[2]Lidong Wang

# Li-Yorke chaos for maps on $G$-Spaces

Yingcui Zhao  Nan Wang

###### Abstract

We introduce the definition of Li-Yorke chaos for the map $f$ on $G$-spaces,
and show $G$-Li-Yorke chaos is iterable for $f$. Li-Yorke chaos implies
$G$-Li-Yorke chaos, while the converse is not true. Then we give a sufficient
condition for $f$ to be chaotic in the sense of $G$-Li-Yorke. Also, we prove
if $f$ is $G$-transitive and there exists a common fixed point for $f$ and all
the maps in $G$, then $f$ is chaotic in the sense of $G$-Li-Yorke.

##  1 Introduction

Chaos comes from practical problems such as physics, biology, economics,
medicine and chemistry. The theory of chaotic dynamical system is one of the
challenging issue of dynamical systems. By convention, ``chaos'' means ``a
state of confusion without order". While in the theory of chaotic dynamical
systems, ``chaos'' is a technical term. The term ``chaos'' in discrete
dynamical systems was introduced firstly by Li and Yorke in 1975 [1].

Let $X$ be a metric $G$-space and $f:X\rightarrow X$ be a continuous map.
$Y\subset X$ is said to be a Li-Yorke chaotic set of $f$, if it contains at
least two points and any pair of distinct points $x,y\in Y$ is a Li-Yorke
pair, i. e.,

$\liminf_{n\rightarrow\infty}d(f^{n}(x),f^{n}(y))=0,\
\limsup_{n\rightarrow\infty}d(f^{n}(x),f^{n}(y))>0.$

Moreover, if there exists an uncountable Li-Yorke chaotic set $S\subset X$ of
$f$, $f$ is said to be chaotic in the sense of Li-Yorke.

In the last fifty years, a large number of papers have been devoted to the
study of chaos in the sense of Li and Yorke. Xiong[2] showed if $f$ is
transitive and there exists a fixed point for $f$, then $f$ is chaotic in the
sense of Li-Yorke. Wang[6] gave a sufficient condition for $f$ to be chaotic
in the strong sense of Li-Yorke. Seeing [7, 8, 9, 10] for more results about
Li-Yorke chaos.

In addition to the topology domain, chaos theory is also being researched in
other settings. Recently, chaos is being defined and studied for group
actions, set-valued maps, iterated funtion system and so on. The present paper
focuses on $G$-space.

Ekta studied Devaney's chaos for maps on $G$-space in [11]. Ali and Ekta
defined and studied the notion of chain transitivity for maps on G-spaces in
[12]. Phinao and Khundrakpam investigated the $G$-mixing, $G$-sensitive and
$G$-shadowing property of $f$ in [13]. Ji Zhanjiang introduced the concepts of
$G$-chain mixing, $G$-mixing, and $G$-chain transitivity in metric $G$-space,
and then researched them in [14]. Raad and Iftichar studied the sequence
$G$-asymptotic average shadowing property with $G$-chain transitive in [15].

The aim of present paper is to define and study Li-Yorke chaos for maps on
$G$-spaces. The specific layout of the present paper is as follows. Some
preliminaries and definitions are introduced in the next section. Then we will
give some propositions in Sec. 3. In the last section, the main conclusions
are given.

##  2 Preliminaries and Basic Concepts

Throughout this paper, $\mathbf{N}=\\{0,1,2,\cdots\\}$,
$\mathbf{Z^{+}}=\\{1,2,3,\cdots\\}$ and
$\mathbf{Z}=\\{\cdots,-2,-1,0,1,2,\cdots\\}$. Let $(X,d)$ be a metric
$G$-space, $G$ be a topological group and $\varphi$ be a continuous map from
$G\times X$ to $X$[4]. The $(X,G,\varphi)$ or $X$ is called to be a metric
$G$-space if the following conditions are satisfied:

  * (1)

$\varphi(e,x)=x$ for all $x\in X$ where $e$ is the identity of $G$;

  * (2)

$\varphi(g_{1},\varphi(g_{2},x))=\varphi(g_{1}g_{2},x)$ for all $x\in X$ and
all $g_{1},g_{2}\in G$.

If $X$ is compact, the $X$ is also said to be compact metric $G$-space. For
the convenience of writing $\varphi(g,x)$ is usually abbreviated as $gx$. For
$x\in X$, the $G-orbit$ of $x$, denoted by $G_{f}(x)$, is given as the set
$\\{gf^{k}(x)|g\in G,k\geq 0\\}$.

The notion of equivariant map was introduced in [3]. We recall the definition.

######  Definition 2.1.

Let $(X,d)$ be a metric $G$-space and $f$ be a continuous map from $X$ to $X$.
We say the map $f$ is an equivariant map if $f(gx)=gf(x)$ for all $x\in X$ and
all $g\in G$.

In [16], the notion of topological transitivity on $G$-space was defined and
studied. We recall the definition.

######  Definition 2.2.

Let $(X,d)$ be a metric $G$-space and $f$ be a continuous map from $X$ to $X$.
$f$ is called $G$-transitive if for any non-empty open subsets $U$ and $V$ of
$X$, there exist $n\in\mathbf{Z^{+}}$ and $g\in G$ such that $gf^{n}(U)\bigcap
V\neq\emptyset$.

Naturally, we give the notion of $G$-transitive point of $f$ as follows.

######  Definition 2.3.

Let $(X,d)$ be a metric $G$-space and $f$ be a continuous map from $X$ to $X$.
$x\in X$ is called a $G$-transitive point of $f$ if $\\{gf^{k}(x)|g\in
G,k\in\mathbf{N}\\}$ is dense in $X$.

Then we define the notion of $G$-recurrent point and $G$-Li-Yorke chaos for
maps on $G$-spaces.

######  Definition 2.4.

Let $(X,d)$ be a metric $G$-space and $f$ be a continuous map from $X$ to $X$.
A point $x\in X$ is called a $G$-recurrent point of $f$ if there exist
sequence $\\{n_{i}\\}_{i=1}^{\infty}$ and $\\{g_{i}\\}_{i=1}^{\infty}\subset
G$ such that $\lim_{i\rightarrow\infty}g_{i}f^{n_{i}}(x)=x.$ The set of all
$G$-recurrent points of $f$ is denoted by $R_{G}(f).$

######  Definition 2.5.

Let $(X,d)$ be a metric $G$-space and $f$ be a continuous map from $X$ to $X$.
$Y\subset X$ is called to be a $G$-Li-Yorke chaotic set of $f$, if it contains
at least two points and any pair of distinct points $x,y\in Y$ is a Li-Yorke
pair, i. e., there exists $\\{g_{n}\\}_{n=1}^{\infty}\subset G$ such that

$\liminf_{n\rightarrow\infty}d(g_{n}f^{n}(x),g_{n}f^{n}(y))=0,\
\limsup_{n\rightarrow\infty}d(g_{n}f^{n}(x),g_{n}f^{n}(y))>0.$

Moreover, if there exists an uncountable $G$-Li-Yorke chaotic set $S\subset X$
of $f$, $f$ is said to be chaotic in the sense of $G$-Li-Yorke.

##  3 Propositions

In this section, we show some lemmas and propositions. Before this, we give a
well known lemma.

######  Lemma 3.1.

Let $\\{n_{i}\\}$ be a sequence of positive integers. For any $n>0$, there
exist $r$, $0\leq r<n$, a subsequence $\\{n_{i_{j}}\\}$ of $\\{n_{i}\\}$ and a
sequence $\\{q_{j}\\}$ of non-negative integers such that
$n_{i_{j}}=nq_{j}+r$.

######  Proposition 3.2.

Let $(X,d)$ be a metric $G$-space. Suppose $f$ is an equivariant map. $f$ is
chaotic in the sense of $G$-Li-Yorke if and only if for any
$N\in\mathbf{Z^{+}}$, $f^{N}$ is chaotic in the sense of $G$-Li-Yorke.

  * Proof.

Let $N=1$, sufficiency is obvious. Next we show the necessity.

Let $N\in\mathbf{Z^{+}}$. Suppose $f$ is chaotic in the sense of $G$-Li-Yorke.
Then there exists an uncountable $G$-Li-Yorke chaotic set $S\subset X$ of $f$.
Set distinct points $x,y\in S$, there exists
$\\{g_{n}\\}_{n=1}^{\infty}\subset G$ such that

$\liminf_{n\rightarrow\infty}d(g_{n}f^{n}(x),g_{n}f^{n}(y))=0,\
\limsup_{n\rightarrow\infty}d(g_{n}f^{n}(x),g_{n}f^{n}(y))>0.$

    * (1)

By $\liminf_{n\rightarrow\infty}d(g_{n}f^{n}(x),g_{n}f^{n}(y))=0$, there exist
$x_{0}\in X$ and infinite sequence $\\{n_{i}\\}$ such that
$\lim_{n_{i}\rightarrow\infty}g_{n_{i}}f^{n_{i}}(x)=x_{0}$ and
$\lim_{n_{i}\rightarrow\infty}g_{n_{i}}f^{n_{i}}(y)=x_{0}$. By Lemma 3.1,
there exist $r,0\leq r<N$ and sequence $\\{q_{i}\\}$ such that

$\lim_{{q_{i}}\rightarrow\infty}g_{Nq_{i}+r}(f^{N})^{q_{i}}(f^{r}(x))=x_{0},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \text{and}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\lim_{{q_{i}}\rightarrow\infty}g_{Nq_{i}+r}(f^{N})^{q_{i}}(f^{r}(y))=x_{0}.$

