Conjugacy problem of strictly monotone maps with only one jump discontinuity

TL;DR

This paper studies the conjugacy problem for strictly monotone maps with a single jump discontinuity, providing conditions for conjugacy, methods for construction, and criteria for smoothness.

Contribution

It offers new necessary and sufficient conditions for conjugacy and methods to construct all conjugacies for these maps, including smoothness criteria.

Findings

Established conditions for conjugacy of maps with one jump discontinuity.

Developed methods to construct all conjugacies explicitly.

Provided criteria for the $C^1$ smoothness of conjugacies.

Abstract

The conjugacy problem is one of the central questions in iteration theory. As far as we, for discontinuous strictly monotone maps there is no complete result. In this paper, we investigate the conjugacy problem of strictly monotone maps with only one jump discontinuity. We give some sufficient and necessary conditions for the conjugacy relationship. And we present some methods to construct all conjugacies. Furthermore, we present the conditions to guarantee $C^1$ smoothness of these conjugacies.

Equations

$$\phi \circ f=g\circ \phi .$$

$$\phi (\underset{x\to t^{-}}{\lim }f(x))=\underset{x\to t^{-}}{\lim }g(\phi (x)).$$

$$\text{(i)}f(t)<t\mathrm{and}g(s)<s,\text{(ii)}f(t)<t\mathrm{and}g(s)>s,$$

$$\text{(iii)}f(t)>t\mathrm{and}g(s)<s,\text{(iv)}f(t)>t\mathrm{and}g(s)>s.$$

$$f_l(x):$$

$$f_r(x):$$

$$g_l(x):$$

$$g_r(x):$$

$$(a,t]={\cup }_{n=0}^{+\infty }{I}_n,[t,b)={\cup }_{n=0}^{+\infty }{J}_n,$$

$${\phi }_l(x):$$

$${\phi }_r(x):$$

$${\phi }_l(f_l(x))=g_l^{n+1}\circ {\phi }_{l_0}\circ f_l^{-n-1}(f_l(x))=g_l\circ g_l^n\circ {\phi }_{l_0}\circ f_l^{-n}(x)=g_l({\phi }_l(x)),$$

$${\phi }_l(f_l(a))=g({\phi }_l(a)).$$

$$\underset{h\to 0^{+}}{\lim }{\phi }_l(f_l(t)+h)=\underset{h\to 0^{+}}{\lim }{\phi }_{l_0}(f_l(t)+h)={\phi }_{l_0}(f_l(t))={\phi }_l(f_l(t)),$$

$$\underset{h\to 0^{+}}{\lim }{\phi }_l(f_l(t)-h)=\underset{h\to 0^{+}}{\lim }{g}_l\circ {\phi }_{l_0}\circ f_l^{-1}(f_l(t)-h)=g_l\circ {\phi }_{l_0}\circ f_l^{-1}(f_l(t))={\phi }_l(f_l(t)).$$

$$\underset{h\to 0^{+}}{\lim }{\phi }_l(f_l^{n+1}(t)+h)=\underset{h\to 0^{+}}{\lim }{\phi }_{l_n}(f_l^{n+1}(t)+h)={\phi }_{l_n}(f_l^{n+1}(t))={\phi }_l(f_l^{n+1}(t))$$

$$\underset{h\to 0^{+}}{\lim }{\phi }_l(f_l^{n+1}(t)-h)$$

$$\underset{n\to \infty }{\lim }{\phi }_l(f_l^n(t))=\underset{n\to \infty }{\lim }{g}_l^n({\phi }_l(t))=\underset{n\to \infty }{\lim }{g}_l^n(s)=c=\phi (a).$$

