Topological Equiconjugacy for Unimodal Nonautonomous Discrete Dynamical Systems with Limit Property

Abstract

The goal of this article is to study how combinatorial equivalence implies topological equiconjugacy. For that we introduce kneading sequences for a particular class of nonautonomous discrete dynamical systems and show that these sequences are a complete invariant for topological equiconjugacy classes.

Appendix A. The Multimodal Case

In this appendix, we work with NDSs where each map $f_n:J^\mathcal {F}\rightarrow J^\mathcal {F}$ is a multimodal map. Let $(J^\mathcal {F}, \mathcal {F})$ be the NDS defined as follows: let $J^\mathcal {F}=[a^\mathcal {F}, b^\mathcal {F}]$ be a compact interval, and let $f_n:J^\mathcal {F}\rightarrow J^\mathcal {F}$ be a piecewise monotone continuous map such that either $f_n(a^\mathcal {F})=a^\mathcal {F}$ for each $n\ge 1$ or $f_n(a^\mathcal {F})=b^\mathcal {F}$ for each $n\ge 1$ and either $f_n(b^\mathcal {F})=a^\mathcal {F}$ for each $n\ge 1$ or $f_n(b^\mathcal {F})=b^\mathcal {F}$ for each $n\ge 1$. Furthermore, there exists $\ell \ge 1$ so that for each $n\ge 1$ there are $a^\mathcal {F}<c_n^\mathcal {F}(1)<\cdots<c_n^\mathcal {F}(\ell )<b^\mathcal {F}$ such that the intervals $I_n^\mathcal {F}(1)=\left[ a^\mathcal {F}, c_n^\mathcal {F}(1)\right] , I_n^\mathcal {F}(2)=\left[ c_n^\mathcal {F}(1), c_n^\mathcal {F}(2)\right] , \ldots , I_n^\mathcal {F}(\ell +1)=\left[ c_n^\mathcal {F}(\ell ), b^\mathcal {F}\right]$ are the largest intervals in which $f_n$ is strictly monotone. The points $\{c_n^\mathcal {F}(1), \ldots , c_n^\mathcal {F}(\ell ):\; n\ge 1\}$ are the turning points of the NDS. We call the NDS defined above $\ell$-modal nonautonomous discrete dynamical system (short $\ell$-MNDS)

Consider the alphabet $\mathcal {A}_\mathcal {F}=\{I_n^\mathcal {F}(1), c_n^\mathcal {F}(1), I_n^\mathcal {F}(2), \ldots , c_n^\mathcal {F}(\ell ), I_n^\mathcal {F}(\ell +1):\; n\ge 1\}$.

Address of a point: Let $n\ge 1$. The address of a point $x\in J^\mathcal {F}$ on the level n is the letter $i_{\mathcal {F},n}(x)\in \mathcal A_\mathcal {F}$ defined by

$$\begin{aligned} i_{\mathcal {F},n}(x)= \left\{ \begin{array}{ll} I_n^\mathcal {F}(j) &{} \text { if } x\in I_n^\mathcal {F}(j) \text { and is not a turning point, } \\ c_n^\mathcal {F}(j) &{} \text { if } x=c_n^\mathcal {F}(j). \end{array} \right. \end{aligned}$$

Itinerary of a point: Let $n\ge 1$. The itinerary of a point $x\in J^\mathcal {F}$ on the level n is the sequence $I_{\mathcal {F},n}(x)\in \mathcal A_\mathcal {F}^{\{0,1,2,\ldots \}}$ defined by

$$\begin{aligned} I_{\mathcal {F},n}(x)=(i_{\mathcal {F},n}(x),i_{\mathcal {F},n+1}(f_n^1(x)),\ldots , i_{\mathcal {F},n+\ell }(f_n^\ell (x)),\ldots ). \end{aligned}$$

Kneading sequences: The kneading sequences of $(J^\mathcal {F},\mathcal {F})$ are the sequences $\mathbb {V}^\mathcal {F}(j)=\{\mathbb {V}_n^\mathcal {F}(j)\}_{n\ge 1}$, where $\mathbb {V}_n^\mathcal {F}(j):=I_{\mathcal {F},n}(c_n^\mathcal {F}(j))$ and $j=1, \ldots , \ell$.

