Strong Pairs of Periodic Segments

Abstract

We introduce the notion of a strong pair of periodic segments over [0, T] and we show its applications in detecting chaotic dynamics. We prove a various number of fixed point index formulas concerning periodic points of the Poincaré map.

1 Introduction

The machinery of periodic segments was introduced by Srzednicki in order to investigate the existence of periodic points of the Poincaré map associated to periodic in time ODE [10]. It is based on the Ważewski retract method and closely related to the Conley index theory [1, 11]. A geometric method for detecting chaotic dynamics based on the existence of special configurations of periodic segments was introduced in [12] and developed in [6,7,8,9, 13, 16, 18, 20]. We very briefly describe the notion of periodic segment. If $v:{\mathbb {R}}\times M\longrightarrow TM$ is a smooth T-time periodic vector field on manifold M then the system of equations

$$\begin{aligned} {\dot{t}}=1, \quad {\dot{x}}=v(t,x) \end{aligned}$$

generates a local flow on the extended phase space ${\mathbb {R}}\times M$. Let $W\subset [0,T]\times M$ be such that $W_0=W_T$, where $W_t\subset M$ is a t-section of W. By $W^-$ we denote the exit set of W i.e., $W^-$ is the set of boundary points of W at which the vector field (1, v) is pointing out with respect to W. We say that W is a periodic segment over [0, T] iff the following conditions hold

• W and $W^-$ are compact ENRs,

• there exists a compact subset $W^{--}$ of $W^-$ (called the essential exit set) such that

  $$\begin{aligned} W^-=W^{--}\cup \big (\{T\}\times W_T\big ), \quad W^-\cap \big ([0,T)\times X\big )\subset W^{--}, \end{aligned}$$

• $(W_0, W_0^{--})=(W_T, W_T^{--})$,

• there exists a homeomorphism $h:[0,T]\times W_0 \longrightarrow W$ such that $\pi _1\circ h=\pi _1$ and

  $$\begin{aligned} h([0,T]\times W_0^{--})=W^{--}. \end{aligned}$$

A homeomorphism h in a natural way induce a corresponding monodromy homeomorphism $m_W:W_0\longrightarrow W_0$. It follows that $m_W(W_0^{--})=W_0^{--}$. We put ${\overline{m}}_W:=m_W|_{W_0^{--}}$. Let P be the Poincaré map associated to the vector field v. Srzednicki proved that if $L(m_W)\ne L({\overline{m}}_W)$ (L is a Lefschetz number) then P has a fixed point $x\in W_0\setminus W_0^{--}$. Moreover, the orbit of x with respect to $\phi$ is contained in W.

Motivated by the results in [12, 13, 16, 18] we introduce the notion of a strong pair of periodic segments. We say that (W, Z) is a strong pair of periodic segments if the following conditions hold:

• Z, W are T-periodic segments over [0, T],

• $Z\subset W$, $Z_0=W_0$ and $Z_0^{--}=W_0^{--}$,

• $m_Z=\textrm{id}_{W_0}$.

We very briefly describe the way in which a strong pair of periodic segments (W, Z) is related to symbolic dynamics. Let $I\subset W_0$ be the set of all points $x\in W_0$ whose full orbit with respect to $\phi$ are contained in the translated copies of the bigger segment W. It follows that I is compact and invariant for the Poincaré map P. Let $\Sigma _2$ be the space of the sequences of two symbols. There is a natural, geometric way to define a continuous map $q:I\longrightarrow \Sigma _2$ such that

$$\begin{aligned} q\circ P|_I=\sigma \circ q, \end{aligned}$$

where $\sigma :\Sigma _2\longrightarrow \Sigma _2$ is the shift map.

Problem 1

Let $c=(c_0,\ldots ,c_{k-1})\in \Sigma _2$ be a k-periodic sequence. Does $g^{-1}(c)$ contains a k-periodic point of the Poincaré map P?

This is a natural question in the context of chaotic dynamics. We recall that if q is surjective then P is chaotic in the sense of Devaney [3]. One way to achieve this is to show that q(I) contains a dense set of periodic sequences. We use the fixed point index theory and the Lefschetz fixed point theorem [4] to attack Problem 1. It turns out that the homeomorphism ${\overline{m}}_W:W_0^{--}\longrightarrow W_0^{--}$ plays a key role here. It follows by Theorem 16 that the answer to the Problem 1 is positive provided

$$\begin{aligned} L(({\overline{\mu }}_{W}-I_{H(W_0^{--})})^{c_W})\ne 0, \end{aligned}$$

where $c_W:=\textrm{card}\, \{s\in \{0,\ldots , k-1\}: c_s=1\}\ge 1$.

