of such (semi-) autonomous technologies have
started to permeate almost all aspects of our personal and professional lives.
These include increased close-quarter collaborations with robotic devices –
from industry assembly lines [1] to mobility aides [2]. Assistive robotic arms
are a particularly useful and versatile subset of collaborative technologies
with varied applications in different fields, e.g., [3, 4].

Yet, new challenges arise when robots are tasked with (semi-) autonomous
actions, resulting in additional stress for end-users if not correctly
addressed during the design process [5]. Pollak et al. highlight the decreased
feeling of control users experienced when using a robot’s autonomous mode.
Switching to manual mode allowed their study participants to regain control
and decrease stress significantly. These findings are corroborated by Kim et
al. whose comparative study of control methods resulted in markedly higher
user satisfaction for the manual mode cohort [6].

A proposed solution from previous work [7] to these challenge are adaptive
controls – referred as Adaptive DoF Mapping Controls (ADMCs) – which merge the
advantages of (semi-) autonomous actions with the flexibility of manual
controls. They combine multiple DoF dynamically for a specific scenario to
assist in controlling the robot. In our concept, a CNN interprets a camera’s
video feed of the environment and dynamically combines the most likely DoF for
a suggested movement. Building on this, we already showed that such ADMC
combinations of the robot’s DoF can lead to a significantly lower number of
mode switches compared to standard control methods [8]. However, our study
could not show that this may also improve task completion time or reduce
cognitive load. Also, challenges concerning the understanding of DoF mappings
were raised during the study.

Based on these previous findings, the present study evaluates two novel ADMC
methods for an assistive robotic arm. We compare the variants _Continuous_ and
_Threshold_ , differing in the time at which suggestions are communicated to
the user, against a _classic_ control method. In detail, we examine possible
effects on task completion time, number of necessary mode switches, perceived
workload, and subjective user experience. Our contribution is two-fold:

  1. 1.

We demonstrate that both ADMC methods significantly reduce the task completion
time, the average number of mode switches, and the perceived workload of the
user.

  2. 2.

Further, we establish that for _Continuous_ and _Threshold_ , each has
specific advantages which some users may prefer over the other, raising the
need for customizable configurations.

##  II Related Work

Collaborative robotic solutions have received much attention in recent years.
Previous work has generally focused on (a) different designs of robot motion
intent and most recently (b) ADMC for robots. The latter requires a critical
yet seldom addressed topic in how collaborative robots can effectively
communicate recommended movement directions to their user.

###  II-A Robot Motion Intent

Advance knowledge of the intended robot behavior and subsequent movements
within the physical world are critical for effective collaboration when humans
and robots occupy the same space and need to coordinate their actions [9]. In
previous work, we analyzed existing techniques of communicating robot motion
intent and identified different _intent types_ as well as several intent
properties, such as _location_ and _information_ or the placement of the
technology [10]. Users generally prefer to have the robot’s future movements
represented visually [11]. To convey detailed robot motion intent, researchers
often rely on AR [12, 13, 14], as “with the help of AR, interaction can become
more intuitive and natural to humans” [15].

Effective communication of robot motion intent is particularly relevant when
using ADMC for assistive robotic arms, as in such a shared or traded control
environment each interaction needs to be precisely coordinated.

###  II-B Adaptive DoF Mapping Controls

Traditionally, robot control methods include individual commands for each DoF,
requiring frequent mode switches for controlling translations, rotations, and
gripper functionality. Herlant et al. called into question the suitability of
these standard control methods as task completion time markedly increases by
using user-initiated compared to time-optimal mode switches [16].

To tackle this issue, we proposed in previous work the concept of ADMC – a
dynamic combination of multiple DoF, thus adjusted to specific scenarios or
tasks [7]. This streamlining decreases the need for constant mode switching,
resulting in faster and more efficient task fulfillment. In [7] we implemented
a CNN as control unit to provide these dynamic DoF mappings and gave the user
a triggering mechanism to request an update. In a 2D simulation study which
had a 4-DoF robot control mapped to a 2-DoF input device, we found promising
results.

