# On bifurcation of statistical properties of partially hyperbolic
endomorphisms

Masato Tsujii  Department of Mathematics, Kyushu university, 744 Motooka,
Nishi-ku, Fukuoka, 819-0395, JAPAN  and  Zhiyuan Zhang  CNRS, Institut Galilée
Universitat Paris 13, 99, Avenue Jean-Baptiste clément 93430 - Villetaneuse,
FRANCE

###### Abstract.

We give an example of a path-wise connected open set of $C^{\infty}$ partially
hyperbolic endomorphisms on the $2$-torus, on which the SRB measure exists for
each system and varies smoothly depending on the system, while the sign of its
central Lyapunov exponent does change.

MT is supported by JSPS KAKENHI Grant Number 21H00994.

##  1\. Introduction

We give an example of a path-wise connected open set of $C^{\infty}$ partially
hyperbolic endomorphisms111We refer [3] for a few technical terms that are
used in this paper such as partially hyperbolic endomorphism, Lyapunov
exponents, etc. on the $2$-torus, on which the SRB measure exists for each
system and varies smoothly depending on the system, while the sign of its
central Lyapunov exponent does change. Since the Lyapunov exponents of the SRB
measure are major characteristics that describe local geometric structure of
dynamics almost everywhere, we tend to think that the dynamical systems
exhibit drastic bifurcations when the Lyapunov exponents of the SRB measure
change their signs. However our example tells that this is not always the case
and suggests that bifurcations of global statistical properties of partially
hyperbolic dynamical systems may be much milder than what we expect from our
knowledge on low dimensional dynamical systems with one dimension of
unstability such as Hénon maps.

The open subset of $C^{\infty}$ partially hyperbolic endomorphisms in our
example is a small open neighborhood of a one-parameter family of skew
products of circle endomorphisms over an angle-multiplying map. Smooth
dependence of the SRB measure in a similar setting is already studied in [4]
partly based on the argument in [1, 2]. In this paper, we construct an example
in which we can observe the switching of the sign for the central Lyapunov
exponent of the SRB measure. In the last section, we present some results of
numerical computations that illustrate the situation in our example.

##  2\. Result

We write $\mathbb{T}=\mathbb{R}/\mathbb{Z}$ for the unit circle and
$\mathbb{T}^{2}$ for 2-dimensional torus. We consider the iteration of a
$C^{\infty}$ locally diffeomorphic map $F:\mathbb{T}^{2}\to\mathbb{T}^{2}$ as
a discrete dynamical system. The Perron-Frobenius operator

(1) $\mathcal{P}:C^{r}(\mathbb{T}^{2})\to
C^{r}(\mathbb{T}^{2}),\quad\mathcal{P}u(p)=\sum_{p^{\prime}\in\mathbb{T}^{2}:F(p^{\prime})=(p)}\frac{u(p^{\prime})}{|\det
DF(p^{\prime})|}$

expresses the action of $F$ on the space of densities, where
$C^{r}(\mathbb{T}^{2})$ denotes the space of $C^{r}$ functions on
$\mathbb{T}^{2}$.

An invariant Borel probability measure $\mu$ is said to be an SRB measure if
almost every point on $\mathbb{T}^{2}$ with respect to the Lebesgue measure is
generic for $\mu$. We consider a partially hyperbolic endomorphism $F$ on
$\mathbb{T}^{2}$ and suppose that $F$ admits an ergodic SRB measure $\mu_{F}$.
Then the Lyapunov exponents take constant values

$\chi^{c}(\mu_{F})\lneq\chi^{u}(\mu_{F})\quad\text{with}\quad\chi^{u}(\mu_{F})>0$

at almost every point with respect to $\mu_{F}$ and also with respect to the
Lebesgue measure.

Our main result is stated as follows.

######  Theorem 1.

For any $r>0$, there exists a path-wise connected $C^{\infty}$ open subset
$\mathcal{U}$ of $C^{\infty}(\mathbb{T}^{2},\mathbb{T}^{2})$ that consists of
locally diffeomorphic partially hyperbolic endomorphisms, a Hilbert space

(2) $C^{\infty}(\mathbb{T}^{2})\subset\mathcal{H}\subset
C^{r}(\mathbb{T}^{2})$

and a constant $0<\rho<1$ such that

  * (a)

The Perron-Frobenius operator $\mathcal{P}_{F}$ for $F\in\mathcal{U}$
restricts to a bounded operator

(3) $\mathcal{P}_{F}:\mathcal{H}\to\mathcal{H}.$

  * (b)

The restriction (3) has a simple eigenvalue $1$ and the rest of its spectral
set is contained in the disk $|z|<\rho<1$.