Problem 2

Give the conditions ensuring that the answer to Problem 1 is positive for each periodic sequence $c\in \Sigma _2$.

Unfortunately, $L(({\overline{\mu }}_{W}-I_{H(W_0^{--})})^{c_W})=0$ in many natural cases, so we need a different approach. We will use the connected components of $Z^{--}$. We show (Theorem 40) that it is sufficient to assume that $\chi (W_0)\ne \chi (W_0^{--})$ and there exists a connected component E of $Z^{--}$ such that

$$\begin{aligned} {\overline{m}}_W(E_0)\subset W_0^{--}\setminus E_0,\quad \chi (E_0)\ne 0. \end{aligned}$$

The main tool in the proof is the fixed point index formula (5) in Theorem 31.

Assume that $E(1), \ldots , E(n)$ are the connected components of $Z^{--}$. If the orbit of $x\in I$ leaves the smaller segment Z it has to do it through one of components of $Z^{--}$. This gives us a possibility to define geometrically a continuous map $q:I\longrightarrow \Sigma _{n+1}$ ([18]).

Problem 3

Let $c=(c_0,\ldots ,c_{k-1})\in \Sigma _{n+1}$ be a k-periodic sequence. Does $g^{-1}(c)$ contains a k-periodic point of the Poincaré map P?

We prove (Proposition 44) that the answer to the Problem 3 is positive for each periodic sequence $c\in \Sigma _3$ provided $Z^{--}$ has exactly two connected components E(1), E(2) such that the following conditions hold

$$\begin{aligned} {\overline{m}}_W(E(1)_0)=E(2)_0, \quad {\overline{m}}^2_W=\textrm{id}_{W_0^{--}}, \end{aligned}$$

and $2\chi (E(1)_0)\notin \{0,\chi (W_0)\}$.

We compare our main results to the techniques developed in [10,11,12,13, 18, 19]. We first focus on the geometric method for detecting fixed points of the Poincaré map P presented in [10, 11]. Assume that (W, Z) is a strong pair of periodic segments and $L({\overline{\mu }}_W)=0$. It follows by Theorem 7.7 in [11] (compare Theorem 7.1 in [10]) that P has a fixed point provided $\chi (W_0^{--})\ne 0$. Theorem 24 allows us to show a multiplicity result. More precisely, if $W_0^{--}$ has k connected components that are not invariant for ${\overline{m}}_W$ and having non-zero Euler characteristic then P has at least k fixed points.

The continuation method was used in [18] to show that the Poincaré map P associated to the local process $\Phi _0$ generated by the planar periodic system

$$\begin{aligned} {\dot{z}}=(1+e^{i\kappa t}|z|^2){\overline{z}} \end{aligned}$$

is $\Sigma _3$-chaotic for $\kappa >0$ sufficiently small. The proof of the main Theorem 8 in [18] rely on the following ingredients:

• (i) The existence of a strong pair of periodic segments (W, Z) such that $W_0^{--}$ consists of two contractible, connected components $E_0$, $F_0$ and ${\overline{m}}_W(E_0)=F_0$,

• (ii) A construction of a model semi-process $\Phi _1$ having the same strong pair of periodic segments (W, Z) (Theorem 20 in [18]),

• (iii) The existence of some appropriate homotopy between processes $\Phi _0$ and $\Phi _1$ (see Remark 33 for more details),

• (iv) Theorem 10 in [18] concerning the continuation of the involved fixed point indices,

• (v) A computation of the fixed point indices of the model Poincaré map for $\Phi _1$.

The proof of Theorem 10 given in [18] is quite difficult and technically complicated. Fortunately, it turns out that it is a simple consequence of our Corollary 32. Moreover, Proposition 44 shows that all what is needed in order to get a conclusion of Theorem 8 in [18] is the point (i). In particular, we can avoid the challenging points (ii–iii) (see Sect. 6 in [18]). This is particularly important from the point of view off the construction of model processes, which may be particularly difficult for the general strong pairs of periodic segments.