We then extended this approach into a 3D VR simulation, thereby mapping a
7-DoF robot control to a 2-DoF input device [8]. We evaluated two ADMC methods
– differing in their respective movement suggestion concept – against the
baseline control method _Classic_. Simulating the effect of a CNN, our work
relied on a task-specific script to provide DoF mappings based on the relative
position and orientation between gripper and target. This removed the
potentially confounding effect of a suboptimal CNN implementation. Results
showed that the number of mode switches was significantly reduced compared to
_Classic_ , but task completion time was unaffected. Users reported high
cognitive demand and difficulties understanding the mapping to 2 different
input DoF. In addition, the system felt difficult to predict and required
trial and error [8].

##  III Adaptive DoF Mapping Controls

Building on our previous work [8], we created a VR simulation of a HRI
experimental setup to compare different ADMC methods to a non-adaptive
baseline condition _Classic_. Like in previous work [8] we applied a task-
specific script to explore our ADMC methods. We tackle previous issues by 1)
visualizing not only the current but also the forthcoming DoF mapping
suggestion (improving predictability) and 2) reducing the input to a single
DoF (reducing cognitive demand). We propose two approaches as different trade-
offs between control fidelity and cognitive demand.

The VR simulation includes a virtual model of the _Kinova Jaco 2_ 111Kinova
Robotic arm. https://assistive.kinovarobotics.com/product/jaco-robotic-arm,
last retrieved August 28, 2024. – a commercially available assistive robotic
arm frequently used in HRI studies, e.g., [4, 16]. Our proposed visual
feedback mimics AR, with directional cues registered in 3D space. This allows
the user to understand different movement directions for the actual control
and the suggested DoF combinations. To simplify understanding, we use _arrows_
, a straightforward and common visualization technique to communicate motion
intent [9, 17, 18].

As a control method for the ADMC, we implemented a task-specific script. This
removed any potential bias that a more generic but currently still technically
limited approach such as a CNN-based control method may introduce. Of course,
our approach only works in a controlled experimental setting. The task-
specific script evaluates the gripper’s current position, rotation, and finger
position relative to a target. The DoF mapping system then suggests five
different movement options (referred in the following to as _modes_) – in
order of assumed usefulness – to the user.

  1. 1.

_Optimal Suggestion:_ Combining translation, rotation, and finger movement
[opening and closing] into one suggestion, causing the gripper to move towards
the target, pick it up, or release it on the intended surface.

  2. 2.

An orthogonal suggestion based on (1) but excluding the finger movement.
Allows the users to adjust the gripper’s position while still being correctly
orientated.

  3. 3.

A pure translation towards the next target, disregarding any rotation.

  4. 4.

A pure rotation towards the next target without moving the gripper.

  5. 5.

Opening or closing of the gripper’s fingers.

During movement, the ADMC system re-calculates the best DoF combinations to
fulfill the specific task, which are then presented as new suggestions. Users
cycle through these modes – by pressing a button on the input device – to
select a suitable one or continue moving with the previous active suggestion
(see Figure 1). A suggestion indicator is visible above the gripper when users
are not moving the robot to distinguish between the modes. Five slanted cubes
represent the possible suggestions. The cubes appear gray if no suggestion is
active and turn blue to indicate that a new suggestion is selected. The cube
corresponding to the selected mode increases in size. In contrast to our
previous work [8] and to the dual axis system of the baseline control method
(see Figure 2), only one input axis is required to control the robotic arm.
Consequently, the cognitive demand on the users is reduced as they can focus
on evaluating one movement rather than two simultaneous suggestions.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 1: Suggestions as visualized in the ADMC methods, (a) Continue previous
movement, (b) Optimal Suggestion, (c) Adjustment Suggestion, (d) Pure
Translation, (e) Pure Rotation, (f) Open / Close Fingers.