Let $l=N-r$. Since $f$ is an equivariant map, as $i\rightarrow\infty$,

$f^{l}(g_{Nq_{i}+r}(f^{N})^{q_{i}}(f^{r}(x)))=g_{Nq_{i}+r}(f^{N})^{q_{i}+1}(x)\rightarrow
f^{l}(x_{0}),$
$f^{l}(g_{Nq_{i}+r}(f^{N})^{q_{i}}(f^{r}(y)))=g_{Nq_{i}+r}(f^{N})^{q_{i}+1}(y)\rightarrow
f^{l}(x_{0}).$

    * (2)

By $\limsup_{n\rightarrow\infty}d(g_{n}f^{n}(x),g_{n}f^{n}(y))>0$, there exist
distinct points $a,b\in X$ and infinite sequence $\\{n_{i}\\}$ such that
$\lim_{n_{i}\rightarrow\infty}g_{n_{i}}f^{n_{i}}(x)=a$ and
$\lim_{n_{i}\rightarrow\infty}g_{n_{i}}f^{n_{i}}(y)=b$. By Lemma 3.1, there
exist $r,0\leq r<N$ and sequence $\\{q_{i}\\}$ such that

$\lim_{{q_{i}}\rightarrow\infty}g_{Nq_{i}+r}(f^{N})^{q_{i}}(f^{r}(x))=a,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \text{and}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\lim_{{q_{i}}\rightarrow\infty}g_{Nq_{i}+r}(f^{N})^{q_{i}}(f^{r}(y))=b.$

There exist a subsequence $\\{q_{i_{j}}\\}$ of $\\{q_{i}\\}$ and
$a_{1},b_{1}\in X$ such that

$\lim\limits_{j\rightarrow\infty}g_{Nq_{i_{j}}}(f^{N})^{q_{i_{j}}}(x)=a_{1},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \text{and}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\lim\limits_{j\rightarrow\infty}g_{Nq_{i_{j}}}(f^{N})^{q_{i_{j}}}(y)=b_{1}.$

Since $f$ is an equivariant map, as $j\rightarrow\infty$,

$f^{r}g_{Nq_{i_{j}}}(f^{N})^{q_{i_{j}}}(x)=g_{Nq_{i_{j}}}(f^{N})^{q_{i_{j}}}(f^{r}(x))\rightarrow
f^{r}(a_{1})=a,$
$f^{r}g_{Nq_{i_{j}}}(f^{N})^{q_{i_{j}}}(y)=g_{Nq_{i_{j}}}(f^{N})^{q_{i_{j}}}(f^{r}(y))\rightarrow
f^{r}(b_{1})=b.$

Then $a_{1}\neq b_{1}$.

By $(1)$ and $(2)$, there exists $\\{h_{n}\\}_{n=1}^{\infty}\subset G$ such
that

$\liminf_{n\rightarrow\infty}d(h_{n}f^{n}(x),h_{n}f^{n}(y))=0,\
\limsup_{n\rightarrow\infty}d(h_{n}f^{n}(x),h_{n}f^{n}(y))>0.$

So, $(x,y)$ is a $G$-Li-Yorke pair of $f^{N}$. Then, $S$ is an uncountable
$G$-Li-Yorke chaotic set of $f^{N}$. Consequently, $f^{N}$ is chaotic in the
sense of $G$-Li-Yorke. ∎

The following example shows the condition "$f$ is an equivariant map" in
Proposition 3.2 can't be removed.

######  Example 3.3.

Consider $f,g_{1},g_{2}:[-2,2]\rightarrow[-2,2]$ as, $f(x)=-|x|$,

$g_{1}(x)=\begin{cases}\frac{2}{3}x-\frac{2}{3},&-2\leq x\leq-\frac{1}{2},\\\
2x,&-\frac{1}{2}<x\leq 0,\\\ x,&0<x\leq 2.\end{cases}$
$g_{2}(x)=\begin{cases}-2x-2,&-2\leq x\leq-\frac{1}{2},\\\
-\frac{2}{5}x-\frac{6}{5},&-\frac{1}{2}<x\leq 2.\end{cases}$

Suppose $G$ is generated by any combination and iteration of
$g_{1},g_{2},\phi,g_{1}^{-1},g_{2}^{-1}$, where $\phi$ is an identify map, i.
e. $\phi(x)=x,x\in[-2,2]$. Note that $g_{1}f(\frac{1}{2})=-1$, while
$fg_{1}(\frac{1}{2})=-\frac{1}{2}$. So, $f$ is not an equivariant map. As we
know, tent map $h_{1}(x)=1-|1-2x|$, $x\in[0,1]$ is chaotic in the sense of Li-
Yorke. Suppose $h(x)=-x$, $x\in[-1,0]$ and

$h_{2}(x)=\begin{cases}-2x-2,&-1\leq x\leq-\frac{1}{2},\\\
2x,&-\frac{1}{2}<x\leq 0.\end{cases}$

Then it can be verified $h$ is the topological conjugation from $h_{1}$ to
$h_{2}$. So, $h_{2}$ is also chaotic in the sense of Li-Yorke. Furthermore,
$[-1,0]$ is an uncountable $G$-Li-Yorke chaotic set of $f$. That is $f$ is
chaotic in the sense of $G$-Li-Yorke.

However, $f^{2}(x)=|x|$ is not chaotic in the sense of $G$-Li-Yorke.

Since $G$ is a group, it contains identity element. Then we have the following
theorem.

######  Theorem 3.4.

Let $(X,d)$ be a metric $G$-space and $f$ be a continuous map from $X$ to $X$.
If $f$ is chaotic in the sense of Li-Yorke, then it is chaotic in the sense of
$G$-Li-Yorke.

However, the converse of the Theorem 3.4 is not true.

######  Example 3.5.

Consider $f,g_{1},g_{2}:[-2,2]\rightarrow[-2,2]$ , $f(x)=x$,

$g_{1}(x)=\begin{cases}x,&-2\leq x\leq 0,\\\ 2x,&0<x\leq\frac{1}{2},\\\
\frac{2}{3}x+\frac{2}{3},&\frac{1}{2}<x\leq 2.\end{cases}$
$g_{2}(x)=\begin{cases}-\frac{2}{5}x+\frac{6}{5},&-2\leq x\leq\frac{1}{2},\\\
-2x+2,&\frac{1}{2}<x\leq 2.\end{cases}$

It is easy to know $f$ is not chaotic in the sense of Li-Yorke. Suppose $G$ is
generated by any combination and iteration of
$g_{1},g_{2},f,g_{1}^{-1},g_{2}^{-1}.$ As we know, tent map $h(x)=1-|1-2x|$,
$x\in[0,1]$ is chaotic in the sense of Li-Yorke. So, $[0,1]$ is an uncountable
$G$-Li-Yorke chaotic set of $f$. Furthermore, $f$ is chaotic in the sense of
$G$-Li-Yorke. Consequently, Li-Yorke chaos $\nRightarrow$ $G$-Li-Yorke chaos.

Now, we give a sufficient condition for $f$ to be chaotic in the sense of
$G$-Li-Yorke.

######  Proposition 3.6.

Let $(X,d)$ be a metric $G$-space. Let $f$ be an equivariant map. Let
$\\{n_{i}\\}_{i=1}^{\infty}$ be a sequence of positive integers and
$\\{A_{i}\\}_{i=0}^{\infty}$ and $\\{B_{i}\\}_{i=0}^{\infty}$ be decreasing
sequences of compact sets satisfying

$\bigcap^{\infty}_{i=0}A_{i}=\\{a\\},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \bigcap^{\infty}_{i=0}B_{i}=\\{b\\},$

where $a,b\in X,a\neq b$. If for any $i\geq 0$, there exists $g_{i}\in G$ such
that

$A_{i+1}\bigcup B_{i+1}\subset g_{n_{i}}f^{n_{i}}(A_{i})\bigcap
g_{n_{i}}f^{n_{i}}(B_{i}),$

then $f$ is chaotic in the sense of $G$-Li-Yorke.

  * Proof.

Let

$\mathcal{C}=\\{c=C_{0}C_{1}\cdots|C_{i}\in\\{A_{i},B_{i}\\},i=0,1,\cdots\\},$

and
$E\subset\Sigma_{2}=\\{x=x_{0}x_{1}\cdots|x_{i}\in\\{0,1\\},i=0,1,\cdots\\}$
be an uncountable set such that for any distinct points
$s=s_{0}s_{1}\cdots,t=t_{0}t_{1}\cdots\in E,s_{n}=t_{n}$ for infinitely many
$n$ and $s_{m}\neq t_{m}$ for infinitely many $m$. Denote
$\varphi(x):E\rightarrow\mathcal{C}$ by $\varphi(x)=C_{0}C_{1}\cdots$, where

$C_{i}=\begin{cases}A_{i},&x_{i}=0,\\\ B_{i},&x_{i}=1,\end{cases}$

for any $i=0,1,2,\cdots$. Let $D=\varphi(E)$. It is easy to see $\varphi$ is
injective. Since $E$ is uncountable, $\varphi(E)$ is uncountable. Let
$c=C_{0}C_{1}\cdots\in D$ and $k>1$, then there exists $x_{k}\in C_{0}$ such
that for every $i=0,1,\cdots,k-1$, we have

$g_{n_{i}}f^{n_{i}}g_{n_{i-1}}f^{n_{i-1}}\cdots g_{n_{0}}f^{n_{0}}(x_{k})\in
C_{i+1}.$

Let $x_{c}$ be anyone of limit points of $\\{x_{k}\\}_{k=0}^{\infty}$ and
$S=\\{x_{c}|c\in D\\}$, then $S$ is uncountable and $S$ is a $G$-Li-Yorke
chaotic set of $f$. So, $f$ is chaotic in the sense of $G$-Li-Yorke. ∎

##  4 Main Result and Proof

Before presenting the main result, we list several lemmas which play an
important role in the proof of the main result.