The fixed point index formula in Theorem 31 is purely topological and is not a consequence of analytical results in [18, 19]. It allows us to replace the construction of a model processes with combinatorial and number-theoretic problems involving the fixed point indices.

2 Periodic Segments

In this section we give a brief exposition of the Srzednicki method. For the proofs we refer the reader to [12, 13]. Let X be a metric space and let $D\subset [0,\infty )\times X$ be an open set. A continuous map $\phi :D\longrightarrow X$ is called a local semi-flow on X if for every $x\in X$ the set $\{t\in {\mathbb {R}}:(t,x)\in D\}$ is equal to an interval $I_x=[0,\omega _x)$ with $0<\omega _x\le \infty ,$ and the following conditions hold:

$$\begin{aligned} \phi (0,x)=x, \end{aligned}$$

(1)

and if $(t,x)\in D$, $(s,\phi (t,x))\in D$ then $(t+s,x)\in D$ and

$$\begin{aligned} \phi (t+s,x)=\phi (s,\phi (t,x)). \end{aligned}$$

(2)

Obviously, $\phi$ is called a semi-flow if $D=[0,\infty )\times X$. We say that $\phi$ is a local flow if $D\subset {\mathbb {R}}\times X$ is an open set, $I_x=(\alpha _x,\omega _x)$ with some

$$\begin{aligned} -\infty \le \alpha _x<0<\omega _x\le \infty , \end{aligned}$$

and the Eqs. (1) and (2) hold. The positive semi-trajectory of $x\in X$ is defined as the set

$$\begin{aligned} \phi ^+(x):=\{\phi (t,x):t\in [0,\omega _x)\}. \end{aligned}$$

Let $W\subset X$. Define the exit set of W as

$$\begin{aligned} W^{-}:=\Big \{x\in W: \phi ([0,t],x)\not \subset W\, \text {for all}\, t\in (0,\omega _x)\Big \}. \end{aligned}$$

We call W a Ważewski set for $\phi$ if it is closed and its exit set $W^-$ is closed as well. A compact Ważewski set is called a block. In the case $\phi$ is a local flow we say that a block W is isolating if the boundary W is equal to the union of $W^-$ and the entry set $W^+$ defined as the exit set of W with respect to the local flow with reversed time. The isolating blocks are basic notion for the Conley index theory of the isolated invariant sets [1, 2, 11].

If W is a Ważewski set and

$$\begin{aligned} W^*:=\Big \{x\in W: \phi (t,x)\notin W\, \text {for some} \, t\in (0,\omega _x)\Big \}, \end{aligned}$$

then the escape-time map

$$\begin{aligned} \sigma _W:W^*\ni x \longrightarrow \sup \Big \{t\in [0,\omega _x):\phi ([0,t],x)\subset W\Big \}\in (0,\infty ) \end{aligned}$$

is continuous. Consequently, $W^-$ is a strong deformation retract of $W^*$. Deformation is given by the flow. This is a famous Ważewski retract theorem: if $W^-$ is not a strong deformation retract of a Ważewski set W then $\phi ^+(x)\subset W$ for some $x\in W$.

By a local semi-process on X we mean a continuous map $\Phi : D\longrightarrow X$, where D is an open subset of $[0,\infty )\times ({\mathbb {R}}\times X)$, such that the map

$$\begin{aligned} \phi :D \ni \big (t,(\sigma , x)\big )\longrightarrow \big (\sigma +t, \Phi (t, (\sigma , x))\big )\in {\mathbb {R}}\times X \end{aligned}$$

is a local semi-flow on ${\mathbb {R}}\times X$. We write $\Phi _{(\sigma , t)}(x)$ instead of $\Phi (t, (\sigma , x))$. It follows that

$$\begin{aligned} \Phi _{(\sigma ,0)}=\textrm{id}, \quad \Phi _{(\sigma , s+t)}=\Phi _{(\sigma +s,t)}\circ \Phi _{(\sigma , s)}, \end{aligned}$$

whenever it is defined.