Continuous: This control method uses continuous feedback of robot motion
intent to increase oversight of updated movement suggestions. Continuous
feedback enables users to move in a direction and constantly evaluate the
updated optimal suggestion by the ADMC system. If found fitting, users can
switch to a new suggestion and move the robot in the updated path to fulfill
the task. Here, two directional indicators are virtually attached to the
robotic arm’s gripper: a light blue and a dark blue arrow. The former
represents the currently selected movement option (mode) mapped to the input
axis. The forward movement of the input axis moves the gripper in the
direction the arrow is pointing; engaging it backward moves the gripper in the
arrow’s reverse direction. The dark blue arrow represents the currently
optimal suggestion at a given time. Users can only move the robot along the
dark blue arrow if they switch to that suggestion first – which causes both
arrows to overlap. While this approach increases transparency, users might be
distracted by the constantly updating suggestions, potentially leading to more
mode switches and perceived workload.

Threshold: In contrast to _Continuous_ , _Threshold_ uses time-discrete and
multimodal feedback to indicate optimized movement suggestions. Again, a light
blue arrow maps the selected movement option (mode) to the input axis. New
suggestions are only shown to the users if the optimal mode differs – by a set
degree – from the current movement. We followed Singhal et al. and used a
cosine between-vector similarity measure to calculate this threshold [19],
ranging from exact alignment [0%] to total opposite direction [100%]. In
pretests, we determined a 20% difference between the current and optimal
vector as the suggestion threshold. If exceeded, a short vibration pulse to
the input device and a 1kHz sound inform the users of an updated suggestion.
In addition, a dark blue arrow appears which visualizes the new suggested
movement. Users can continue the active movement, switch to the new
suggestion, or cycle through the other four modes before deciding on one.
Unlike with _Continuous_ , users can therefore entirely focus on the movement
they are currently performing until explicitly notified and directed to a new
suggestion. We expect _Threshold_ to reduce perceived workload compared to
_Continuous_ as it does not require constant evaluation of the visual
feedback. However, we expect task completion time to increase, as _Threshold_
systematically interrupts the users’ workflow. Additionally, _Threshold_ might
result in a perceived loss of control, potentially negatively influencing
usability.

##  IV Study Method and Materials

To explore the effectiveness of our ADMC methods, we conducted a supervised,
controlled experiment as a VR simulation study with 24 participants. We
compared our ADMC methods to _Classic_ , which relies on mode switching to
access and control all DoF one after another. Approaches as _Classic_ are well
established (e.g., when driving a car) and are predictable and transparent for
the user. Comparing ADMC methods to _Classic_ allows HRI researchers to
disentangle their respective advantages and disadvantages.

###  IV-A Study Design

We applied a within-participant design with _control method_ as an independent
variable with three conditions: (1) _Classic_ , (2) _Continuous_ , and (3)
_Threshold_. Every participant performed eight training trials and 24 measured
trials per condition, resulting in 72 measured and 24 training trials per
participant and 1,728 measured trials in total. To counter learning and
fatigue effects, the order of conditions was fully counter-balanced. We
measured the following dependent variables:

  1. 1.

Average Task Completion Time For each trial, we measured the time in seconds
needed to pick an object and place it on the target surface.

  2. 2.

Average Number of Mode Switches For each trial, we recorded every mode switch
conducted by pressing a button on the input device.

  3. 3.

Perceived Workload After completing each condition, we measured cognitive
workload with the RTLX questionnaire [20].

  4. 4.

Subjective Assessment After completing each condition, we measured the five
dimensions of the QUEAD [21]. After completing all trials, participants were
further asked to rank the three conditions.

After each condition, participants were prompted with several open questions
regarding their experience, their understanding of the control methods and the
directional cues, plus any issue of interest they considered noteworthy.
Additionally, participants were asked how they proceeded in situations when
they could not solve the task at first.

Video and audio recordings of the interviews with the entire study cohort were
assessed independently by two researchers. Open coding was applied to gather
participants’ opinions of the different control methods. We used Miro222Miro.
https://miro.com, last retrieved August 28, 2024. – an online whiteboard [22]
– to complete an affinity diagram of the open codes. Codes were then organized
into themes (see Section V-F).