  * (c)

$F\in\mathcal{U}$ admits a unique SRB measure $\mu_{F}=\rho_{F}\mathrm{Leb}$
where $\rho_{F}\in\mathcal{H}$ is the eigenfunction of $\mathcal{P}_{F}$ for
the simple eigenvalue $1$.

  * (d)

The SRB measure $\mu_{F}$ depends on $F\in\mathcal{U}$ smoothly in the sense
that, for any $C^{\infty}$ one-parameter family $G_{t}$ of maps in
$\mathcal{U}$ and $\psi\in C^{\infty}(\mathbb{T}^{2})$, the correspodence
$t\mapsto\int\psi d\mu_{G_{t}}$ is $C^{r}$.

  * (e)

There are $F_{\sigma}\in\mathcal{U}$ for $\sigma\in\\{+,-\\}$ such that the
central Lyapunov exponent $\chi^{c}(\mu_{F_{\sigma}})$ has the same sign as
$\sigma$.

The claims of the theorem above imply that, if we take any $C^{\infty}$ one-
parameter family $G_{t}$ that connects $F_{-}$ and $F_{+}$ in $\mathcal{U}$,
we observe that the SRB measure $\mu_{G_{t}}$ changes smoothly with respect to
$t$ while the central Lyapunov exponent will change its sign at some
parameter.

##  3\. Circle endomorphisms

We first consider the doubling map on the circle $\mathbb{T}$:

$f_{0}:\mathbb{T}\to\mathbb{T},\quad f_{0}(y)=2y\quad\mod\mathbb{Z}.$

Below we deform the map $f_{0}$ in order to make a neutral fixed point in a
small neighborhood of $0\in\mathbb{T}$.

Let $\varphi:\mathbb{R}\to\mathbb{R}$ be a $C^{\infty}$ map with the following
properties:

  1. (i)

$0\leq\varphi(y)\leq 1$ and $|\varphi^{\prime}(y)|\leq 4/3$ for
$y\in\mathbb{R}$,

  2. (ii)

$\varphi(y)=0$ for $y\notin[1/10,1]$,

  3. (iii)

$\varphi(1/2)=1/2$, $\varphi^{\prime}(1/2)=1$, $\varphi^{\prime\prime}(1/2)<0$
and

  4. (iv)

$\varphi(y)<y$ for $y\in(0,1)\setminus\\{1/2\\}$.

For a small real number $\varepsilon>0$, we define

$f_{\varepsilon}:\mathbb{T}\to\mathbb{T},\quad
f_{\varepsilon}(y)=\begin{cases}f_{0}(y)-\varepsilon\cdot\varphi(\varepsilon^{-1}y),&\quad\text{if
$y\in[0,\varepsilon]$};\\\ f_{0}(y),&\quad\text{otherwise.}\end{cases}$

For the dynamics of $f_{\varepsilon}$, we observe that there are only two
fixed points $0$ and $P=\varepsilon/2$: $0$ is a hyperbolic repelling fixed
point and $P=\varepsilon/2$ is a one-sided attracting neutral fixed point with
immediate basin $(0,P]$.

We henceforth suppose that the parameter $\varepsilon>0$ is sufficiently
small, say $0<\varepsilon<1/100$. Then, for $a\in\mathbb{R}$, we set

$f_{\varepsilon,a}:\mathbb{T}\to\mathbb{T},\quad
f_{\varepsilon,a}(y)=f_{\varepsilon}(y)+a\varepsilon.$

From the assumption (iv), we have

$\frac{2}{3}\leq f^{\prime}_{\varepsilon,a}(y)\leq\frac{10}{3}\quad\text{for
any }y\in\mathbb{T}.$

Hence, if $a\geq 1$, we have that
$f_{\varepsilon,a}^{-1}([0,\varepsilon])\cap(0,\varepsilon)=\emptyset$ and
hence

(4) $(f_{\varepsilon,a}^{2})^{\prime}(y)\geq
2\cdot\frac{2}{3}=\frac{4}{3}>1\quad\text{for any }y\in\mathbb{T}.$

The family $a\mapsto f_{\varepsilon,a}$ exhibits the saddle-node bifurcation
of the fixed point $0$ at the parameter $a=0$. It is not difficult to check
that $f_{\varepsilon,a}$ is uniformly expanding if $0<a\leq 2$. If $a<0$ and
$|a|$ is sufficiently small, then $f_{\varepsilon,a}$ admits three fixed
points