######  Lemma 4.1.

[2] Let $H$ be a family of real-valued functions from $X$ to $[0,+\infty)$.
Denote $\mu:X\rightarrow[0,+\infty)$: for any $x\in X$,

$\mu(x)=inf\\{h(x)|h\in H\\}.$

If each $h\in H$ is semicontinuous, then $g$ is semicontinuous.

######  Lemma 4.2.

[2] Suppose that for each integer $i\geq 1$, $h_{i}:X\rightarrow[0,+\infty]$
is semicontinuous. Let $a\in[0,+\infty]$,

$g(x)=\liminf_{i\rightarrow\infty}h_{i}(x),x\in X.$

If the set $g^{-1}([0,a])$ is dense in $X$, then it must be a dense
$G_{\delta}$ set of $X$.

$x\in X$ is called a periodic point of $f$ if there exists $n>0$ such that
$f^{n}(x)=x$. The smallest such positive integer $n$ is called the period of
$x$. And the periodic point $x$ with period $n$ is called a $n$-periodic
point. When $n=1$, $n$-periodic point $x$ is called a fixing point of $f$.

######  Lemma 4.3.

Let $(X,d)$ be a metric $G$-space. Suppose $f$ is an equivariant map. If there
exist a $G$-transitive point of $f$ and a fixed point of $f$ $\vartheta\in X$
which satisfies $g(\vartheta)=\vartheta$ for any $g\in G$, then there exists a
dense $G_{\delta}$ set $B$ in $X\times X$ satisfying for any $(x_{1},x_{2})\in
B$,

$\liminf_{n\rightarrow\infty}d(f^{n}(u),f^{n}(v))=0,\forall u\in
G(x_{1}),\forall v\in G(x_{2}).$

  * Proof.

Let $F:X\times X\rightarrow[0,+\infty)$ denoted by for any $(u,v)\in X\times
X$,

$F(x,y)=\liminf_{n\rightarrow\infty}d(f^{n}(u),f^{n}(v)).$

Let $\omega\in X$ be a $G$-transitive point of $f$. Then there exist sequence
$\\{n_{j}\\}_{j=3}^{\infty}$ and $\\{g_{j}\\}_{j=3}^{\infty}\subset G$ such
that $\lim_{j\rightarrow\infty}g_{j}f^{n_{j}}(\omega)=\vartheta$. For any
$Z=(z_{1},z_{2})\in G_{f}(\omega)\times G_{f}(\omega)$, there exist
$k_{1},k_{2}\in\mathbf{N}$ and $g_{1},g_{2}\in G$ such that
$g_{1}f^{k_{1}}(\omega)=z_{1},g_{2}f^{k_{2}}(\omega)=z_{2}.$ So for $i=1,2$ we
have

$\lim_{j\rightarrow\infty}g_{j}f^{n_{j}}(z_{i})=\lim_{j\rightarrow\infty}g_{j}f^{n_{j}}(g_{i}f^{k_{i}}(\omega))=\vartheta.$

Thus, $F(z)=0$. Since $G_{f}(\omega)\times G_{f}(\omega)$ is dense in $X\times
X$, by Lemma 4.2, $F^{-1}(0)$ is a dense $G_{\delta}$ set in $X\times X$. ∎

######  Lemma 4.4.

Let $(X,d)$ be a metric $G$-space and $f$ be a continuous map from $X$ to $X$.
If the $G$-recurrent points set $R_{G}(f)$ is dense in $X$, then $R_{G}(f)$ is
a dense $G_{\delta}$ set.

  * Proof.

Denote $F:X\rightarrow[0,+\infty)$ by for any $x\in X$,

$F(x)=\liminf_{n\rightarrow\infty}\inf_{g\in G}d(gf^{n}(x),x).$

Then $x\in R_{G}(f)\Leftrightarrow F(x)=0$. So, $F^{-1}(0)$ is dense in $X$.
For any $n\geq 1$ and any $g\in G$, $d_{n}(x):=d(gf^{n}(x),x)$ is
semicontinuous. By Lemma 4.1, for any $n\geq 1$, $h_{n}(x)=\inf_{g\in
G}d_{n}(x)$ is semicontinuous. Then by Lemma 4.2 $R_{G}(f)=F^{-1}(0)$ is a
dense $G_{\delta}$ set in $X$. ∎

######  Lemma 4.5.

Let $(X,d)$ be a metric $G$-space and $f$ be a continuous map from $X$ to $X$.
If $f$ is $G$-transitive, then $R_{G}(f\times f)$ is a dense $G_{\delta}$ set
in $X\times X$.

  * Proof.

Since $X$ is a compact metric space with a metric $d$, there exist numerable
topological basis $\\{U_{n}\\}_{n=1}^{\infty}$. By

$\\{x\in
X|\overline{G_{f}(x)}=X\\}=\bigcap_{n=1}^{\infty}\bigcup_{m=0}^{\infty}\bigcup_{g\in
G}gf^{-m}(U_{n}),$

and Proposition 2.4 of [5], open set $\bigcup_{m=0}^{\infty}\bigcup_{g\in
G}gf^{-m}(U_{n})$ is dense in $X$. By Baire Theorem, $\\{x\in
X|\overline{G_{f}(x)}=X\\}$ is dense in $X$. So, there exists $\omega\in X$
being a $G$-transitive point of $f$. Then, $\\{gf^{k}(\omega)|g\in G,k\geq
0\\}$ is dense in $X$. Furthermore,

$\\{gf^{k}(\omega)|g\in G,k\geq 0\\}\times\\{gf^{k}(\omega)|g\in G,k\geq
0\\}\subset R_{G}(f\times f).$

Thus, $R_{G}(f\times f)$ is dense in $X\times X$. By Lemma 4.4, $R_{G}(f\times
f)$ is a dense $G_{\delta}$ set in $X\times X$. ∎

######  Theorem 4.6.

Let $(X,d)$ be a metric $G$-space. Suppose $f$ is an equivariant map. If $f$
is $G$-transitive and there exists a fixed point of $f$ $\vartheta\in X$ which
satisfies $g(\vartheta)=\vartheta$ for any $g\in G$, then there exists a $c$
dense $G$-Li-Yorke set in $X$ for $f$. Specifically, $f$ is chaotic in the
sense of $G$-Li-Yorke.

  * Proof.

Set $D=R_{G}(f\times f)\bigcap B$, in which $B$ is the same with the $B$ in
Lemma 4.3. Therefore, by Lemma 4.3 and Lemma 4.5 $D$ is a residual set in
$X\times X$. Since $X$ is compact, then it is complete and separable. Then by
[2] there exists a dense Mycielski set $K$ such that for any distinct
$x_{1},x_{2}\in K,(x_{1},x_{2})\in D$.

So, $K$ satisfies the condition of the present theorem. ∎

The following example shows the condition "The fixed point of $f$
$\vartheta\in X$ satisfies $g(\vartheta)=\vartheta$ for any $g\in G$" in
Theorem 4.6 can't be removed.

######  Example 4.7.

Suppose $\mathcal{S}^{1}$ is a unit circumference on complex plane. Denote $f$
by $f(z)=z$, $\forall z\in\mathcal{S}^{1}$, and $g(z)=ze^{2\pi i\lambda}$,
$\forall z\in\mathcal{S}^{1}$, in which $\lambda$ is an irrational number. $G$
is generalized by $g$, i.e., $G=\\{g^{n}|n\in\mathbf{Z}\\}$, in which
$g^{0}=f$. $f$ is an equivariant map. Since $g$ is transitive, $f$ is
$G$-transitive. For any $z\in\mathcal{S}^{1}$, $g(z)\neq z$. Since $g$ is an
isometric mapping, $f$ is not chaotic in the sense of $G$-Li-Yorke.

######  Corollary 4.8.

Let $(X,d)$ be a metric $G$-space. Suppose $f$ is an equivariant map. If $f$
is $G$-transitive and there exists a $n$-periodic point of $f$ $\varrho\in X$
which satisfies $g(\varrho)=\varrho$ for any $g\in G$, then there exists a $c$
dense $G$-Li-Yorke set in $X$ for $f$. Specifically, $f$ is chaotic in the
sense of $G$-Li-Yorke.

  * Proof.