###  IV-B Hypotheses

Overall, we expected ADMC methods to reduce not just mode switches (as in
prior work [8]) but – due to the advances in our designs – also improve on
task completion time and workload.

  1. H1:

_Continuous_ and _Threshold_ lead to a lower task completion time compared to
_Classic_. However, we expect _Continuous_ to perform faster compared to
_Threshold_ , as the latter systematically interrupts the user during
interaction.

  2. H2:

_Continuous_ and _Threshold_ result in fewer mode switches compared to
_Classic_. We expect _Continuous_ to require more mode switches than
_Threshold_ , as users have no clear guidance about when to switch modes. This
may cause them to oversteer or accept new suggestions inefficiently.

  3. H3:

_Continuous_ and _Threshold_ cause lower perceived workload compared to
_Classic_. However, we expect _Continuous_ to cause a higher workload compared
to _Threshold_ , as it requires constant evaluation of the visual feedback
while _Threshold_ allows the user to relax until further notification.

###  IV-C Apparatus

Developing and testing new concepts for a robotic arm involves inherent
challenges associated with a real robot’s physical bulk and complexity.
Quickly changing the experimental setup, adding feedback components, or
providing information to the user further complicate testing regimes. We
created a 3D testbed environment for HRI studies in VR to address these
challenges. This testbed contains a simulated robotic arm (a virtual model of
the _Kinova Jaco 2_) with multiple control mechanisms and a standardized pick-
and-place task. Visual feedback mimics AR, with directional cues registered in
3D space. A _Meta Quest_ motion controller is used as an input device to
control the robotic arm.

Photogrammetry scans of an actual room were used to design the VR environment,
which was created using the _Unreal Engine 4.27_ and optimized for usage with
a _Meta Quest_ VR HMD (see Figure 2). During the study, user behavior was
recorded with appropriate software on a _Schenker XMG Key 17_ laptop with
_Windows 10 64-bit_ and _Oculus Link_ connected to the VR headset.

For our implementation of the baseline control method _Classic_ , users cycled
through four distinct modes to access all seven robot DoF, as they are mapped
on a two-DoF joystick, such as the control-stick on a _Meta Quest_ motion
controller:

  1. 1.

X-Translation + Y-Translation

  2. 2.

Z-Translation + Roll

  3. 3.

Yaw + Pitch

  4. 4.

Open/Close fingers

We illustrate the current mapping between the robot’s DoF and the input device
through two arrows attached to the gripper. Light blue arrows indicate the
robot’s DoF assigned to the first, dark blue arrows to the second input axis.
Looking at the joystick in VR, the same color-coded visualization is applied.

Users press a button on the input device – the A-Button of the _Meta Quest_
motion controller – to switch between modes, cycling back to the first one at
the end. Four blue spheres – in contrast to the slanted cubes used in our ADMC
methods – above the robotic arm’s gripper indicate the total number of
available and the currently active mode when users are not moving the robot.
The sphere representing the active mode is bigger and brighter than the
spheres of inactive modes.

Figure 2: Virtual environment consisting of (left to right): a virtual canvas,
the motion controllers, a table with the blue object and red target, and a
_Kinova JACO_ with an arrow-based visualization

###  IV-D Participants

A total of 24 participants took part in our study (7 female, 17 male). The
participants were aged 19 to 37, with a mean age of 26 years (SD $=$ 4.85
years). No one declared any motor impairments that might influence reaction
times. Five participants had prior experience with controlling a robotic arm.
Participants were recruited from a university campus and an online appointment
form.

###  IV-E Procedure

Utilizing the benefits of a standardized and portable VR simulation
environment, the study was conducted in multiple comparable physical
localities. Before commencing, participants were fully informed about the
project objective and the various tasks they had to complete. Every
participant gave their full and informed consent to partake in the study, have
video and audio recordings taken, and have all the relevant data documented.