$P_{0}=-a\varepsilon<P_{-}<P_{+}$

in a small neighborhood of $0$, where $P_{0}$ and $P_{+}$ are hyperbolic
repelling while $P_{-}$ is hyperbolic attracting. The immediate basin of the
hyperbolic attracting fixed point $P_{-}$ is the interval $B=(P_{0},P_{+})$
and we have

$\lim_{a\to-0}P_{0}=0,\quad\lim_{a\to-0}P_{-}=\lim_{a\to-0}P_{+}=\frac{\varepsilon}{2}.$

\begin{overpic}[scale={.10}]{intermittent.eps} \put(85.0,21.0){$\varepsilon$}
\put(81.0,20.0){$\frac{\varepsilon}{2}$} \put(20.0,-2.0){$0$}
\put(-3.0,22.0){$0$} \end{overpic} Figure 1. The graph of the function
$f_{\varepsilon}$.

##  4\. Skew products over angle multiplying maps

We consider the dynamics of perturbations of the skew product

$F_{\varepsilon,a,\delta,m}:\mathbb{T}^{2}\to\mathbb{T}^{2},\quad
F_{\varepsilon,a,\delta,m}(x,y)=(mx,f_{\varepsilon,a}(y)+\delta\varepsilon\cos
2\pi x)$

where $m$ is a positive integer and $\delta>0$ is a positive real parameter.
In the following, we suppose that $r>0$ is a given integer. We suppose that
the constants $\varepsilon>0$ and $\delta>0$ are small, say
$\varepsilon,\delta\in(0,1/100)$. We will also fix $m$ as a large constant so
that the conclusion of Theorem 2 below holds true. Since we regard
$F_{\varepsilon,a,\delta,m}$ as a one parameter family with parameter
$a\in[-2\delta,2]$, we henceforth write $F_{a}$ for
$F_{\varepsilon,a,\delta,m}$.

###  4.1. Quasi-compactness of $\mathcal{P}$

We adapt the argument in [4] to get the next theorem. Since the situation is
only a little different from that in [4], we give a brief account on its proof
in Section 5.

######  Theorem 2.

If we let $m$ be sufficiently large depending on the parameters $r$,
$\varepsilon$, $\delta$ and a given $0<\rho_{0}<1$, there exists a Hilbert
space $\mathcal{H}$ satisfying (2) and a $C^{\infty}$ neighborhood
$\mathcal{U}\subset C^{\infty}(\mathbb{T}^{2},\mathbb{T}^{2})$ of the family
$\mathcal{F}=\\{F_{a}=F_{\varepsilon,a,\delta,m},a\in[-2\delta,2]\\}$, such
that the Perron-Frobenius operator $\mathcal{P}_{F}:\mathcal{H}\to\mathcal{H}$
for $F\in\mathcal{U}$ is bounded and its essential spectral radius is bounded
by $\rho_{0}$.

Further, if $1$ is a simple eigenvalue of the Perron-Frobenius operator
$\mathcal{P}_{F}:\mathcal{H}\to\mathcal{H}$ for every $F\in\mathcal{F}$, then,
by letting the neighborhood $\mathcal{U}$ be smaller, we may suppose that the
same is true for all $F\in\mathcal{U}$ and the positive eigenfunction
$\rho_{F}\in\mathcal{H}$ for the simple eigenvalue $1$, determined by the
condition $\int\rho_{F}d\mathrm{Leb}=1$, depends on $F$ smoothly in the
following sense: for any $C^{\infty}$ one-parameter family $G_{t}$ of maps in
$\mathcal{U}$ and $\psi\in C^{\infty}(\mathbb{T}^{2})$, the correspodence
$t\mapsto\int\psi d\mu_{G_{t}}=\int\psi\rho_{G_{t}}d\mathrm{Leb}$ is $C^{r}$.

######  Remark 3.

We can not let $r=\infty$ in our construction because it is essential to take
$m$ large enough depending on $r$.

###  4.2. Simplicity of the eigenvalue $1$

We show the following theorem for the family $\mathcal{F}=\\{F_{a}\mid
a\in[-2\delta,2]\\}$.

######  Theorem 4.

For any $a\in[-2\delta,2]$, the principal eigenvalue $1$ of
$\mathcal{P}_{F}:\mathcal{H}\to\mathcal{H}$ is simple and there is no other
eigenvalue on the unit circle. The eigenfunction $\rho_{a}\in\mathcal{H}$ for
the simple eigenvalue $1$ satisfying $\int\rho_{a}d\mathrm{Leb}=1$ is the
density of the SRB measure $\mu_{a}$ with respect to the Lebesgue measure.