By $G$-transitivity of $f$, there exists $\omega\in X$ being a $G$-transitive
point of $f$. Set

$X_{0}=\overline{\\{gf^{n}(\omega)|g\in G,n\in\mathbf{N}\\}}.$

For any given nonempty subsets $U,V$ of $X_{0}$, there exist $g_{1},g_{2}\in
G$, and $n_{1},n_{2}\in\mathbf{N}$ such that $g_{1}f^{n_{1}}(\omega)\in U$ and
$g_{2}f^{n_{2}}(\omega)\in V$. Without loss of generality, let $n_{1}\geq
n_{2}$. Then,
$(g_{1}g_{2}^{-1}f^{n_{1}-n_{2}})(g_{2}f^{n_{2}}(\omega))=g_{1}f^{n_{1}}(\omega)\in
U$. So,

$g_{1}g_{2}^{-1}f^{n_{1}-n_{2}}(V)\bigcap U\neq\emptyset.$

That is, $f^{n}$ is $G$-transitive. By Theorem 3.4 $f^{n}$ is chaotic in the
sense of $G$-Li-Yorke. By Proposition 3.2 $f$ is chaotic in the sense of
$G$-Li-Yorke. ∎

##  5 Conclusions

We introduce the definition of Li-Yorke chaos for the map on $G$-spaces. And
we show:

  * (1)

If $f$ is equivariant, then its Li-Yorke chaos is iterable. And the condition
that $f$ is equivariant can't be removed.

  * (2)

Li-Yorke chaos $\Rightarrow\nLeftarrow$ $G$-Li-Yorke chaos.

  * (3)

A sufficient condition for $f$ to be $G$-Li-Yorke chaotic is given.

The above conclusion are the generalization of the continuous maps of general
compact metric space. They will senrich the theory of $G$-space and iterated
system. It provided the theoretical basis and scientific foundation for the
application of chaos in computational mathematics and biological mathematics.

## References

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# Evolution of emotion semantics

Aotao Xu<EMAIL_ADDRESS>Jennifer E. Stellar  Yang Xu  Department of
Computer Science, University of Toronto  Department of Psychology, University
of Toronto  Cognitive Science Program, University of Toronto

###### Abstract

Humans possess the unique ability to communicate emotions through language.
Although concepts like anger or awe are abstract, there is a shared consensus
about what these English emotion words mean. This consensus may give the
impression that their meaning is static, but we propose this is not the case.
We cannot travel back to earlier periods to study emotion concepts directly,
but we can examine text corpora, which have partially preserved the meaning of
emotion words. Using natural language processing of historical text, we found
evidence for semantic change in emotion words over the past century and that
varying rates of change were predicted in part by an emotion concept’s
prototypicality—how representative it is of the broader category of “emotion”.
Prototypicality negatively correlated with historical rates of emotion
semantic change obtained from text-based word embeddings, beyond more
established variables including usage frequency in English and a second
comparison language, French. This effect for prototypicality did not
consistently extend to the semantic category of birds, suggesting its
relevance for predicting semantic change may be category-dependent. Our
results suggest emotion semantics are evolving over time, with prototypical
emotion words remaining semantically stable, while other emotion words evolve
more freely.

###### keywords:

emotion; semantic change; semantic stability; prototype theory; word embedding

††journal: Cognition

20cm(3.5cm,3cm) To appear in Cognition. This preprint may contain errors not
present in the published version.

##  1 Introduction

Much like emotion concepts vary in their meaning across cultures [1, 2, 3], it
is possible emotion words can take on different meanings over time.111In our
study, we use the terms “emotion concept” and “emotion word” interchangeably
to refer to emotions that are lexicalized in natural language. For instance,
the English word awe in the 18th century may not represent the same feeling it
does today, after a century of evolving perspectives on power and beauty [4].
Although we cannot travel to earlier historical periods to study emotion
concepts directly, we do have access to text corpora which have partially
preserved the meaning of emotion words. These words do not reflect the
entirety of an emotion concept, which includes expressive, experiential, and
physiological components, but they do offer insight into its shared meaning
within a society. Here we use computational linguistic analyses to investigate
the evolution of emotion semantics.

If the meaning of different emotion words like awe or joy are changing over
time, are they changing at the same rate or are there features of an emotion
word that might predict its rate of semantic change? We propose that an
emotion’s conceptual prototypicality is one such feature. Prototypicality is a
graded measure of the goodness of a concept’s membership in a semantic
category [5, 6]. In the case of emotions, joy is considered a more
prototypical concept than optimism. Prototypical emotion concepts may have
clearer biological and cultural functions and more distinct features than less
prototypical ones. For instance, prototypical concepts like fear and disgust
are particularly suited to solving evolutionary challenges or taking advantage
of opportunities that faced early humans [7], and they may have particularly
strong social or cultural scripts [8, 5, 9]. These emotion concepts are often
more clearly marked by distinctive expressions, experience, and patterns of
activation in the body [10], and prototypical members may even help define the
meaning of their less prototypical counterparts [11] (see Supplementary
Information for further evidence). We hypothesize that these well-defined
functions and features of prototypical emotion concepts could promote semantic
stability. As a result, the meaning of words for more prototypical concepts
like joy may tend to resist change, more so than words for less prototypical
ones like optimism; see Figure 1 for an illustration.

Although prototypicality has been discussed in other semantic categories, we
do not expect prototypicality to predict semantic stability in every category.
The basis of prototypicality and thus its ability to predict semantic change
may differ in the classic example of birds [12]. The prototypicality of a bird
name is primarily based on differences in biological taxonomies [13] and
features grounded in sensory or visual perception [14]. As such, the features
that define more (e.g., sparrow) or less prototypical birds (e.g., penguin)
are equally well-defined, so the meanings of prototypical bird names do not
help define the meanings of less prototypical bird names (see Supplementary
Information), in contrast to the category of emotion words. We expect that
while prototypicality should correlate with semantic stability of emotion
words, it should not correlate with semantic stability of bird names.

Figure 1: An illustration of the relation between prototypicality and semantic
stability of emotion words. Each blue dot represents an emotion word, and the
size of the dot is proportional to its predicted semantic stability; the
smaller the dot, the higher its rate of semantic change over time. The
contours indicate degrees of prototypicality. Visually, an emotion word close
to the center has high prototypicality, and vice versa.

Our hypothesis augments general principles of lexical evolution and semantic
change. It has been shown that usage frequency is a general determiner of
stability in English verb regularization [15], lexical replacement [16, 17,
18], loan word borrowing [19], and semantic change [20, 21]. If explained
through the lens of communication, frequency should predict semantic
stability: when speakers change the meaning of a highly frequent lexical item,
they would face a higher number of misunderstandings than if they change a
low-frequency item [22, 23]. As a result, we expect frequent emotion words to
change less in meaning than other emotion words. We examine the
prototypicality of emotion concepts as an additional predictor of semantic
stability beyond usage frequency.

Our hypothesis differs from diachronic prototype semantics [24], which states
that more prototypical senses of a word tend to stay prototypical over time
and exhibit more stability than peripheral senses. Although this theory is
consistent with our hypothesis regarding the pattern that prototypicality
offers stability, we focus on explaining rates of semantic change among
concepts in a lexical field, as opposed to characterizing principles of change
among senses of an individual word [24, 25]. Previous studies have examined
the theory of diachronic prototype semantics over the whole lexicon and found
the prototypicality of words in statistical clusters (formed in meaning space)
to negatively correlate with rates of semantic change [26, 21]. However, these
studies do not explain how semantic change relates to prototypicality in the
scope of a specific category such as emotions or birds.

We present a methodology for modeling emotion semantics and its evolution by
building on work from machine learning and natural language processing in word
embedding [27, 28, 29] and its historical extensions [30, 20, 31, 32]. We
model emotion semantics using a vector-space representation trained on
historical text corpora of natural language use, and we use this
representation to model human judgments of prototypicality and semantic change
of emotion words. Vector-space models of word meaning have been used within
affective science for reconstructing human emotion ratings on dimensions such
as valence and arousal [33], sentiment analysis [34], and analyzing emotion
categories in documents [35], but not for investigating the open question of
the evolution of emotion semantics.

##  2 Methodologies for quantifying rates of semantic change

Quantifying the rate of semantic change for a word requires records of its
meaning from two distinct time periods and a quantitative metric that compares
these records. One type of methods that constructs word meanings and enables
comparisons over time is based on word embeddings [27, 28]. The embedding of a
word is a real-valued vector that represents its meaning through a high-
dimensional space; vectors for words with similar meanings tend to be close in
this space, such as compassion and sympathy. Word embeddings are constructed
from co-occurrence statistics in large text corpora. We thus obtain meaning
representations from two distinct time periods by constructing word embeddings
based on historical text corpora from the corresponding periods [20].

Existing methods for computing rates of semantic change often rely on the
cosine distance between two embeddings [20, 21]. According to this metric, a
large cosine distance between historical and contemporary embeddings implies a
high rate of semantic change, and vice versa. However, this metric tends to
bias the correlation between rate of semantic change and frequency [21]. For
this reason, we use an alternate neighbourhood-based metric to compare word
embeddings across time [36]. This metric quantifies the rate of semantic
change for a word $w$ between periods $t_{1}$ and $t_{2}$ via the Jaccard
distance between sets of $k$-nearest neighbours in meaning space:

$rate(w,t_{1},t_{2})=1-\frac{|kNN(w,t_{1})\,\cap\,kNN(w,t_{2})|}{|kNN(w,t_{1})\,\cup\,kNN(w,t_{2})|}$
(1)

where $kNN(w,t)$ contains the $k$ words whose embeddings are the closest to
the embedding of $w$ in terms of cosine similarity. Intuitively, we say a word
underwent semantic change if the composition of its semantic neighbourhood has
changed. Following [36], the part of speech (POS) of the members of $kNN(w,t)$
is always the same as the POS of $w$, and we also set $k$ to 100. In
Supplementary Information, we show that this measure is robust to variations
in $k$. Compared to the cosine metric, this metric enables more transparent
interpretation on rates of change because we can inspect and evaluate the sets
of semantic neighbours (see Supplementary Information for examples of emotion
semantic change).