Definition A.1

Let $(J^\mathcal {F},\mathcal {F})$ and $(J^\mathcal {G},\mathcal {G})$ be two $\ell$-MNDSs. We say that

• (1) $(J^\mathcal {F},\mathcal {F})$ and $(J^\mathcal {G},\mathcal {G})$ have the same kneading sequences if

  $$\begin{aligned} \mathbb {V}^\mathcal {F}(j)=\mathbb {V}^\mathcal {G}(j),\ \text {for each } j=1, \ldots , \ell . \end{aligned}$$

• (2) $(J^\mathcal {F},\mathcal {F})$ is monotonically equivalent to $(J^\mathcal {G},\mathcal {G})$ if for all $n\ge 1$ and each $j=1, \ldots , \ell +1$:

  $\bullet$:

  either $f_n\restriction _{I_n^\mathcal {F}(j)}$ and $g_n\restriction _{I_n^\mathcal {G}(j)}$ are strictly increasing;

  $\bullet$:

  or $f_n\restriction _{I_n^\mathcal {F}(j)}$ and $g_n\restriction _{I_n^\mathcal {G}(j)}$ are strictly decreasing.

The limit property for $\ell$-MNDSs has a little change: if f is the $\ell$-modal map such that $f_n$ converges uniformly to f, then $c^f(j)$ is not periodic and $c^f(j)\notin \{f^{-m}(c^f(i)): m\ge 1\}$ for all $j,i=1,\ldots ,\ell$. The other items remain the same. With these more general definitions, we have the following.

Theorem A.2

Let $(J^\mathcal {F},\mathcal {F})$ and $(J^\mathcal {G},\mathcal {G})$ be two monotonically equivalent $\ell$-modal nonautonomous discrete dynamical systems, and assume that both satisfy the limit property. Then $(J^\mathcal {F},\mathcal {F})$ and $(J^\mathcal {G},\mathcal {G})$ are topologically equiconjugate if and only if they have the same kneading sequences.

We shall give only a sketch of the proof of above theorem since it follows the same ideas and arguments as in the proof of Theorem 1.1, and rewriting it would probably only increase the technicalities.

Sketch of proof

The proof of necessary condition is identical to the proof of Proposition 2.1 since $(J^\mathcal {F},\mathcal {F})$ is monotonically equivalent to $(J^\mathcal {G},\mathcal {G})$. The proof of the reverse implication requires some minor changes. For $n,k\ge 1$ we put $\mathscr {C}^k_n(\mathcal {F})=\{x\in J^\mathcal {F}: f_n^m(x)= c_{n+m}^\mathcal {F}(j) \text { for some } 0\le m\le k-1 \text { and } 1\le j\le \ell \}$. Now the definitions of $\mathscr {P}_{n}^k(\mathcal {F})$ and $\mathscr {A}_n^k(\mathcal {F})$ are as before. So there exists $h_n^k:\mathscr {C}_n^k(\mathcal {F})\rightarrow \mathscr {C}_n^k(\mathcal {G})$, and hence we can define $h_n:\mathscr {C}_n(\mathcal {F})\rightarrow \mathscr {C}_n(\mathcal {G})$ like in conclusion of Theorem 3.2. The limit property implies now that each $h_n$ could be extended to a homeomorphism satisfying $h_{n+1}\circ f_n=g_n\circ h_n$, for each $n\ge 1$ and the families $(h_n^{\pm 1})_{n\ge 1}$ are equicontinuous. Therefore $(J^\mathcal {F},\mathcal {F})$ and $(J^\mathcal {G},\mathcal {G})$ are topologically equiconjugate. $\square$