$$\phi (x):$$

$${\tilde{\phi }}_l(x):$$

$${\tilde{\phi }}_r(x):$$

$$\tilde{\phi }(x):$$

$$\text{(i)}f(t)<t\mathrm{and}g(s)<s,\text{(ii)}f(t)<t\mathrm{and}g(s)>s,$$

$$\text{(iii)}f(t)>t\mathrm{and}g(s)<s,\text{(iv)}f(t)>t\mathrm{and}g(s)>s.$$

$$f_e(x):$$

$$f_v(x):$$

$$g_e(x):$$

$$g_v(x):$$

$$(a,b)={\cup }_{k=0}^{+\infty }[f_e^{2k+2}(t),f_e^{2k}(t)]\cup [t,f_e(t)]\cup {\cup }_{k=0}^{+\infty }[f_e^{2k+1}(t),f_e^{2k+3}(t)].$$

$${\psi }_1(x):$$

$${\psi }_2(x):$$

$$\psi (x):$$

$${\tilde{f}}_e(x):$$

$${\tilde{f}}_v(x):$$

$${\tilde{g}}_e(x):$$

Full text

1 Introduction

Let $I$, $J$ be two closed intervals, and $f:I\to I$ and $g:J\to J$ be two self-maps. We say that $f$ and $g$ are topologically conjugate if there exists a homeomorphism $\varphi:I\to J$ satisfying the conjugacy equation

[TABLE]

Here $\varphi$ is called a conjugacy from $f$ to $g$.

The conjugacy relation is an important equivalent relation in dynamical systems, and it is used for topological classification and simplification of dynamical systems and functional equations. In general, the conjugacy problem contains three aspects. The first one is how to determine any two maps in some family to be topologically conjugate. The second is how to construct all conjugacies. The last is what about the smooth of every conjugacy. Many works are devoted to the conjugacy problem of continuous interval maps (i.e., [1, 6, 7, 8, 10, 12, 13]).

However, there is a few work for the conjugacy problem of discontinuous maps, for example, Lorenz map, cf. [2, 3, 4, 5, 9, 11]. As far as we, even for discontinuous strictly monotone maps there is no complete result. Consider a family of simple discontinuous strictly monotone maps with only one jump discontinuity. It is interesting to consider how this jump discontinuity affects the conjuacy relationship. One can see that there exist three main factors: (i) the value of functions at the jump discontinuity, (ii) the position relationship between the image of functions and the diagonal, (iii) the right and left limits at the discontinuity.

In this paper, we firstly give some necessary conditions for the cojugacy relationship in the next section. Further, we distinguish between increasing conjugacies and decreasing conjugacies. Section 3 is devoted for sufficient and necessary conditions for these discontinuous maps, and construct all conjugacies. In section 4, we give some sufficient conditions under which these conjugacies are $C^{1}$ smooth. In the final section, two examples are presented.

2 Preliminaries

We only consider the following representative cases, others can be discussed similarly. Let $I:=[a,b]$ and $\mathcal{A}_{t}(I)$ denote the family of strictly increasing functions $f$ with only one jump discontinuous point $t\in(a,b)$ satisfying (i) $f(t-0)\leq f(t)\leq f(t+0)$ and (ii) $f$ has exactly two attractive fixpoints $a$ and $b$. See Figs 1-4.

Let $\mathcal{B}_{t}(I)$ denote the family of strictly decreasing functions $f$ with only one jump discontinuous point $t\in(a,b)$ satisfying (i) $f(t-0)\leq f(t)\leq f(t+0)$ and (ii) $f$ has exactly two periodic points $a$ and $b$, both of which are period 2 points. See Figs 5-8.

Let $J:=[c,d]$ and $c<s<d$. Some necessary conditions for the conjugacy relationship are given below.

Lemma 2.1

Let $f\in\mathcal{A}_{t}(I)$ and $g\in\mathcal{A}_{s}(J)$ be topologically conjugate via a homeomorphism $\varphi$. Then $\varphi(f^{n}(t))=g^{n}(s)$ for $n=0,1,2,...$.