To implement this metric at scale, we used pretrained historical word
embeddings and POS tags from HistWords [20]. Specifically, we used
300-dimensional Word2Vec (SGNS) embeddings obtained from the Skip-Gram model
[28] and trained on the corpora Google Books Ngrams English and French. We
used historically most frequent POS tags from the same sources. This provided
us with historical word embeddings and most frequent POS tags for 100,000
English words and 100,000 French words, for every decade between 1800 and
2000.

##  3 Analyses of emotion concepts

In the first set of analyses, we provide evidence for our hypothesis that the
well-defined features and functions of prototypical emotion words promote
semantic stability. Specifically, we test against the null hypothesis that
prototypicality does not predict semantic stability in English and French
emotion words over the past century.222We focused on these two languages
because 1) we want to test if our analysis generalizes beyond a single
language, and 2) there is a limited cross-linguistic variety of empirical
studies on emotion prototypicality and of the historical data provided by
HistWords. We describe resources that provide us with lists of English and
French emotion words, emotion prototypicality ratings, and historical
frequency estimates. We then describe our methods for estimating
prototypicality ratings historically and for hypothesis testing, which is
followed by a presentation of our results.

###  3.1 Materials

We obtained a list of English emotion words and their corresponding
prototypicality ratings from [6]. The authors produced the list by obtaining
213 emotion nouns from a collection of emotion concepts. They produced emotion
prototypicality ratings by asking 112 American university students to rate
each of these nouns on a 4-point scale, where 4 means the noun is definitely
an emotion, and 1 means the noun is definitely not an emotion. Following this
work, our analyses focused on nouns that have prototypicality ratings at least
2.75 with the addition of surprise and exclusion of abhorrence, ire,
malevolence, and titillation; we additionally included the word awe. We also
obtained the valence of these emotion words from the study, which was
originally derived from applying multidimensional scaling to similarity
judgments [6].

We also obtained a list of French emotion words with their corresponding
prototypicality ratings [37]. The authors produced the list by translating 237
Italian emotion words from an earlier study into French. They produced emotion
prototypicality ratings by asking 319 French university students to rate each
of these words on a 10-point scale, where 10 means the word is certainly an
emotion, and 1 means it is not an emotion. To be consistent with the English
list, we kept emotion words whose most frequent POS tag is noun in the final
decade of our historical POS data. We also obtained the valence of these
emotion words from the study, which was originally obtained by asking 300
French university students to rate the words on a scale of -5 (very
unpleasant) to 5 (very pleasant) [37].

We obtained historical frequency data from HistWords [20], which is based on
the corpora Google Books Ngrams English and French. This yielded historical
frequency data for 682,459 English words and 213,686 French words, for every
decade between 1800 and 2000. We intersected the word lists with historical
word embeddings, POS tags, and frequency from HistWords. We noticed that more
emotion words were unavailable when we increased the span between flanking
decades than otherwise: if $t_{1}=1890$ and $t_{2}=1990$, only 9 words from
the English list are unavailable in HistWords and the HTE, but if we used
$t_{1}=1800$, the number increased to 28; similarly in French, the shorter
time span resulted in 32 unavailable words, but the longer one resulted in 58
unavailable words. Consequently, we decided to use the decades of 1890 and
1990 as the flanking decades for our analysis (i.e. $t_{1}=1890$,
$t_{2}=1990$), and we used historical frequency data from the 1890s. After the
intersection, we had a total of 123 English emotion words and 112 French
emotion words.

###  3.2 Methods

Since we cannot go back in time to measure the prototypicality of emotion
concepts in the past, we needed a method for estimating historical
prototypicality. Let $x$ represent the word embedding of a concept in category
$c$. Following previous work in prototype theory [38, 39], we estimated the
prototypicality of $x$ as the unnormalized conditional probability $p(c|x)$,
which can be computed using an isotropic Gaussian via Bayes rule:

$p(c|x)\propto p(x|c)\sim N(\mu,I)$ (2)

where $\mu=\frac{1}{|E_{c}|}\sum_{v\in E_{c}}v$ and $E_{c}$ is the set of
embeddings for members of $c$; $I$ is an identity matrix. Intuitively, we
estimated the prototypicality of $x$ by computing its distance from the
category centroid $\mu$; the closer they are, the higher its estimated
prototypicality. To estimate the prototypicality of an emotion concept in
history, we used its historical embedding $x$ and the embeddings of other
emotion concepts to compute $p(x|c=emotion)$. We evaluated this method by
computing the correlation between our empirical prototypicality ratings
obtained from [6, 37] and our estimated prototypicality based on embeddings
from the 1980s and 1990s, the decades closest to the publication of those
studies.

To test against the null hypothesis, we computed the rate of change for every
emotion concept $x$, $rate(x,1890,1990)$ using Equation (1) and historical
embeddings and POS tags from HistWords. Separately for English and French, we
then computed the Pearson correlations between the emotion concepts’ rates of
change and prototypicality estimated for the 1890s. To evaluate whether the
prototypicality of emotion concepts predicts rates of change beyond frequency,
we performed multiple linear regressions for English and French using the
following regression formula:

$rate(x,1890,1990)\sim p(x|c=emotion)+freq(x)+val(x)$ (3)

where for every concept $x$, we denote its usage frequency as $freq(x)$ and
its valence as $val(x)$, which we added to control for unequal numbers of
negative and positive emotion concepts in our datasets. We fitted the model
using ordinary least squares implemented by statsmodel [40]; we also used this
package to compute relevant test statistics. Following previous work [20], we
performed a log transformation on frequency.

###  3.3 Results

Figure 2 shows the Pearson correlation between estimated prototypicality from
English word embeddings and ratings from English speakers [6]: $\rho=0.428$,
$p<0.001$, $n=123$. We obtained similar results with French word embeddings
for a set of French emotion concepts [37]: $\rho=0.438$, $p<0.001$, $n=112$.
These initial results show our estimated degrees of prototypicality for
emotion concepts capture human judgments to some extent. For this reason, we
used the same method to estimate historical prototypicality ratings and
evaluated them as predictors of semantic stability.

Figure 2: Word embedding reconstruction of emotion prototypicality in a)
English and b) French. Scatter plots compare estimated prototypicality
computed from Equation 2 against empirical ratings. Each dot corresponds to an
emotion concept (a sample of concepts annotated), and each band shows a 95%
confidence interval for the line of best fit.

Figure 3 shows a significant negative correlation between emotion
prototypicality and degree of semantic change: $\rho=-0.580$, $p<0.001$,
$n=123$. On average, emotion concepts rated prototypical such as anger, joy,
fear underwent less change in meaning compared to words denoting less
prototypical concepts such as zest, exhilaration and hysteria (see annotated
word samples in Figure 3). Similar results hold for French: $\rho=-0.576$,
$p<0.001$, $n=112$. Supplementary Information provides additional examples of
English and French emotion concepts from the most changing to the most
semantically stable, along with their semantic neighbours retrieved from our
methods.

Figure 3: Scatter plots showing the negative correlations between emotion
prototypicality and rates of emotion semantic change between the 1890s and
1990s, in a) English and b) French. Each dot corresponds to an emotion word
(with a sample set of words annotated), and each band shows a 95% confidence
interval for regressions between emotion prototypicality and rates of semantic
change.

Figure 4 shows our results for multiple regression. The adjusted $R^{2}$ of
the model for English is $0.680$, with $p<0.001$, $n=123$; mean regression
coefficients for prototypicality ($\beta=-0.417$, $p<0.001$) and frequency
($\beta=-0.0451$, $p<0.001$) are significant, but for valence ($\beta=0.0053$,
$p=0.208$) it is insignificant. For French, the adjusted $R^{2}$ of the model
is $0.538$, with $p<0.001$, $n=112$; mean regression coefficients for
prototypicality ($\beta=-0.6363$, $p<0.001$) and frequency ($\beta=-0.0331$,
$p<0.001$) are significant, but for valence ($\beta=0.0019$, $p=0.454$) it is
insignificant. These results show that frequency predicts semantic stability,
which confirms the previous findings [20, 21]. Beyond frequency, we find that
prototypicality plays an important role in predicting semantic stability of
emotion words, manifested in its significant and negative effect. This
provides evidence for our hypothesis that prototypical emotion words tend to
be semantically stable over time.

Figure 4: Predictor coefficients from multiple regressions on rates of emotion
semantic change. Error bars show standard error, and “n.s.”, “*”, “**”, “***”
denote no significance at $p<0.05$, and $p<0.05$, $p<0.01$, $p<0.001$
respectively. a) shows results for English, and b) shows results for French.

Supplementary Information includes three more analyses that further
corroborate our findings. The first analysis repeats the multiple regression
but restricts the neighbourhoods to emotion concepts only when computing
$rate(w,1890,1990)$; the results rule out the possibility that our findings
are an artifact of the non-emotion senses of polysemous emotion concepts
(e.g., zest). The second analysis extends the multiple regression for English
by including additional predictors based on hypernymy-hyponymy, age of
acquisition, and degrees of polysemy, which could potentially subsume the
effects of prototypicality; our results show that this is not the case. The
third analysis repeats the multiple regression for English emotion concepts,
except the rates of change are computed as $rate(w,1980,1990)$ and empirical
prototypicality from [6] were used; these results provide evidence that the
effect of prototypicality is not caused by potential artifacts in our
estimation of prototypicality based on Equation 2.