A study administrator observed the experiment on a laptop and briefed
participants on using the hardware as well as the general functionalities of
the study environment. Once set up, users followed command prompts embedded in
the virtual simulation environment. For each of the three conditions, the
following steps were performed:

  1. 1.

Participants were given a written and standardized explanation of the control
method used in the current condition.

  2. 2.

Participants conducted eight training trials for familiarization with the
respective control method.

  3. 3.

Participants then conducted 24 measured trials.

  4. 4.

Interview and questionnaires.

After completing all conditions, participants ranked the three control methods
from most to least preferred and explained the reasoning behind their
decision. The study concluded with a de-briefing. The average session lasted
for 90 minutes and participants were compensated with 30 EUR.

###  IV-F Experimental Task

The experimental task is based on our previous work and resembles a common
pick-and-place scenario [8]. A blue object appears on a table in front of the
participant, which signals the start of a trial. The user has to control the
robot from its starting position to pick the object and place it on a red
target surface, also located on the table. To change the DoF mapping – for
trial fulfillment – users could switch modes. Upon completion, the blue object
disappears, and the robot automatically returns to the original starting
position. A new blue object appears when this position is reached, and a new
trial commences. For each trial, the position of the blue object is placed in
one of eight possible locations spaced evenly around the red target surface.
Each position occurred once during training and thrice during measured trials.
However, the order of appearance was randomized. We used a neutral block shape
rather than specific objects to avoid bias and ensure trial comparability.

##  V Results

The study comprises 1,728 (24 participants $\times$ 3 control methods $\times$
24 trials) measured trials. Training trials were excluded from the analysis.

We explored the distribution of the data through QQ-plots and either applied
parametric RMANOVA or non-parametric Friedman tests. For the latter, post-hoc
pairwise comparisons using Wilcoxon signed-rank test with Bonferroni
correction followed the omnibus test. Relevant effect sizes were calculated
with r: $>$0.1 small, $>$0.3 medium, and $>$0.5 large effect.

###  V-A Task Completion Time

Mean task completion time calculated per participant and control method (see
Fig. 3) resulted in _Threshold_ $=$ 16.54s (SD $=$ 4.09s); _Continuous_ $=$
16.61s (SD $=$ 4.77s); and _Classic_ $=$ 30.96s (SD $=$ 4.89s). Outliers [N
$=$ 3] with average times $\geq$ 2.2 $*$ IQR of the mean task completion time
in at least one control method were excluded [23]. The QQ-plot of the
remaining 21 participants followed a normal distribution.

Figure 3: Raincloud Plots for Average Task Completion Time and Mode Switches

A RMANOVA found a significant main effect (F(2, 36) = 130.92, p $\leq$0.001).
A post-hoc pairwise comparison (Bonferroni corrected) showed a significant
difference between _Continuous_ and _Classic_ (p $\leq$0.001) as well as
between _Threshold_ and _Classic_ (p $\leq$0.001). No significant difference
was found between _Continuous_ and _Threshold_ (p $\geq$0.999).

###  V-B Mode Switches

We used a non-parametric Friedman test, as our data was not normally
distributed, to determine differences between the average number of necessary
mode switches between control methods. Two outliers – based on $\geq$ 2.2 $*$
IQR of the mean value – were excluded prior to further analysis. This resulted
in mean numbers of mode switches for _Threshold_ $=$ 9.28 (SD $=$ 1.26);
_Continuous_ $=$ 9.93 (SD $=$ 1.47); and _Classic_ $=$ 19.55 (SD $=$ 2.93) for
N $=$ 22\. We found a significant main effect ($\chi^{2}$(2) $=$ 33.82, p
$\leq$0.001, N $=$ 22). Post-hoc pairwise comparisons showed a significant
difference between _Continuous_ and _Classic_ (Z $=$ $-$4.11, p $\leq$0.001, r
$=$ 0.62) as well as _Threshold_ and _Classic_ (Z $=$ $-$4.11, p $\leq$0.001,
r $=$ 0.62). Again, we found no significant difference between the two ADMC
methods (Z $=$ $-$1.51, p $=$ 0.131, r $=$ 0.28) (see Fig. 3).