Proof. Since $\varphi\circ f$ is not continuous at point $t$, and $g\circ\varphi=\varphi\circ f$, $g$ is not continuous at $\varphi(t)$. Therefore $\varphi(t)=s$. Further, $\varphi(f(t))=g(\varphi(t))=g(s)$. By induction, $\varphi(f^{n}(t))=g^{n}(s)$ for $n=0,1,2,...$. $\Box$

Lemma 2.2

Let $f\in\mathcal{A}_{t}(I)$ and $g\in\mathcal{A}_{s}(J)$ be topologically conjugate. Then there are only two cases:

(C1)

$f(t-0)<f(t)<f(t+0)$* and $g(s-0)<g(s)<g(s+0)$;*

(C2)

$f(t)=f(t-0)$* and $g(s)=g(s-0)$, or $f(t)=f(t-0)$ and $g(s)=g(s+0)$, or $f(t)=f(t+0)$ and $g(s)=g(s-0)$, or $f(t)=f(t+0)$ and $g(s)=g(s+0)$.*

Proof. We only prove the case $f(t-0)<f(t)<f(t+0)$ and $g(s)=g(s-0)$. Others are similar. Without loss of generality, assume that $\varphi$ is an increasing conjugacy from $f$ to $g$. One can see that $\varphi(f(t-0))=g(\varphi(t-0))=g(s)=g(s-0)$. It follows from Lemma 2.1 that $\varphi(f(t))=g(\varphi(t))=g(s)=g(s-0)$. Then $\varphi(f(t))=\varphi(f(t-0))$. Since $\varphi$ is a homeomorphism, we have $f(t-0)=f(t)$. This is a contradiction. $\Box$

Lemma 2.3

Let $f\in\mathcal{A}_{t}(I)$ and $g\in\mathcal{A}_{s}(J)$ be topologically conjugate via a homeomorphism $\varphi$.

(i)

If $(f(t)-t)(g(s)-s)>0$, then $\varphi$ is increasing and goes through points $(a,c)$, $(b,d)$, $(t,s)$, $(f(t),g(s))$, $(f(t-0),g(s-0))$ and $(f(t+0),g(s+0))$.

(ii)

If $(f(t)-t)(g(s)-s)<0$, then $\varphi$ is decreasing and goes through points $(a,d)$, $(b,c)$, $(t,s)$, $(f(t),g(s))$, $(f(t+0),g(s-0))$ and $(f(t-0),g(s+0))$.

(iii)

If $f(t)=f(t\pm 0)$ and $g(t)=g(t\pm 0)$, then $\varphi$ is increasing and goes through points $(a,c)$, $(b,d)$, $(t,s)$, $(f(t-0),g(s-0))$ and $(f(t+0),g(s+0))$.

(iv)

If $f(t)=f(t\pm 0)$ and $g(t)=g(t\mp 0)$, then $\varphi$ is decreasing and goes through points $(a,d)$, $(b,c)$, $(t,s)$, $(f(t+0),g(s-0))$ and $(f(t-0),g(s+0))$.

Proof. We only prove the fact (i). Others are similar. For fact (i), we will prove that $\varphi$ is increasing by contradiction. We only consider the case $f(t)<t$ and $g(s)<s$, the other case is similar. Assume that $\varphi$ is decreasing, then $\varphi(f(t))>\varphi(t)$. Thus $g(\varphi(t))>\varphi(t)$, i.e., $g(s)>s$, which contradicts the condition that $g(s)<s$. Therefore, $\varphi$ is increasing.

From $\varphi(f(a))=g(\varphi(a))$, we have $\varphi(a)=g(\varphi(a))$. Thus $\varphi(a)=c$ or $\varphi(a)=d$. Since $\varphi$ is strictly increasing, it is clear that $\varphi(a)=c$. Similarly, we have $\varphi(b)=d$.

Since $\lim_{x\rightarrow t^{-}}\varphi(f(x))=\lim_{x\rightarrow t^{-}}g(\varphi(x))$, we have

[TABLE]

Then $\varphi(f(t-0))=g(s-0)$. Similarly, we get $\varphi(f(t+0))=g(s+0)$. $\Box$

For the family $\mathcal{B}_{t}$, we can obtain the corresponding results by the similar arguments.