Let $T>0$. A local semi-process is called T-periodic if $\Phi _{(\sigma , t)}=\Phi _{(\sigma +T, t)}$ for each $\sigma$ and t. In that case the map $P=\Phi _{(0,T)}$, called the Poincaré map, satisfies

$$\begin{aligned} \Phi _{(0,nT)}=\Phi ^n_{(0,T)}. \end{aligned}$$

Definition 4

A set $W\subset [0,T]\times X$ is called a periodic segment over [0, T] if it is a block with respect to $\phi$ such that the following conditions hold:

• (S0) there exists a compact subset $W^{--}$ of $W^-$ (called the essential exit set) such that

  $$\begin{aligned} W^-=W^{--}\cup \big (\{T\}\times W_T\big ), \quad W^-\cap \big ([0,T)\times X\big )\subset W^{--}, \end{aligned}$$

• (S1) $(W_0, W_0^{--})=(W_T, W_T^{--})$,

• (S2) there exists a homeomorphism $h:[0,T]\times W_0 \longrightarrow W$ such that $\pi _1\circ h=\pi _1$ and

  $$\begin{aligned} h([0,T]\times W_0^{--})=W^{--}. \end{aligned}$$

Obviously, W and $W^{--}$ are topological periodic sets over [0, T]. Let $m_W:W_0\longrightarrow W_0$ be the corresponding monodromy homeomorphism given by

$$\begin{aligned} m_W (x):=\pi _2 h(T, \pi _2 h^{-1}(0,x)). \end{aligned}$$

It follows that $m_W(W_0^{--})=W_0^{--}$. We put

$$\begin{aligned}{} & {} {\overline{m}}_W:=m_W|_{W_0^{--}}:W_0^{--}\longrightarrow W_0^{--}, \\{} & {} {\tilde{m}}_W:(W_0, W_0^{--})\ni x\longrightarrow m_W(x)\in (W_0, W_0^{--}). \end{aligned}$$

In particular, W and $W^{--}$ are the topological periodic sets over [0, T]. A homeomorphism ${\tilde{m}}_W$ induces the isomorphism in the singular homologies (with a coefficients in ${\mathbb {Q}}$):

$$\begin{aligned} {\tilde{\mu }}_W:=H({\tilde{m}}_W):H(W_0,W_0^{--})\longrightarrow H(W_0, W_0^{--}). \end{aligned}$$

It can be proved that a different choice of the homeomorphism h provides the monodromy map homotopic to ${\tilde{m}}_W$, so $\mu _W$ is an invariant of the segment W [17]. We put

$$\begin{aligned} \mu _W=H(m_W), \quad {\overline{\mu }}_W=H({\overline{m}}_W). \end{aligned}$$

If W and $W^{--}$ are ENRs (i.e., euclidean neighborhood retracts), then $H(W_0, W_0^{--})$ is of finite type and the Lefschetz number

$$\begin{aligned} L({\tilde{\mu }}_W)=\sum _{i\ge 0}(-1)^{i}\textrm{tr}\, H_i ({\tilde{\mu }}_W) \end{aligned}$$

is correctly defined. In particular,

$$\begin{aligned} L({\tilde{\mu }}_W)= & {} L(\mu _W)-L({\overline{\mu }}_W), \\ L(\textrm{id}_{H(W_0, W_0^{--})})= & {} \chi (W_0, W_0^{--})=\chi (W_0)-\chi ( W_0^{--}), \end{aligned}$$

where $\chi$ is the Euler-Poincaré characteristic.

Let $k\in {\mathbb {Z}}$. By $\tau _k$ we denote the translation

$$\begin{aligned} \tau _k:{\mathbb {R}}\times X\ni (t,x)\longrightarrow (t+kT,x)\in {\mathbb {R}}\times X. \end{aligned}$$

For a T-periodic segment W we define a Ważewski set $W^{\infty }\subset [0,\infty )\times X$ by:

$$\begin{aligned} W^{\infty }_{[kt,(k+1)T]}=\tau _k (W), \end{aligned}$$

i.e., $W^{\infty }$ is obtained by gluing together the translated copies of W. For a T-periodic local proces $\Phi$ and a periodic segment W over [0, T] we define the set of all points in $W_0$ whose full trajectories are contained in $W^{\infty }$ i.e.,

$$\begin{aligned} I_W:=\Big \{x\in W_0:\Phi _{(0, t)}(x)\in W^{\infty }_t\; \text {for all}\; t\ge 0\Big \}. \end{aligned}$$