###  V-C Perceived Workload

RTLX [20] scores [scale from 1 to 100] for all participants resulted in mean
task load values of _Threshold_ $=$ 22.67 (SD $=$ 13.86); _Continuous_ $=$
23.23 (SD $=$ 13.26); and _Classic_ $=$ 34.24 (SD $=$ 14.65). We applied a
Friedman test which revealed a significant main effect for perceived task
load: ($\chi^{2}$(2) $=$ 9.87, p $=$ 0.007, N $=$ 24). Post-hoc pairwise
comparisons show significant differences between _Continuous_ and _Classic_ (Z
$=$ $-$3.03, p $=$ 0.002, r $=$ 0.44), _Threshold_ and _Classic_ (Z $=$
$-$2.76, p $=$ 0.006, r $=$ 0.40), but not between _Continuous_ and
_Threshold_ (Z $=$ $-$0.21, p $=$ 0.830, r $=$ 0.03).

###  V-D Evaluation of Physical Assistive Devices

The QUEAD encompasses five individual scales (3 to 9 items each, 7-point
Likert). Friedman tests for individual dimensions revealed significant main
effects for _Perceived Usefulness (PU)_ , _Perceived Ease of Use (PEU)_ ,
_Emotions (E)_ , and _Comfort (C)_ , but not for _Attitude (A)_. Post-hoc
pairwise comparisons indicate significant differences between _Continuous_ and
_Classic_ for _PU_ , _PEU_ , and _C_ as well as between _Threshold_ and
_Classic_ for _PU_ and _PEU_ (refer to Table I for detailed scores).

Table I: Statistics for individual QUEAD dimensions: Perceived Usefulness (PU), Perceived Ease of Use (PEU), Emotions (E), Attitude (A), and Comfort (C).  | PU | PEU | E | A | C
---|---|---|---|---|---
Descriptive Statistics
$M_{\,Classic}$ | 4.98 | 4.87 | 5.00 | 4.81 | 5.65
$SD_{\,Classic}$ | 1.39 | 1.20 | 1.71 | 1.75 | 1.71
$M_{\,Continuous}$ | 5.68 | 5.80 | 5.90 | 5.42 | 6.44
$SD_{\,Continuous}$ | 1.05 | 1.04 | 1.25 | 1.48 | 0.78
$M_{\,Threshold}$ | 5.77 | 5.90 | 5.68 | 5.44 | 6.13
$SD_{\,Threshold}$ | 1.02 | 0.97 | 1.43 | 1.58 | 1.14
Friedman Tests
$\chi^{2}$(2) | 7.49 | 15.22 | 7.20 | 1.76 | 6.39
$p$ | 0.022 |  $\leq$0.001 | 0.026 | 0.422 | 0.040
$N$ | 24 | 24 | 24 | 24 | 24
Pairwise Comparisons
Classic vs. Continuous |  |  |
$|Z|$ | 2.32 | 2.47 | 1.85 | — | 2.29
$p$ | 0.021 | 0.014 | 0.064 | — | 0.022
$r$ | 0.33 | 0.36 | 0.27 | — | 0.33
Classic vs. Threshold |  |  |
$|Z|$ | 2.68 | 2.90 | 1.28 | — | 1.23
$p$ | 0.007 | 0.003 | 0.202 | — | 0.220
$r$ | 0.39 | 0.43 | 0.18 | — | 0.18
Continuous vs. Threshold |  |  |
$|Z|$ | 0.62 | 0.38 | 1.03 | — | 1.70
$p$ | 0.538 | 0.706 | 0.302 | — | 0.089
$r$ | 0.09 | 0.05 | 0.15 | — | 0.25

###  V-E Individual Ranking

Participants ranked the control methods in order of preference from 1 =
_favorite_ to 3 = _least favorite_. Mean values in ascending order are
_Continuous_ $=$ 1.67; _Threshold_ $=$ 2.04; and _Classic_ $=$ 2.29. A
Friedman test revealed no significant main effect ($\chi^{2}$(2) $=$ 4.75, p
$=$ 0.100, N $=$ 24).