Lemma 2.4

Let $f\in\mathcal{B}_{t}(I)$ and $g\in\mathcal{B}_{s}(J)$ be topologically conjugate via a homeomorphism $\varphi$. Then $\varphi(t)=s$ and $\varphi(f(t))=g(s)$.

Lemma 2.5

Let $f\in\mathcal{B}_{t}(I)$ and $g\in\mathcal{B}_{s}(J)$ be topologically conjugate. Then there are only two cases:

(D1)

$f(t-0)<f(t)<f(t+0)$* and $g(s-0)<g(s)<g(s+0)$;*

(D2)

$f(t)=f(t-0)$* and $g(s)=g(s-0)$, or $f(t)=f(t-0)$ and $g(s)=g(s+0)$, or $f(t)=f(t+0)$ and $g(s)=g(s-0)$, or $f(t)=f(t+0)$ and $g(s)=g(s+0)$.*

Lemma 2.6

Let $f\in\mathcal{B}_{t}(I)$ and $g\in\mathcal{B}_{s}(J)$ be topologically conjugate via a homeomorphism $\varphi$.

(i)

If $(f(t)-t)(g(s)-s)>0$, then $\varphi$ is increasing and goes through points $(a,c)$, $(b,d)$, $(t,s)$, $(f(t),g(s))$, $(f^{2}(t-0),g^{2}(s-0))$, $(f^{2}(t+0),g^{2}(s+0))$, $(f(t+0),g(s+0))$, and $(f(t-0),g(s-0))$.

(ii)

If $(f(t)-t)(g(s)-s)<0$, then $\varphi$ is decreasing and goes through points $(a,d)$, $(b,c)$, $(t,s)$, $(f(t),g(s))$, $(f^{2}(t+0),g^{2}(s-0))$, $(f^{2}(t-0),g^{2}(s+0))$, $(f(t-0),g(s+0))$, and $(f(t+0),g(s-0))$.

(iii)

If $f(t)=f(t\pm 0)$ and $g(t)=g(t\pm 0)$, then $\varphi$ is increasing and goes through points $(a,c)$, $(b,d)$, $(t,s)$, $(f^{2}(t-0),g^{2}(s-0))$, $(f^{2}(t+0),g^{2}(s+0))$, $(f(t+0),g(s+0))$, and $(f(t-0),g(s-0))$.

(iv)

If $f(t)=f(t\pm 0)$ and $g(t)=g(t\mp 0)$, then $\varphi$ is decreasing and goes through points $(a,d)$, $(b,c)$, $(t,s)$, $(f^{2}(t+0),g^{2}(s-0))$, $(f^{2}(t-0),g^{2}(s+0))$, $(f(t-0),g(s+0))$, and $(f(t+0),g(s-0))$.

3 Construction of topological conjugacy

In this section, we give the sufficient and necessary conditions for the conjugacy in the family $\mathcal{A}_{t}$ and $\mathcal{B}_{t}$, and construct all conjugacies.

3.1 The family $\mathcal{A}_{t}$

Theorem 3.1

Suppose that $f\in\mathcal{A}_{t}(I)$ and $g\in\mathcal{A}_{s}(J)$. Then $f$ and $g$ are topologically conjugate if and only if $f$ and $g$ are in case (C$1$) or both in (C$2$).

Proof. The necessarity follows from Lemma 2.2. It suffices to prove the sufficiency.

We only consider case (C1). The other is similar. For case (C1), we have the following four subcases:

[TABLE]

We only consider subcases (i)-(ii). Other subcases can be proved similarly.