###### Proof.

We consider the following two cases for $a\in[-2\delta,2]$ separately:

$\textrm{(i)}\;a+\delta>0,\quad\textrm{(ii)}\;a+\delta\leq 0.$

### Case (i)

First we prove

######  Lemma 5.

In Case (i), we have $U_{\infty}:=\cup_{n\geq 0}F_{a}^{n}(U)=\mathbb{T}^{2}$
for any non-empty open subset $U$ on $\mathbb{T}^{2}$.

###### Proof.

Since $F_{a}$ is expanding in the horizontal (or $x$-) direction, we have that
$U_{\infty}\cap(\\{0\\}\times\mathbb{T})\neq\emptyset$. The map $F_{a}$
restricted to $\\{0\\}\times\mathbb{T}$ can be identified with
$f_{\varepsilon,a+\delta}$. From the assumption, we have $a+\delta>0$ and
hence $f_{\varepsilon,a+\delta}$ is uniformly expanding, provided that
$\delta>0$ is sufficiently small.

######  Remark 6.

The last claim is not completely obvious but easy to check. Let
$f=f_{\varepsilon,a}$. To show that $f$ is uniformly expanding, it is enough
to show that there exists $n>0$ for any $x\in\mathbb{T}$ such that
$(f^{n})^{\prime}(x)>1$. This holds obviously with $n=1$ for $x$ on the
outside of $(0,\varepsilon)$. For a point $x\in(0,\varepsilon)$, we let $m$ be
the smallest integer such that $f^{m}(x)\notin(0,\varepsilon)$. By the
elementary estimates on intermittent one dimensional map, we see that
$(f^{m})^{\prime}(x)>h>0$ for some constant $h>0$ independent of $a>0$ and
$\varepsilon>0$ (as far as they are sufficiently small). By letting
$\varepsilon>0$ be sufficiently small, we may suppose that the orbit starting
from $f^{m}(x)$ will not return to $(0,\varepsilon)$ for arbitrarily long time
and therefore we can find $n>m$ such that $(f^{n})^{\prime}(x)>1$.

Hence we have $U_{\infty}\supset\\{0\\}\times\mathbb{T}$. Again, using the
fact that $F_{a}$ is expanding in the horizontal direction, we obtain the
claim $U_{\infty}=\mathbb{T}^{2}$. ∎

Suppose that $\rho\in\mathcal{H}$ is an eigenfunction for an eigenvalue on the
unit circle. Then we have $|\mathcal{P}^{n}\rho|=|\rho|=\mathcal{P}^{n}|\rho|$
for $n\geq 1$. From the last lemma, this holds only if
$\rho=e^{i\theta}|\rho|$ for some $\theta\in[0,2\pi)$ and therefore we may
suppose $\rho\geq 0$. This implies that there is no eigenvalue on the unit
circle other than $1$. By the same reason, the geometric multiplicity of the
eigenvalue $1$ should be $1$. Further, since $\mathcal{P}$ preserves the
integral of functions with respect to the Lebesgue measure, we conclude that
the algebraic multiplicity is not greater than $1$.

Let $\rho_{F_{a}}\in\mathcal{H}\subset C^{r}(\mathbb{T}^{2})$ be the
eigenfunction of $\mathcal{P}_{F_{a}}$ for the simple eigenvalue $1$. We may
and do suppose that $\rho_{F_{a}}$ is non-negative and
$\int\rho_{F_{a}}\,d\mathrm{Leb}=1$. Then the measure
$\nu_{F_{a}}:=\rho_{F_{a}}\,\mathrm{Leb}$ is ergodic since $\mathcal{P}^{n}u$
converges to a constant multiple of $\rho_{F_{a}}$ for any $u\in\mathcal{H}$.
Since $\rho_{F_{a}}\in C^{r}(\mathbb{T})$, there is an open subset
$U\subset\mathbb{T}^{2}$ on which $\rho_{F_{a}}>0$ and therefore almost every
point in $U$ is generic for $\mu_{F_{a}}$. As $F_{a}$ is locally
diffeomorphic, almost every point on $F_{a}^{n}(U)$ with $n\geq 0$ is generic
for $\mu_{F_{a}}$. Since $\cup_{n\geq 0}F_{a}^{n}(U)=\mathbb{T}^{2}$ as we
showed in Lemma 5, we conclude that almost every point on $\mathbb{T}^{2}$ is
generic for $\mu_{F_{a}}$. This finishes the proof of the theorem in Case (i).