Assume that X is an ENR and $f:U\longrightarrow X$ is a continuous map, where $U\subset X$ is open. If the set of fixed points of f is compact then the fixed point index $\textrm{ind}(f)$ is well defined [4]. Sometimes, it is more convenient to use the notation $\textrm{ind}\, (f, \textrm{Fix}, (f))$, where $\textrm{Fix}\, (f)=\{x\in U:f(x)=x\}$ is the set of fixed points of f. Then U is an open set isolating a compact set of fixed points $\textrm{Fix}\, (f)$. We will use the following properties of the fixed point index:

• (i1) If X is a compact ENR, then $\textrm{ind}(f)=L(f)$ (Lefschetz fixed point theorem),

• (i2) If $f:U\longrightarrow X$, $U=U_1\cup U_2$, $U_1$, $U_2$ are open and disjoint, then $\textrm{ind}(f)=\textrm{ind}(f|_{U_{1}})+\textrm{ind}(f|_{U_{2}})$ (additivity).

Theorem 5

([13]) Let W be a T-periodic segment over [0, T] and W, $W^{--}$ are ENRs. Then the set

$$\begin{aligned} U_W=\Big \{x\in W_0: \Phi _{(0,t)}(x)\in W_t\setminus W_t^{--} \, for \, all\, t\in [0, T]\Big \} \end{aligned}$$

is open in $W_0$, the set of fixed points of the restriction $P|_{U_W}:U_W\longrightarrow W_0$ is compact and

$$\begin{aligned} \textrm{ind}(P|_{U_W})=L({\tilde{\mu }}_W). \end{aligned}$$

In particular, if $L({\tilde{\mu }}_W)\ne 0$ then P has a fixed point in $U_W$.

3 Strong Pair of Periodic Segments

Let $\Phi$ be a local semi-process on a metric space X.

Lemma 6

Assume that Z, W are T-periodic segments over [0, T] such that $Z\subset W$, $Z_0=W_0$ and $Z_0^{--}=W_0^{--}$. Then there exists a homotopy $H:[0,1]\times W_0 \longrightarrow W_0$ such that

• (a) $H(0,\cdot )=m_W$ and $H(1,\cdot )=m_Z$,

• (b) $H(t,\cdot ):W_0\longrightarrow W_0$ ($t\in [0,1]$) is a homeomorphism on its image.

Proof

For $s\in [0,T]$ we define

$$\begin{aligned}{} & {} m_s:W_s\ni x\longrightarrow \pi _2h_W(T,\pi _2h_W^{-1}(s,x))\in W_0, \\{} & {} m^s:W_0\ni x\longrightarrow \pi _2h_Z(s,\pi _2h_Z^{-1}(0,x))\in Z_s. \end{aligned}$$

It follows that

$$\begin{aligned} m^T=m_Z, \quad m_0=m_W, \quad m^0=m_T=\textrm{id}_{W_0}. \end{aligned}$$

We define a homotopy

$$\begin{aligned} H:[0,1]\times W_0\ni (t,x)\longrightarrow m_{tT}(m^{tT}(x))\in W_0. \end{aligned}$$

It follows that

$$\begin{aligned} H(0,\cdot )=m_W,\quad H(1,\cdot )=m_Z, \end{aligned}$$

so the proof is complete. $\square$

Definition 7

We say that (W, Z) is a strong pair of periodic segments over [0, T] if the following conditions hold:

• (i) Z, W are T-periodic segments over [0, T],

• (ii) $Z\subset W$, $Z_0=W_0$ and $Z_0^{--}=W_0^{--}$,

• (iii) $m_Z=\textrm{id}_{W_0}$,

• (iv) if E is a sum of some connected components of $Z^{--}$ and n is a natural number such that ${\overline{m}}_W^n(E_0)=E_0$ then ${\overline{m}}_W^n|_{E_0}=\textrm{id}_{E_0}$,

• (v) $W_0$ and $W_0^{--}$ are ENRs.

We assume a technical condition (iv) in order to simplify formulation of some further results.

Corollary 8

If (W, Z) is a strong pair of periodic segments then

$$\begin{aligned} L({\tilde{\mu }}_W)=\chi (W_0)-L({\overline{\mu }}_W). \end{aligned}$$

Moreover, if the sequence $L({\overline{\mu }}_W^n)$ is non-constant, then the Poincaré map has a periodic point in $W_0{\setminus } W_0^{--}$.