For subcase (i). Firstly, define functions $f_{l}$, $f_{r}$, $g_{l}$ and $g_{r}$ as follows

[TABLE]

Then define two strictly monotone sequences $\{f_{l}^{n}(t)\}$ and $\{f_{r}^{n}(t)\}$, where $n=0,1,2,\cdots$. Obviously, $f_{l}^{n}(t)\rightarrow a$ and $f_{r}^{n}(t)\rightarrow b$ as $n\rightarrow+\infty$. With these sequences, give a partition of $I$, i.e.,

[TABLE]

where $I_{n}=(f_{l}^{n+1}(t),f_{l}^{n}(t)]$ and $J_{n}=[f_{r}^{n}(t),f_{r}^{n+1}(t))$. By means of Lemma 2.3, arbitrarily choose two increasing homeomorphisms $\varphi_{l_{0}}:[f_{l}(t),t]\rightarrow[g_{l}(s),s]$ and $\varphi_{r_{0}}:[t,f_{r}(t)]\rightarrow[s,g_{r}(s)]$ as initial functions such that $\varphi_{l_{0}}(f(t))=g(s)$. With the iterative constructive method and Lemma 2.3, define for $n=1,2,...$

[TABLE]

[TABLE]

Thus $\varphi_{l}:[a,t]\to[c,s]$ is an increasing conjugacy from $f_{l}$ to $g_{l}$ and $\varphi_{r}:[t,b]\to[s,d]$ is an increasing conjugacy from $f_{r}$ to $g_{r}$. In fact, for $x\in I_{n}$, it follows that

[TABLE]

[TABLE]

Now, it suffices to show $\varphi_{l}$ is strictly increasing and continuous on the interval $[a,t]$. One can see $\varphi_{l}$ is strictly increasing and continuous on each open interval $(f_{l}^{n+1}(t),f_{l}^{n}(t))$ for $n\in\mathbb{N}$. Thus it suffices to show $\varphi_{l}$ is continuous at each transitional point $f_{l}^{n}(t)$ for $n\in\mathbb{N}$ and right-continuous at $x=a$. By induction, for $n=1$, we have

[TABLE]

[TABLE]

Then assume $\varphi_{l}$ is continuous at the point $f_{l}^{n}(t)$ for some positive integer $n$. We shall show $\varphi_{l}$ is continuous at the transitional point $f_{l}^{n+1}(t)$. For convenience, we denote $g_{l}^{n}\circ\varphi_{l_{0}}\circ f_{l}^{-n}$ by $\varphi_{l_{n}}$. Thus $\varphi_{l}=\varphi_{l_{n}}$ on the interval $(f_{l}^{n+1}(t),~{}f_{l}^{n}(t)]$. Since

[TABLE]

and

[TABLE]

Thus $\varphi_{l}$ is continuous at the transitional point $f_{l}^{n+1}(t)$. Therefore, $\varphi_{l}$ is continuous at the point $f_{l}^{n}(t)$ for $n\in\mathbb{N}$. Finally, we shall show that $\varphi_{l}$ is right-continuous at $x=a$. Since $\varphi_{l}(t)=\varphi_{l_{0}}(t)=s$ and $\lim_{n\rightarrow\infty}g_{l}^{n}(s)=c$, we have $\varphi_{l}(f_{l}^{n}(t))=g_{l}^{n}(\varphi(t))$ and

[TABLE]

Thus that $\varphi_{l}$ is right-continuous at $x=a$. Therefore $\varphi_{l}$ is continuous on $[a,t]$.

With the similar argument, we can prove that $\varphi_{r}:[t,b]\to[s,d]$ is an increasing conjugacy from $f_{r}$ to $g_{r}$.

Then define $\varphi$ on $[a,b]$ by

[TABLE]

It is clear that $\varphi:I\to J$ is an increasing conjugacy from $f$ to $g$.

Finally, we will show all conjugacies can be obtained in this manner.