### Case (ii)

Note that $a\leq-\delta<0$ in this case. The region

$W=\mathbb{T}\times((\delta-a)\varepsilon,\varepsilon/2)$

satisfies $F_{a}(W)\subset W$ and the iteration of $F_{a}$ is (non-uniformly)
contracting on the fibers $\\{x\\}\times((\delta-a)\varepsilon,\varepsilon/2)$
for $x\in\mathbb{T}$.

######  Remark 7.

The choice of the interval $(\delta-a)\varepsilon,\varepsilon/2)$ in the
definition of $W$ is made as follows: The left end point
$y_{-}=(\delta-a)\varepsilon$ is the unique point in $(0,\varepsilon/10)$
satisfying $f_{\varepsilon,a-\delta}(y_{-})=y_{-}$. (Recall the condition (ii)
in the definition of the function $\varphi$.) The right end point
$y_{+}=\varepsilon/2$ is the neutral fixed point of $f_{\varepsilon,0}$, which
satisfies $f_{\varepsilon,a+\delta}(y_{+})\leq y_{+}$ when $a+\delta\leq 0$.

Hence there exists a unique mixing $F_{a}$-invariant measure $\mu_{F_{a}}$
supported in $W$ such that Lebesgue almost every point on $W$ is generic for
$\mu_{F_{a}}$.

Writing $\pi_{2}:\mathbb{R}^{2}\to\mathbb{R}$ for the projection to the second
component, we have $\pi_{2}\partial_{y}(F_{a}\circ F_{a})(p)>1$ on the
complement of $W$, with only one exception $p=(0,\varepsilon/2)$ when
$a+\delta=0$. Hence the intersection of the complement

$C=\mathbb{T}^{2}\setminus\cup_{n\geq 0}F_{a}^{-n}(W)$

with any fiber $\\{x\\}\times\mathbb{T}$ can not contain any non-trivial
interval.

We next show that the complement $C$ is of null Lebesgue measure. Suppose that
$C$ has positive Lebesgue measure and write $\mathbf{1}_{C}$ for the
characteristic function of it. Then we can find a weak limit point $\rho$ of
the sequence $(1/n)\sum_{k=0}^{n-1}\mathcal{P}^{k}\mathbf{1}_{C}$. By
approximating $\mathbf{1}_{C}$ by $C^{\infty}$ function in $L^{1}$ sense and
using the spectral property of $\mathcal{P}$ in Theorem 2, we see that $\rho$
belongs to $\mathcal{H}\subset C^{r}(\mathbb{T}^{2})$ and is supported on $C$.
But this is impossible because $C$ has no interior point.

Since the complement $C$ is of null Lebesgue measure, almost every point on
$\mathbb{T}^{2}$ is generic for the mixing measure $\mu_{F}$. Clearly this
implies the conclusion of the theorem. ∎

Finally we prove the following theorem on the central Lyapunov exponent of the
SRB measure $\mu_{F_{a}}$ for $F_{a}$ with $a\in[-2\delta,2]$. Note that we
always assume that $\varepsilon>0$ and $\delta>0$ are small.

######  Theorem 8.

(a) If $a+\delta<0$, the central Lyapunov exponent $\chi^{c}(\mu_{F_{a}})$ is
negative.
(b) If $a\geq 1$, the central Lyapunov exponent $\chi^{c}(\mu_{F_{a}})$ is
positive.

###### Proof.

(a) As we observed in the proof of Theorem 4 in Case (ii), there is a unique
SRB measure $\mu_{F_{a}}$ whose support is contained in $W$ and its central
Lyapunov exponent is negative.
(b) By (4), the map $F_{a}$ is expanding along the fibers in this case and
therefore the central Lyapunov exponent of the SRB measure is positive,
provided that $\delta>0$ is sufficiently small. ∎

##  5\. The proof of Theorem 2

We can obtain the proof of Theorem 2 by following the argument in [4] with
slight modifications. Below we explain briefly how we modify the argument in
[4].