###  V-F Qualitative Insights

Overall, the open coding process led to the identification of five main
themes, as discussed below.

####  V-F1 Familiarization

While all three control methods included a training phase, comments suggest
that in particular the ADMC methods required familiarization. Here,
participants felt the controls were sometimes “inverted” (P3) and wanted to
“move the stick in the direction the arrow was pointing at” (P6). They also
reported that “it takes a while to get used to” (P24), but “routine set in
fast” (P18).

####  V-F2 Handling Adaptive DoF Mapping Suggestions

The study cohort showed a relatively uniform response to the two ADMC methods
with clear distinctions between _Threshold_ and _Continuous_. In _Threshold_ ,
many participants “trusted the system” (P23) and switched to the new
suggestion as soon as they perceived the multimodal indicator. They “did not
have to think a lot” (P4) and “relied on what the suggestion says” (P7). This
dependence on the system caused some to “draw a blank when something went
wrong because [they] forgot they had other options” (P8). One participant even
tried using the _Threshold_ control method with eyes closed, which “worked
surprisingly well” (P7).

In contrast, participants evaluated the suggestions in _Continuous_ more
thoroughly, as they had to decide when to switch without the help of
threshold-based indicators. Some participants waited for suggestions with
relatively simple direction cues, such as “straight arrows” (P6, P16) as an
indication to switch modes, while others trusted their “gut feeling” (P23).
Uncertainties of “How do I approach this?” (P23) were more frequent in this
control method than _Threshold_. Participants dealt with problems in both ADMC
conditions in one of two ways to find alternative suggestions that better
align with their needs. They cycled through the further offered suggestions
for an alternative option or reversed their current movement direction until a
different suggestion was offered.

####  V-F3 Visualization

Overall, participants understood the different visualizations. Yet,
difficulties arose in all three conditions relating to depth perception and
understanding if the gripper is positioned correctly to pick or place the
object. Some participants suggested a “laser pointer” (P16) to indicate the
gripper’s position above the table for improved depth perception. This is a
known problem for robot teleoperation. In the past, researchers have suggested
and explored AR _Visual Cues_ to counter that, which include similar
approaches as the ones mentioned by our participants [24, 25].

Interestingly, some participants “manipulated” the second mode of _Classic_
(X- and Y-Translation) to mimic this effect, as that mode shows straight up-
and downward pointing arrows as directional cues along the y-axis.

####  V-F4 Multimodal Feedback

As described above, most participants used _Threshold_ as intended, switching
to the next suggestion when they received the multimodal feedback. However,
some participants experienced the haptic and audio indicators as “irritating”
(P20) or “weird and horrible” (P17). The poignant statement “If I had to do
this for five more minutes, it would be too annoying.” (P7) reveals some
participants’ strong reactions to this control method. As a possible
mitigation, one participant suggested implementing multiple thresholds of
varying intensity instead of a singular one that “instantly beeps loudly at me
and says ’Do this now!’” (P24).

####  V-F5 Control vs. Comfort

Participants reported substantial differences in the level of control and
comfort between _Classic_ , _Continuous_ , and _Threshold_. By nature,
_Classic_ offers the highest control level but requires participants to decide
individually on every task step. In contrast, _Threshold_ allowed participants
to perform tasks “entirely brainlessly” (P16) and only press “forward, then A,
then forward, then A” (P17). Many participants expressed that they “felt too
directed by [_Threshold_]” (P8), attesting _Continuous_ a higher level of
comfort or “freedom to experiment” (P24). Overall, participants described
_Continuous_ as a reasonable compromise or “the golden middle” (P14) between
the comfortable execution in _Threshold_ and the high level of control in
_Classic_.