Figure 5 illustrates our main finding with two example words: disgust and awe.
These words had similar usage frequencies over time, but disgust is rated as a
more prototypical emotion word than awe [6]. Over time, awe has shifted
meaning more substantially than disgust. In particular, both words were in the
neighbourhood of negative emotion words (e.g., sadness, anger, and fear) in
the 1890s. However, while disgust still remained close to these words in the
1990s, awe moved closer to positive emotion words (e.g., love and happiness).

Figure 5: An illustrative comparison of prototypicality, frequency, and
semantic stability in emotion words awe and disgust. Left panels show the
embedding-based prototypicality, frequency and degree of semantic change of
awe and disgust over time. Right panels visualize the rates of change in these
words by placing them in the two principal components [41] of meaning space,
along side prototypical emotion concepts which are annotated based on their
valence (“+” for positive, “-” for negative).

##  4 Analyses of bird names

In this set of analyses, we demonstrate that the sources of prototypicality do
not always provide semantic stability as we have shown for emotion concepts.
Here we repeat our previous analyses on a case study of birds, a frequently
investigated category in prototype theory [12, 42]. As we will see, our
embedding-based estimation of prototypicality does not work well with bird
names, and we will focus our analysis on using empirical ratings from the
1970s.

###  4.1 Materials

We obtained a list of English bird names with prototypicality ratings from
[12]. The author produced the list by consulting previous work so that the
selected names were relatively frequent. They produced bird prototypicality
ratings by asking 209 American university students to rate each of these names
on a 7-point scale, where 1 means the name refers to a very good example of a
bird, and 7 means the name refers to a very poor example. Note that the scale
operates in the opposite direction of our prototypicality ratings for emotion
concepts. For clarity, we multiplied these ratings by -1 so the direction is
the same as our emotion data. Focusing on the 1970s and 1990s, we used
historical data from HistWords [20], which was intersected with the word list
and provided us with 41 bird names.

###  4.2 Methods

Similar to the previous section, we attempted at estimating bird
prototypicality using Equation 2. We then computed the rates of change for
every bird name $w$, $rate(w,1970,1990)$ using Equation 1. We computed the
Pearson correlation between rates of change and prototypicality ratings
obtained from the 1970s, and we performed a multiple regression using the
following formula:

$rate(w,1970,1990)\sim proto(w)+freq(w)$ (4)

where we denote the empirical prototypicality rating of every bird name $w$ as
$proto(w)$.

###  4.3 Results

Figure 6a shows the Pearson correlation between estimated prototypicality and
empirical ratings from [12]: $\rho=0.153$, $p=0.340$, $n=41$. While the same
method reconstructs prototypicality for emotion concepts to some extent, our
text-based method does not explain a significant amount of variance in the
prototypicality of birds which depends more on sensory features [14]. It has
been shown that prototypical birds in our dataset tend to be passerines, small
perching birds that sing (e.g., robin), and less prototypical ones tend to be
non-passerines (e.g., penguin) [13], which our text-based methodology did not
capture. For this reason, we chose to focus on empirical prototypicality
ratings for birds in our analyses.

Figure 6b shows a significant positive correlation between bird
prototypicality and degree of semantic change: $\rho=0.428$, $p=0.005$,
$n=41$. This finding suggests that the relation between semantic change and
prototypicality in bird names is opposite to our previous findings for emotion
words. Figure 7a shows the results for multiple regression. The adjusted
$R^{2}$ is $0.508$, with $p<0.001$, $n=41$; mean regression coefficients for
empirical prototypicality ($\beta=0.0283$, $p=0.011$) and frequency
($\beta=-0.0454$, $p<0.001$) are significant. We observe frequency still
predicts semantic stability, suggesting it is indeed a general predictor of
semantic change. Interestingly, prototypicality of birds not only failed to
predict stability as in the case of emotion concepts, but also pointed to the
opposite trend: in the category of birds, names of prototypical birds tend to
undergo more change than other names.

Figure 6: Analyses of bird names: a) word embedding reconstruction of bird
prototypicality and b) correlations between bird prototypicality and rates of
semantic change between the 1970s and 1990s. Each dot corresponds to a bird
name, and each band shows a 95% confidence interval for the line of best fit.
Figure 7: Multiple regression analysis of bird names: a) predictor
coefficients from multiple regressions on rates of semantic change, following
the same layout as Figure 4; b) Cook’s distance for every bird name, showing
the influence of individual data points on the regression result.

To better understand the implications of this variability to our finding about
bird names, we performed a more in-depth analysis of the data. Unlike the case
of emotion, we observe bird names exhibit high variability in Figure 6b, which
is reflected in the wide confidence region.333Note that the number of
available bird terms for our analysis is substantially lower than that of
emotion terms. This suggests the opposite trend in bird names is influenced by
only a handful of less prototypical birds. We estimated the influence of each
bird name using Cook’s distance, which takes into account the data point’s
residual and leverage. Figure 7b shows the influence of each bird in the
regression analysis: the most influential points are turkey, bat and chicken.
We can observe bat is likely to be influential as it has a much higher rating
(not prototypical) than other bird names; this might be because subjects in
the original study, being university students, were familiar with the
scientific classification of bats. More importantly, despite not being
prototypical birds, turkey and chicken could have important cultural roles
(festive or culinary) in North America so that they provided anchors for their
meaning, thereby contributing to the significant correlation between bird
prototypicality and semantic change.

Figure 8 compares the degrees of semantic change that took place in emotion
concepts and bird names between the 1970s and 1990s. Many prototypical emotion
concepts tend to lie at the lower tail of the density distribution and show
high stability, mirroring the results we have seen previously, but the same
pattern does not hold for birds. We observe that overall bird names tend to
undergo greater change than emotion concepts do. It is possible that
prototypical birds possess the most representative features of the bird
category, which could provide points of attachment for meaning change via
processes such as chaining, in which a word for one object is extended to be
used for another, or metaphor [43, 25]. This general pattern of more rapid
change among bird names together with the additional semantic stability of a
handful of influential bird exemplars may be responsible for the positive
correlation between degrees of bird prototypicality and rates of semantic
change.

Figure 8: Distributions of semantic change in emotion and bird categories.
Each dot corresponds to a word, and the size of the dot is proportional to its
degree of prototypicality. The density plots were obtained using kernel
density estimation; although degrees of change given by Equation 1 are
technically bounded between 0 and 1, we did not bound the support of this
figure for illustrative reasons.

##  5 Conclusion

Language offers a lens into the history of emotion semantics. Our
computational linguistic analyses of semantic change suggest that a new view
of emotion concepts in language may be warranted. Rather than perceiving of
emotion concepts as static, their meaning is evolving over time. The exact
cultural or societal factors responsible for semantic change in emotion words
are difficult to pinpoint, and they may be different for each emotion term.
For example, semantic change in awe may reflect a movement away from its use
in religious contexts, in which it reflects more of a fearful respect, towards
greater use in beautiful artistic and natural contexts that followed the
emergence of romanticism and transcendentalism in the early to middle
nineteenth century. We assessed semantic change over a relatively short
timescale, suggesting that in the centuries to come it is possible that words
like awe may continue to evolve and mean something very different than they do
today.

Further, we found in two languages that more prototypical emotion words [6,
42] showed greater semantic stability than other emotion words over time. The
relation between prototypicality and semantic change depends on its exact
sources, as we observed opposite trends for emotions and birds. The importance
of prototypicality as a predictor in semantic change for other semantic
categories remains an open question and future work should investigate what
features affect the importance of prototypicality. Our study extends research
on emotions to its historical development and offers a computational cognitive
characterization of evolving emotion semantics from natural language use.

## Acknowledgements

We thank Brett Ford and two anonymous reviewers for providing constructive
comments on our manuscript. This work was supported by a University of Toronto
Graduate Entrance Scholarship to AX, a Canada Foundation for Innovation grant
#502426 to JES, a NSERC Discovery Grant RGPIN-2018-05872 and a SSHRC Insight
Grant #435190272 to YX.

##  Appendix A Supplementary material

Code and data used for our analyses are available on GitHub at
https://github.com/johnaot/Emotion_Semantic_Change.

##  Appendix B Word age and prototypicality

Here we further demonstrate the differences between prototypicality in
emotions and prototypicality in birds. Previous work has suggested that
prototypical emotion concepts are well-defined and may have particularly
strong social or cultural scripts [8, 5, 9]. Thus it is conceivable that words
for prototypical emotion concepts exist in a lexicon prior to those less
prototypical emotion words. In contrast, prototypicality of birds is based on
biological taxonomy [13] and grounded in sensory and visual perception [14],
and we do not expect prototypicality to be reflected in the age of a word; it
is likely that a relatively newly documented passerine (e.g., bluebird)
entered the lexicon after well-established non-passerines (e.g., chicken). To
test these ideas, we analyzed the correlation between word age and
prototypicality in the categories of emotion and birds.

Following existing work [44], we obtained the age of a word from the
Historical Thesaurus of English (HTE) [45]. For each word entry, the HTE
provides a list of senses of the word, and for each sense, the HTE provides
the word class associated with that sense of the word and the date of first
appearance of the sense in historical records. We operationalized the date of
(the first) emergence of a word to be the earliest date among the dates of
first appearance across all of its senses. Since our analyses focused on
nouns, we considered only noun senses. We did not analyze the age of French
words due to the unavailability of comparable French dictionaries.