Suppose $\varphi$ is an increasing conjugacy between $f$ and $g$. Putting $\varphi_{l_{n}}:=\varphi$ on $I_{n}$ and $\varphi_{r_{n}}:=\varphi$ on $J_{n}$. By Lemma 2.3, we can verify $\varphi_{l_{0}}$ goes through point $(t,s)$, $(f(t),g(s))$ and $(f_{l}(t),g_{l}(s))$, $\varphi_{r_{0}}$ goes through point $(t,s)$ and $(f_{r}(t),g_{r}(s))$, and $\varphi_{l_{n}}$ (resp. $\varphi_{r_{n}}$) is of form (3.8) (resp. (3.12)). Thus relation (3.18) holds.

For subcase (ii), similarly, but choose any two decreasing homeomorphisms $\tilde{\varphi}_{l_{0}}:[f_{l}(t),t]\rightarrow[s,g_{r}(s)]$ and $\tilde{\varphi}_{r_{0}}:[t,f_{r}(t)]\rightarrow[g_{l}(s),s]$ as initial functions such that $\tilde{\varphi}_{l_{0}}(f(t))=g(s)$. Define for $n=1,~{}2,~{}3,\cdots$,

[TABLE]

[TABLE]

Then define

[TABLE]

which is a conjugacy from $f$ and $g$. $\Box$

3.2 The family $\mathcal{B}_{t}$

Theorem 3.2

Suppose that $f\in\mathcal{B}_{t}(I)$ and $g\in\mathcal{B}_{s}(J)$. Then $f$ and $g$ are topologically conjugate if and only if $f$ and $g$ are in case (D$1$) or both in (D$2$).

Proof. The necessarity follows from Lemma 2.5. It suffices to prove the sufficiency.

We only consider case (D1). The other is similar. For case (D1), we have the following four subcases:

[TABLE]

We also prove subcase (i)-(ii). Other subcases can be proved similarly.

For subcase (i), define functions $f_{e}$, $f_{v}$, $g_{e}$ and $g_{v}$ as follows

[TABLE]

Then define sequence $\{f_{e}^{n}(t)\}$, where $n=0,1,2,\cdots$. Obviously, $f_{e}^{2k}(t)\rightarrow a$, and $f_{e}^{2k+1}(t)\rightarrow b$. With these sequence, give a partition of $I$, i.e.,

[TABLE]

Choose any increasing homeomorphism $\psi_{0}:[f_{e}^{2}(t),t]\rightarrow[g_{e}^{2}(s),s]$ as an initial function such that $\psi_{0}$ goes through points $(f(t),g(s))$ and $(f_{v}(t),g_{v}(s))$.Ddefine for $n=1,~{}2,~{}3,\cdots$

[TABLE]

[TABLE]

and

[TABLE]

One can see that $\psi:[a,b]\to[c,d]$ is an increasing conjugacy from $f$ to $g$, and every conjugacy can be obtained in this manner.

For subcase (ii), define functions $\tilde{f}_{e}$, $\tilde{f}_{v}$, $\tilde{g}_{e}$ and $\tilde{g}_{v}$ as follows

[TABLE]

Then define sequence $\{\tilde{f}_{e}^{n}(t)\}$, where $n=0,1,2,\cdots$. Obviously, $\tilde{f}_{e}^{2k}(t)\rightarrow a$, and $\tilde{f}_{e}^{2k+1}(t)\rightarrow b$. With these sequence, give a partition of $I$, i.e.,

[TABLE]

Choose any decreasing homeomorphism $\tilde{\psi}_{0}:[\tilde{f}_{e}^{2}(t),t]\rightarrow[s,\tilde{g}_{v}^{2}(s)]$ as an initial function such that $\tilde{\psi}_{0}$ goes through points $(f(t),g(s))$ and $(\tilde{f}_{v}(t),\tilde{g}_{e}(s))$. Define for $n=1,~{}2,~{}3,\cdots$

[TABLE]

[TABLE]

and

[TABLE]

We have $\tilde{\psi}:[a,b]\to[c,d]$ is a decreasing conjugacy from $f$ to $g$ and every conjugacy can be obtained in this manner. $\Box$

4 Smoothness of conjugacy

For convenience, we only consider subcase (i) under case (C1) in Theorem 3.1, Others are similar.