Proof

Let

$$\begin{aligned} U^n=U\cup \tau _1 (U)\cup \ldots \cup \tau _{n-1}(U), \quad (n\ge 1), \end{aligned}$$

for $U\in \{Z,W\}$. Then $(W^n,Z^n)$ is a strong pair of periodic segments over [0, nT]. It follows by Theorem 5 and Lemma 6 that the sequence $\textrm{ind}(P^n|_{U_{W^n}})=\chi (W_0)-L({\overline{\mu }}_W^n)$ is non-constant, so $\textrm{ind}(P^n|_{U_{W^n}})\ne 0$ for some $n\ge 1$. $\square$

Assume that W is a periodic segment. Let ${\tilde{m}}_W:(W_0,W_0^{--})\longrightarrow (W_0,W_0^{--})$ be a corresponding monodromy homeomorphism. The fixed point set of $m_W:W_0\longrightarrow W_0$ splits into a disjoint union of fixed point classes - two fixed points are in the same Nielsen fixed point class F if and only if they can be joined by a path which is homotopic (relatively end-points) to its own m-image. More precisely, $x_0, x_1\in \textrm{Fix}\, (m_W)$ are Nielsen related if there exists a continuous path $\alpha :[0,1]\longrightarrow W_0$ such that $\alpha (i)=x_i$ for $i=0,1$ and

$$\begin{aligned} \alpha \simeq m_W\circ \alpha ,\quad \textrm{rel}\, \{0,1\}. \end{aligned}$$

It follows that

$$\begin{aligned} \textrm{Fix}\, (m_W)=F_1\cup \ldots \cup F_s, \end{aligned}$$

where $F_i$ are the Nielsen fixed point classes. We say that a fixed point class F of $m_W$ assume its index in $W_0^{--}$ if

$$\begin{aligned} \textrm{ind}\, (m_W,F)=\textrm{ind}\, ({\overline{m}}_W, F\cap W_0^{--}). \end{aligned}$$

Definition 9

The relative Nielsen number of ${\tilde{m}}_W$on the closure of the complement $N({\tilde{m}}_W,\overline{W_0\setminus W_0^{--}})$ is the number of the Nielsen fixed point classes of $m_W$ that do not assume its index in $W_0^{--}$ [21].

The following was proved in [17].

Theorem 10

Let W be a periodic segment over [0, T] and $W_0$, $W_0^{--}$ are ENRs. Then $P|_{U_W}$ has at least $N({\tilde{m}}_W,\overline{W_0\setminus W_0^{--}})$ fixed points.

Proposition 11

Assume that (W, Z) is a strong pair of periodic segments over [0, T]. Then

$$\begin{aligned} N({\tilde{m}}_W,\overline{W_0\setminus W_0^{--}})= {\left\{ \begin{array}{ll} 1 &{} \text {if} \; L({\overline{\mu }}_W)\ne \chi (W_0),\\ 0 &{} \text {if} \; L({\overline{\mu }}_{W})=\chi (W_0). \end{array}\right. } \end{aligned}$$

Proof

It follows by Lemma 6 that $m_W$ is homotopic to $\textrm{id}_{W_0}$, hence $m_W$ has exactly one fixed point class $F=\textrm{Fix}\, (m_W)$. Moreover, $\textrm{ind} (m_W, F)=L(m_W)=\chi (W_0)$ by the Lefschetz fixed point theorem. On the other hand,

$$\begin{aligned} L({\overline{\mu }}_W)=L({\overline{m}}_W)=\textrm{ind}({\overline{m}}_W, F\cap W_0^{--}), \end{aligned}$$

hence the result follows. $\square$

Remark 12

Proposition 11 shows that if (W, Z) is a strong pair of periodic segments, then the relative Nielsen number $N(m_W,\overline{W_0\setminus W_0^{--}})$ does not give more information about the fixed point of the Poincaré map P than the relative Lefschetz number $L({\tilde{\mu }}_W)=L({\tilde{m}}_W)=\chi (W_0)-L({\overline{m}}_W)$.

4 Fixed Point Index Formula and Chaotic Dynamics

Assume that $\Sigma _n=\{0,\ldots ,n-1\}^{{\mathbb {N}}}$ and $\sigma :\Sigma _n\longrightarrow \Sigma _n$ is the shift map. Let ${\mathbb {P}}\subset \Sigma _n=\{1,\ldots ,n\}^{{\mathbb {N}}}$ be the set of all periodic sequences of n symbols.