We analyzed the same lists of English emotion words and bird names described
in Section 3.1 and Section 4.1, which we intersected with the HTE data. For
English emotion words, the Pearson correlation between emerging date and
prototypicality is $-0.366$, $p<0.001$, $n=135$. This indicates that
prototypical emotion words emerged earlier in the history of English. The same
pattern does not hold for the bird category. For Rosch bird names, the Pearson
correlation between emergence date and prototypicality is $-0.0657$,
$p=0.643$, $n=52$. See Figure 9 for illustration.

Figure 9: Scatter plots showing the relations between prototypicality and word
age for a) emotion words and b) bird names. Each dot corresponds to a word
(with a sample set of words annotated), and each band shows a 95% confidence
interval for regressions between prototypicality and age.

##  Appendix C Evaluation of the nearest-neighbour measure

We show that the nearest-neighbour measure of semantic change described in
Section 2 is 1) robust to variation in neighbourhood size (denoted by $k$),
and 2) interpretive for a word’s semantic change based on nearest neighbours
retrieved at different time points.

###  C.1 Robustness in neighbourhood size $k$

We evaluate whether the nearest-neighbour measure is robust to variation in
$k$. Following Section 3.1, we again quantify semantic change by setting
$t_{1}=1890$ and $t_{2}=1990$ and focus our analysis on the lists of English
and French emotion words we analyzed in the main text. We show that for
$k=20,40,60,80,$ and $100$, resulting degrees of semantic change
$rate(w,t_{1},t_{2})$ are highly correlated. The correlation results are
summarized in Figure 10. We observed that in both English and French, the
degrees of change are strongly correlated between any two of the predetermined
settings of $k$, with a small decrease in correlations for the lowest value of
$k=20$.

Figure 10: Robustness of $kNN$ demonstrated using emotion words. The first
column shows results for English and the second column shows results for
French. Each cell shows the Pearson correlation between changes measured by
$x$-nearest neighbours and by $y$-nearest neighbours. All p-values are
significant and less than 0.001.

###  C.2 Examples of emotion semantic change

Qualitative changes in the nearest neighbours of a word offer interpretability
for semantic change. We provide examples of emotion words that underwent the
most and the least changes and their nearest semantic neighbours in Tables 1,
2, 3, and 4. For example, in Table 1, we observed zest, which used to
primarily convey joy but later became primarily associated with food, is among
the most changing emotion words; on the other hand, in Table 2, we observed
words like surprise has barely changed in meaning. Similarly, in Table 3, we
observed stupéfaction in French become less associated with despair and
anxiety over time; in Table 4, the least changing words on the other hand tend
to preserve similar emotion words as neighbours.

Most Changing | Nearest Neighbours in 1890s | Nearest Neighbours in 1990s
---|---|---
zest | relish, enjoyment, sprightliness | juice, teaspoons, vinegar
infatuation | priestcraft, devastations, misanthrope | inhomogeneity, palates, pleurisy
sentimentality | cant, sentimentalism, rusticity | polyphony, sterne, mandel
optimism | pessimism, aptness, sentimentalism | pessimism, insecurity, enthusiasm
exhilaration | mountebank, festivity, tulip | joy, sadness, excitement
aggravation | misery, symptoms, consequences | proxies, sleeplessness, stressor
exasperation | vehemence, peevishness, pitch | astonishment, amazement, disgust
glee | merriment, wassail, delight | claps, shouts, megaphone
cheerfulness | hopefulness, sprightliness, vivacity | serenity, blasphemies, cannonade
gaiety | sprightliness, vivacity, gayety | pontellier, faints, plaudits
fondness | liking, peevishness, passion | affection, pianists, groanings
hysteria | neurasthenia, simulation, melancholia | neurosis, hypochondriasis, psychosis
dejection | despondency, sullenness, irresolution | pantomime, theseus, disquiet
elation | sullenness, despondency, peevishness | despair, revulsion, dread
ferocity | fierceness, cruelty, prowess | vigor, fury, proverb
revulsion | feeling, disquietude, outburst | disgust, hisses, yearnings
isolation | loneliness, disorganization, seclusion | monger, characterization, coli
alienation | eviction, property, repugnancy | helplessness, blauner, resentment
hopelessness | uselessness, futility, helplessness | helplessness, despair, frustration
rapture | ecstasy, delight, joy | joy, indignation, outcasts
Table 1: Top 20 most changing English emotion words along with their 3 nearest neighbours in the flanking decades. Least Changing | Nearest Neighbours in 1890s | Nearest Neighbours in 1990s
---|---|---
grief | sorrow, anguish, joy | sorrow, sadness, anguish
pity | compassion, love, sympathy | compassion, shame, sadness
misery | wretchedness, miseries, degradation | sorrow, bitterness, anguish
disgust | horror, aversion, indignation | sadness, annoyance, amazement
anger | indignation, resentment, rage | resentment, rage, frustration
surprise | astonishment, amazement, dismay | astonishment, amazement, dismay
sorrow | grief, anguish, sadness | grief, sadness, misery
affection | affections, tenderness, esteem | admiration, sympathy, love
happiness | felicity, prosperity, welfare | prosperity, joy, enjoyment
despair | desperation, dismay, rage | anguish, frustration, sadness
fear | dread, anger, shame | dread, resentment, anger
horror | terror, astonishment, amazement | terror, astonishment, amazement
regret | disappointment, grief, sorrow | disappointment, sadness, bitterness
envy | jealousy, uncharitableness, hatred | jealousy, hatred, resentment
disappointment | mortification, grief, regret | frustration, sadness, regret
rage | fury, anger, indignation | anger, fury, indignation
shame | disgrace, infamy, blush | guilt, pity, humiliation
astonishment | amazement, surprise, dismay | amazement, dismay, surprise
joy | gladness, delight, grief | delight, sorrow, excitement
sympathy | sympathies, compassion, affection | affection, admiration, compassion
Table 2: Top 20 least changing English emotion words along with their 3 nearest neighbours in the flanking decades. Most Changing | Nearest Neighbours in 1890s | Nearest Neighbours in 1990s
---|---|---
stupéfaction | indiscrétion, désespoir, anxiété | désapprobation, émotion, allégresse
suspicion | inculpation, défiance, prévention | méfiance, défiance, incertitude
culpabilité | réussite, identité, présomption | infériorité, persécution, châtiment
déplaisir | étonnement, inquiétude, appréhensions | plaisir, mâle, océans
torpeur | engourdissement, apathie, léthargie | apathie, consternation, stupeur
extase | contemplation, stupeur, somnambulisme | contemplation, joie, angoisse
soupçon | soupçons, équivoque, délit | préjugé, partialité, préjugés
hystérie | épilepsie, diabète, étiologie | névrose, épilepsie, névroses
séduction | adultère, entraînements, cruauté | immédiateté, impiété, éloquence
désolation | épouvante, dévastation, misère | pauvreté, saleté, nausées
froideur | bonhomie, trousseau, bassesse | arrogance, ingratitude, insolence
excitation | irritation, nerfs, nerf | tension, angoisse, agitation
intimidation | violence, ruse, corruption | menaces, coercition, chantage
timidité | fierté, délicatesse, naïveté | docilité, avidité, découragement
tension | volts, pression, potentiel | appareillage, excitation, tensions
intérêt | intérêts, utilité, équité | intérêts, utilité, rentabilité
espérance | espoir, espérances, désir | espoir, espérances, désir
dépit | mépris, défaillances, hésitations | grâce, précocité, conséquence
allégresse | fierté, épouvante, gaieté | joie, émotion, gaieté
effusion | sang, tendresse, larmes | sang, amertume, lucre
Table 3: Top 20 most changing French emotion words along with their 3 nearest neighbours in the flanking decades. Least Changing | Nearest Neighbours in 1890s | Nearest Neighbours in 1990s
---|---|---
tristesse | angoisse, amertume, effroi | amertume, douleur, angoisse
tendresse | bienveillance, sympathie, sollicitude | douceur, compassion, amour
patience | courage, persévérance, prudence | courage, persévérance, audace
orgueil | vanité, amour, ambition | arrogance, insolence, vanité
horreur | honte, effroi, angoisse | opprobre, effroi, tristesse
effroi | tristesse, consternation, terreur | tristesse, consternation, horreur
indignation | admiration, effroi, cri | enthousiasme, cri, admiration
joie | tristesse, douleur, bonheur | tristesse, enthousiasme, douleur
honte | horreur, chagrin, humiliation | opprobre, humiliation, peur
jalousie | haine, ambition, convoitise | arrogance, haine, rancune
colère | désespoir, mécontentement, anxiété | désespoir, fureur, émotion
douleur | chagrin, souffrance, douleurs | souffrance, tristesse, douleurs
stupeur | tristesse, angoisse, effroi | consternation, surprise, effroi
vengeance | haine, ressentiment, fureur | haine, jalousie, orgueil
bonheur | malheur, gloire, joie | malheur, joie, plaisir
chagrin | douleur, tristesse, honte | tristesse, douleur, amertume
terreur | effroi, horreur, haine | horreur, effroi, anarchie
enthousiasme | ardeur, joie, admiration | joie, admiration, indignation
impatience | anxiété, indignation, angoisse | joie, humeur, espoir
souffrance | douleur, tristesse, angoisse | douleur, angoisse, souffrances
Table 4: Top 20 least changing French emotion words along with their 3 nearest
neighbours in the flanking decades.