##  VI Discussion

Adaptive DoF mapping controls have already been indicated to have benefits
over classic methods [7, 8]. Yet, research is still limited, and analysis of
_time-based dimensions_ of directional cues is lacking. In this paper, we
examined to what extent the two ADMC methods, _Continuous_ and _Threshold_ ,
differ from the _Classic_ baseline – and each other – in terms of task
completion time, necessary mode switches, perceived workload, and subjective
assessment.

Significant results for all four metrics partially support our initial
hypotheses. Most strikingly, ADMC methods reduced task completion time (_H1_)
and mode switches (_H2_) by 50% respectively compared to _Classic_. As
previously suggested by Kim et al., this establishes that ADMC methods lead to
faster and less involved execution of pick-and-place tasks [6]. These findings
are in line with previous work [7], underlining the benefits of ADMC compared
to _Classic_ controls.

In contrast to previous results [8], our novel ADMC methods were able to
significantly lower task completion time and perceived workload compared to
the _Classic_ method. The latter finding also partially supports _H3_. This
highlights that ADMC which communicate the suggested recommendation to the
user – irrespective of timing – were able to increase usability. Notably, the
decreased workload of ADMC is particularly meaningful as the end goal should
be the smooth integration of robotic devices into people’s lives and
workflows, not to add stress.

Turning to the second part of our analysis – contrasting different time-based
communication of feed-forward recommendations – we found no significant
differences in the four metrics between _Continuous_ and _Threshold_. The lack
of measurable differences between _Continuous_ and _Threshold_ implies that
both discrete and continuous communication of movement suggestions allows
users to use ADMC methods efficiently. Insights gained by the results of the
QUEAD and our qualitative interviews corroborate these findings, while the
latter also helped to provide a more distinguished analysis.

Overall, participants expressed a positive stance regarding the ADMC methods.
However, individual preferences vary greatly between _Continuous_ and
_Threshold_. While some participants preferred the higher level of control
_Continuous_ allowed, others favored the comfortable execution possible with
_Threshold_. Consequently, future development of ADMC methods should – in
accordance with Burkolter et al. – include individualization options to
increase comfort and end-user acceptance [26]. Customizations would be
particularly beneficial for _Threshold_ -based controls as participants
repeatedly criticized the multimodal feedback. Allowing users to adjust the
modalities, the signal intensity, and even the threshold itself may improve
usability while still offering the advantages of ADMC.

In contrast to expectations derived from our initial hypotheses, qualitative
insights revealed that the _Classic_ control method could still be a valuable
addition in specific situations. Participants felt an apparent lack of control
when the ADMC suggestions did not match their expectations. To improve
usability, ADMC methods could incorporate static suggestions for certain
situations. A potential way to address this could be combining ADMC and static
suggestions using only the most common input-DoF.

However, further experimental studies are needed to disentangle exactly which
factors shape personal preferences and how customizations or crossover methods
can deliver the best results.

###  VI-A Limitations

We explored the proposed ADMC methods in a VR simulation environment. While
the usage of virtual simulations in industrial settings has been successfully
established [27, 28, 29], future work should confirm if our promising findings
can be replicated in the real world with a physical robot.

##  VII Conclusions

Our ADMC methods _Continuous_ and _Threshold_ are promising approaches to
communicate proposed directional cues effectively. We extend our previous work
[8] by demonstrating that ADMC significantly reduce task completion time (1),
the average number of necessary mode switches (2), and the perceived workload
of the user (3). Further, we establish that _Continuous_ and _Threshold_
perform equally well in quantitative measures while qualitative insights
reveal individual preferences.

The observations of this study provide valuable implications for any HRI
researcher involved in designing novel ADMC methods for human-robot
collaborative settings. Future work should focus on disentangling quantitative
and qualitative feedback of focus groups to develop optimal robot motion
control methods, thus increasing usability, safety and – ultimately – end-user
acceptance.

## ACKNOWLEDGMENT

This research is supported by the German Federal Ministry of Education and
Research (BMBF, FKZ: 16SV8563, 16SV8565). Our study is approved by the Ethics
Committee of the Faculty of Business Administration and Economics of the
University of Duisburg-Essen with the ID: 2202007.

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