Theorem 4.1

The conjugacy $\varphi$ in the proof of Theorem 3.1 is continuously differentiable, provided the following hypotheses are added:

(a)

$f_{l}$, $f_{r}$, $g_{l}$ and $g_{r}$ are continuously differentiable, $f_{l}^{\prime}(x)\neq 0$ for $x\in[a,t]$ and $f_{r}^{\prime}(x)\neq 0$ for $x\in[t,b]$; $\varphi_{l_{0}}$ and $\varphi_{r_{0}}$ are continuously differentiable on $[f_{l}(t),t]$ and $[t,f_{r}(t)]$, respectively, satisfying $\varphi_{l_{0}}^{\prime}(t)=\varphi_{r_{0}}^{\prime}(t)$ and

[TABLE]

[TABLE]

(b)

For any $x\in(f_{l}(t),t]$, $\varphi_{l_{0}}^{\prime}(x)\prod_{j=0}^{\infty}\frac{g_{l}^{\prime}(g_{l}^{j}(\varphi_{l_{0}}(x)))}{f_{l}^{\prime}(f_{l}^{j}(x))}=L_{1}$;

(c)

For any $x\in[t,f_{r}(t))$, $\varphi_{r_{0}}^{\prime}(x)\prod_{j=0}^{\infty}\frac{g_{r}^{\prime}(g_{r}^{j}(\varphi_{r_{0}}(x)))}{f_{r}^{\prime}(f_{r}^{j}(x))}=L_{2}$.

Proof. By induction, we can prove $\varphi_{l}$ is continuously differentiable on the interval $(a,t]$. Now it suffices to show $\varphi_{l}$ is continuously differentiable at $x=a$. Since $\varphi_{l}$ is continuously differentiable on the interval $(a,~{}t]$, we have

[TABLE]

It follows from $f_{l}^{\prime}(x)\neq 0$ that

[TABLE]

By induction, we have

[TABLE]

In particular, we choose $x\in(f_{l}(t),t]$ and have

[TABLE]

By the assumption (b), one has $\varphi_{l_{0}}^{\prime}(x)\prod_{j=0}^{n-1}\frac{g_{l}^{\prime}(g_{l}^{j}(\varphi_{l_{0}}(x)))}{f_{l}^{\prime}(f_{l}^{j}(x))}\rightarrow L_{1}$ as $n\rightarrow\infty$ for $x\in(f_{l}(t),t]$. Then

[TABLE]

By the Mean Value Theorem, for any $x\in(a,t]$, there exists $\xi\in(a,x)$ such that

[TABLE]

It follows from (4.3) that

[TABLE]

Therefore, $\varphi_{l}$ is continuously differentiable on $[a,t])$. Similarly, we can prove $\varphi_{r}$ is also continuously differentiable on $[t,b]$ by the condition (c). Thus $\varphi$ is a continuously differentiable conjugacy. $\Box$

Remark 4.1

We can also investigate the smoothness of conjugacies in Theorem 3.2 as Theorem 4.1. More hypotheses are imposed on the initial function $\psi_{0}$.

5 Applications to Lorenz maps

Example 5.1

Consider the following two Lorenz maps

[TABLE]

One can see that $f\in\mathcal{A}_{\frac{1}{4}}([0,1])$ and $g\in\mathcal{A}_{\frac{1}{2}}([0,1])$. By the proof of Theorem 3.1, a topological conjugacy $\phi$ from $f$ to $g$ is constructed in Fig.10.

Example 5.2

Consider the following two Lorenz maps

[TABLE]

It is clear that $f\in\mathcal{B}_{\frac{1}{2}}([0,1])$ and $g\in\mathcal{B}_{\frac{1}{4}}([0,1])$. By the proof of Theorem 3.2, a topological conjugacy $\phi$ from $f$ to $g$ is constructed in Fig.10.