First we check a transversality condition. We consider the constant cones in
the tangent bundle

$\mathbf{C}=\bigcup_{p\in\mathbb{T}^{2}}\mathbf{C}_{p}=\\{(p,v)=((x,y),(v_{x},v_{y}))\in
T\mathbb{T}^{2}\mid|v_{y}|\leq C_{0}\delta\varepsilon|v_{x}|\\}$

where we fix a large constant $C_{0}$ so that
$DF(\mathbf{C})\subset\mathbf{C}$. For given $p\in\mathbb{T}^{2}$ and
$q,q^{\prime}\in\mathbb{T}^{2}$, we write $q\pitchfork q^{\prime}$ if

$DF_{q}(\mathbf{C})\cap DF_{q^{\prime}}(\mathbf{C})=\\{0\\}.$

We define

$\mathbf{m}(F)=\frac{1}{(2/3)\cdot m}\cdot\sup_{p\in\mathbb{T}^{2}}\sup_{q\in
F^{-1}(p)}\\#\\{q^{\prime}\in F^{-1}(p)\mid q^{\prime}\pitchfork q\\}$

where $(2/3)\cdot m$ stands for a lower bound of $\det dF$. We can check the
following lemma by simple computations.

######  Lemma 9.

The quantity $\mathbf{m}(F)$ converges to $0$ when we let $m$ go to infinity
and the convergence is uniform for sufficiently small $\varepsilon>0$,
$\delta>0$ and any $a\in[-2,2]$.

We then follow the argument in [4] almost literally, noting that
$\mathbf{m}(F)$ corresponds to $m(f,1)$ defined in [4, Sec.3] and that we just
consider the first iteration (or the case $n=1$ there). Another difference of
our setting from that in [4] is the point that we consider non-linear
endomorphisms on the fibers while they were rigid rotations in [4]. But, since
we just consider the first iteration, if we take sufficiently fine local
charts and partition of unity in the argument in [4, Sec.4], it is direct to
get a parallel argument in our setting. Then the claim corresponding to [4,
Prop.3] and Hennion’s theorem gives the former claim of Theorem 2. We can
deduce the latter claim using the abstract perturbation theorem in [2, Sec.8]
about perturbation of transfer operators. For this we again follow the
argument in [4, Sec.4.4].

##  6\. Some numerical experiments

We present some results of numerical experiments related to the claim of the
main theorem. For simplicity of computation, we consider a similar but
slightly different setting from that in the previous sections. We consider a
$C^{\infty}$ map $f:\mathbb{T}\to\mathbb{T}$ defined by

$f(y)=2y-\frac{\sin(2\pi y)+\cos(2\pi x)-1}{2\pi}\quad\mod\mathbb{Z}.$

It has a neutral fixed point at $0$ and its dynamics is of very similar nature
to that of $f_{\varepsilon}$ in Section 3. (See also Figure 1.)

Then we consider a family of dynamical systems
$F_{a}:\mathbb{T}^{2}\to\mathbb{T}^{2}$ defined by

$F_{a}(x,y)=\left(7x,f(y)+\delta\cdot\cos(2\pi x)+a\right)\quad\text{for
}a\in[-2\delta,2\delta].$

where we set $\delta=10^{-2}$. In Figure 2, we compute the approximate central
Lyapunov exponent at a randomly chosen point by iterating $F$ for $10^{6}$
times and plot it against the parameters $-0.02\leq a\leq 0.02$ (resp.
$-0.004\leq a\leq 0.004$) with step $10^{-3}$ (resp. $10^{-4}$). We observe
that the (central) Lyapunov exponent varies smoothly and changes its sign at a
parameter $-0.001<a_{0}<0$.

We also plot an orbit of randomly chosen initial point at the parameters
$a=-0.02,-0.006,-0.003,-0.002$. (We draw the orbit from time $10^{3}$ to time
$10^{6}$.) At the parameter $a=-0.02$, we observe that the orbits are trapped
by a horizontal zonal region. When the parameter $a$ crosses the value
$-\delta=-0.01$, we expect that the orbits start to spread over the whole
space $\mathbb{T}^{2}$ and, as the parameter $a$ gets large, the density of
the orbits become more uniform. But when the value of $a$ is close to $-0.01$,
it is difficult to detect this phenomenon because only very small portion of
orbits go out of (the ruin of) the attracting region and return to it again
soon. (See the picture for the parameter $a=-0.006$ in Figure 3.)

|
---|---
Figure 2. The central Lyapunov exponent of the SRB measure as a function of the parameters $a$ close to $0$. We draw the graph with the domain $[-0.02,0.02]$ and $[-0.004,0.004]$ respectively on the left and right pictures. $a=-0.02$ |  $a=-0.006$
---|---
$a=-0.003$ |  $a=-0.002$
Figure 3. Plots of orbits of $F_{a}$ at a few values of the parameter $a$

## References

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