##  Appendix D Additional analyses of emotion semantic change

We describe three additional analyses that corroborate our findings on emotion
semantic change in the main text. The first analysis rules out the possibility
that our findings are an artifact of the non-emotion senses of polysemous
emotion concepts (e.g., zest). The second analysis shows that additional
predictors based on hypernymy-hyponymy and degrees of polysemy do not subsume
the effects of prototypicality on emotion semantic change. The third analysis
provides evidence that our results on emotion concepts were not caused by
artifacts in our estimation of prototypicality.

###  D.1 Category-bounded analysis

We investigate the robustness of our semantic change analyses by considering a
variant of the nearest-neighbour measure discussed in Section 2. Originally in
Section 2, the degree of semantic change of a word $w$ is defined as the
Jaccard distance between its nearest neighbours at time $t_{1}$, $kNN(t_{1})$,
and its nearest neighbours at a later time, $kNN(t_{2})$, where $kNN$ is
restricted to nouns in the entire lexicon and determined by cosine similarity
over word vectors. Since the meaning of every emotion word is represented by
one word vector only, the set $kNN$ might also capture meaning change with
respect to the word’s polysemes and homonyms, i.e., meaning change outside the
category of emotion. To assess how such meaning change might affect our
results, we restricted the set of nearest neighbours so that only the list of
emotion words are included, i.e., a category-bounded analysis of emotion
semantic change. Since the set of emotion words is much smaller than the full
lexicon, we set the size of the neighbourhood to be $k=25$.

We first provide evidence that this variant of the nearest-neighbour measure
is also capable of capturing semantic change by showing 1) this measure is
positively correlated with the original nearest-neighbour measure, and 2) this
measure captures the negative relationship between frequency and semantic
change [20]. We obtain degrees of change under this variant measure by
following the same procedure described in Section 2. In the case of English
emotion words, the Pearson correlation between degrees of semantic change
measured by this variant and degrees obtained by the original measure is
$0.751$, $p<0.001$, $n=123$; the Pearson correlation between degrees of change
measured by the variant and frequency is $-0.489$, $p<0.001$, $n=123$. In the
case of French emotion words, the Pearson correlation between degrees of
change measured by this variant and degrees obtained by the original measure
is $0.604$, $p<0.001$, $n=112$; the Pearson correlation between degrees of
change measured by the variant and frequency is $-0.203$, $p=0.0318$, $n=112$.
These results suggest this variant is capable of replicating patterns of
change identified previously.

After validating this variant measure, we also repeated the analyses on
emotion semantic change described in the main text. Figure 11 shows a
significant negative correlation between prototypicality and degree of
semantic change: for English, $\rho=-0.535$, $p<0.001$, $n=123$; for French,
$\rho=-0.558$, $p=0.002$, $n=112$. Figure 12 shows multiple regression results
similar to the results presented in the main text. The adjusted $R^{2}$ for
English is 0.432, with $p<0.001$, $n=123$. Mean regression coefficients for
prototypicality ($\beta=-0.479$, $p<0.001$) and frequency ($\beta=-0.0356$,
$p<0.001$) remained negative and significant, whereas valence ($\beta=0.0112$,
$p=0.091$) is insignificant. Again, results hold similarly for French with the
adjusted $R^{2}=0.381$, $p<0.001$, $n=112$ (prototypicality $\beta=-0.600$,
$p<0.001$; frequency $\beta=-0.0208$, $p<0.001$; valence $\beta=-0.0015$,
$p=0.599$). Compared to the main results, we observed that prototypicality
remains a competitive predictor of semantic stability relative to frequency.

Figure 11: Scatter plots showing the negative correlations between emotion
prototypicality and rates of semantic change between the 1890s and 1990s, in
a) English and b) French. Each dot corresponds to an emotion term (with a
sample set of words annotated), and each band shows a 95% confidence interval
for regressions between prototypicality and rates of semantic change. Figure
12: Predictor coefficients from multiple regressions on rates of semantic
change. Error bars show standard error, and “n.s.”, “*”, “**”, “***” denote no
significance at $p<0.05$, and $p<0.05$, $p<0.01$, $p<0.001$ respectively. a)
shows results for English, and b) shows results for French.

###  D.2 Other factors of semantic change

In the main text, we tested prototypicality as a predictor of semantic
stability alongside frequency. Here we examine the role of prototypicality in
predicting semantic stability by controlling for three additional predictors:
1) the degree of polysemy of a word, 2) superordinate-subordinate relations
between emotion words, and 3) the age of acquisition (AoA) of a word. Firstly,
similar to frequency, we control for degree of polysemy because it is a
general predictor of semantic change which has been found to negatively
correlate with stability in meaning [20, 43]. Secondly, since one function of
prototypical emotion words is that they can help define more complex emotion
words [11] and this anchoring function may provide relative semantic
stability,444For example, suppose joy is part of the definition of
exhilaration; if the meaning of joy changed, the meaning of exhilaration will
necessarily change as well, but not vice versa. we examine superordinate-
subordinate relations (i.e., the hierarchy in the semantic category of
emotion) as a potential confounding variable influencing both prototypicality
and semantic change. Thirdly, since prototypical emotion words are relatively
well-defined and would be easy to learn, and that AoA is a known predictor of
stability in lexical change [46, 19], we control for AoA as a potential
mediator between prototypicality and semantic change.

Following [43], we operationalized the degree of polysemy of a word as the
number of senses the word had at the starting time $t_{1}=1890$ according to
the HTE [45]. To operationalize superordinate-subordinate relations, we used
WordNet [47] provided by NLTK [48]. Specifically, we constructed a directed
graph based on hypernym-hyponym relations, where the root is the sense for
feeling, and the other nodes correspond to the most frequent sense of an
emotion word (see Figure 13 for illustration). Then, we quantified a word’s
degree of subordination as its depth in the graph. For example, the word
thrill has a depth of 4, while joy has a depth of 2. Furthermore, to match the
historical period of our analyses, we used objective, test-based measurements
of AoA originally published by [49]. A digitized version of this data was
obtained from [50], where each entry contains a word form, its meaning, and
the age at which it was acquired. We computed the AoA of a word by taking the
average over all entries in which it appears. Due to the lack of analogous
French historical data, we only focused on English emotion words. We assumed
these hypernym-hyponym relations and AoA are stable over the past century.
After intersecting WordNet and AoA with our historical resources described in
the main text, we had a total of 109 English emotion words.

Figure 13: WordNet hierarchy of hypernyms and hyponyms for a) positive emotion
words and b) negative emotion words. Valence is determined by our data
described in Section 3.1.

We analyzed these factors using our materials and methods described in Section
3.1 and Section 3.2 of the main text. Specifically, we computed the rates of
change for every emotion concept $x$, $rate(x,1890,1990)$. Then, we performed
multiple regression using the following formula:

$\displaystyle rate(x,1890,1990)\sim$ $\displaystyle
p(x|c=emotion)+freq(x)+poly(x)+$ (5) $\displaystyle val(x)+depth(x)+AoA(x)$
(6)

where $poly(x)$ is the degree of polysemy of the word operationalized by
number of senses, $depth(x)$ is the depth of the concept in the hypernym-
hyponym graph we constructed, and $AoA(x)$ is the age at which the word was
acquired.

Figure 14 shows the multiple regression results suggesting the dominant roles
of prototypicality and frequency. The adjusted $R^{2}$ of the model is
$0.697$, with $p<0.001$, $n=109$; mean regression coefficients for
prototypicality ($\beta=-0.4704$, $p<0.001$) and frequency ($\beta=-0.0460$,
$p<0.001$) are significant, but for valence ($\beta=0.0049$, $p=0.267$),
number of senses ($\beta=0.0029$, $p=0.075$), depth ($\beta=-0.0057$,
$p=0.408$), and AoA ($\beta=0.0036$, $p=0.093$) it is insignificant. We
observe that prototypicality still has a significant, negative effect as
predicted by our hypothesis. We also observe that we can reproduce the finding
by [20] for frequency.

Figure 14: Predictor coefficients from multiple regressions on rates of
semantic change. Error bars follow the same layout as Figure 12.
Prototypicality is estimated using Equation 2 in the main text. Figure 15:
Predictor coefficients from multiple regressions on rates of semantic change.
Error bars follow the same layout as Figure 12. Prototypicality is based on
human ratings.

###  D.3 Human judgements of prototypicality

We repeated the analysis in the previous section by replacing estimated
prototypicality with empirical prototypicality ratings. We computed the rates
of change for every emotion concept $x$, $rate(x,1980,1990)$. Then, we
performed a multiple regression using the following formula:

$rate(x,1980,1990)\sim proto(x)+freq(x)+poly(x)+val(x)+depth(x)$ (7)

where $proto(x)$ is the prototypicality rating of $x$ obtained from [6].

Figure 15 shows the multiple regression results. The adjusted $R^{2}$ of the
model is $0.524$, with $p<0.001$, $n=109$; mean regression coefficients for
prototypicality ($\beta=-0.0887$, $p=0.025$), frequency ($\beta=-0.0793$,
$p<0.001$), number of senses ($\beta=0.0097$, $p=0.002$), and valence
($\beta=0.0222$, $p=0.014$) are significant, but for depth ($\beta=-0.0250$,
$p=0.077$) and AoA ($\beta=0.0062$, $p=0.150$) it is insignificant. We observe
that empirical prototypicality also has a significant, negative effect, albeit
the effect size is smaller than previously.

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