Definition 13

Let $\Phi$ be a T-periodic local semi-process. We say that the corresponding Poincaré map P has a $\Sigma _n$-weak chaotic dynamics on some compact, invariant set I if the following two conditions hold:

• (ch1) there exists a continuous, surjective map $q:I\longrightarrow \Sigma _n$ such that

  $$\begin{aligned} q\circ P|_I =\sigma \circ q, \end{aligned}$$

• (ch2) there exists $D\subset {\mathbb {P}}$ such that D is dense in $\Sigma _n$ and if $c\in D$ is a k-periodic sequence, then $g^{-1}(c)$ contains a k-periodic point of P.

We say that P has a $\Sigma _n$-chaotic dynamics if it has a $\Sigma _n$-weak chaotic dynamic with $D={\mathbb {P}}$.

Let (W, Z) be a strong pair of periodic segments over [0, T]. We define a Ważewski set $W^{\infty }\subset [0,\infty )\times X$ by:

$$\begin{aligned} W^{\infty }_{[kt,(k+1)T]}=\tau _k (W),\quad k\ge 0 \end{aligned}$$

i.e., $W^{\infty }$ is obtained by gluing together the translated copies of W. The set I is the set of all points in $W_0$ whose full trajectories are contained in $W^{\infty }$ i.e.,

$$\begin{aligned} I:=\Big \{x\in W_0:\Phi _{(0, t)}(x)\in W^{\infty }_t\; \text {for all}\; t\ge 0\Big \}. \end{aligned}$$

(3)

It follows that I is compact and invariant for P. For $x\in I$ we define $q(x)\in \Sigma _2$ by the following rule:

• if on the time interval $[iT,(i+1)T]$ the trajectory of x is contained in Z, then $q(x)_i=0$,

• if $P^i (x)$ leaves Z in time less than T, then $q(x)_i=1$.

The map q is continuous and satisfies (ch1) [13].

Corollary 14

Let $g:I\longrightarrow \Sigma _2$ be a continuous map defined above. If (ch2) holds then P has a $\Sigma _2$-weak chaotic dynamics.

Proof

It remains to show that g is surjective. It follows that $D\subset g(I)$, so $g(I)=\Sigma _2$ by density of D in $\Sigma _2$. $\square$

Assume that (W, Z) is a strong pair of periodic segments over [0, T]. Let $c=(c_0,\ldots ,c_{k-1})\in \{Z,W\}^k$ ($k\ge 2$) be a finite sequence. We define

$$\begin{aligned} W[c]=c_0\cup \tau _1 (c_1)\cup \ldots \cup \tau _{k-1}(c_{k-1}). \end{aligned}$$

It follows that $(W[c],Z^k)$ is a strong pair of periodic segments over [0, kT] and

$$\begin{aligned} {\tilde{\mu }}_{W[c]}={\tilde{\mu }}_{c_{k-1}}\circ \ldots \circ {\tilde{\mu }}_{c_0}. \end{aligned}$$

For $S\in \{Z,W\}$ we put

$$\begin{aligned} S^k=S\cup \tau _1 (S)\cup \ldots \cup \tau _{k-1}(S), \quad k\ge 2. \end{aligned}$$

We say that a point $x\in W[c]$ follows a sequence c if the following condition holds: $W[c]|_{[iT, (i+1)T]}=W$ implies that

$$\begin{aligned} \Phi _{(0,t)}(x)\in W_t\setminus Z_t, \end{aligned}$$

for some $t\in (iT, (i+1)T)$. We put

$$\begin{aligned} {\tilde{U}}_{W[c]}:=\big \{x\in U_{W[c]}: x\; \text {follows}\; c\big \}. \end{aligned}$$

Remark 15

It follows that ${\tilde{U}}_{W[c]}$ is open in $W[c]_0$ and the set of fixed points of the restriction $P^k|_{{\tilde{U}}_{W[c]} }:{\tilde{U}}_{W[c]} \longrightarrow W[c]_0$ is compact [13].

We are ready to formulate the main result of this section.

Theorem 16

Assume that (W, Z) is a strong pair of periodic segments and $c=(c_0,\ldots ,c_{k-1})\in \{Z,W\}^k$. Then

$$\begin{aligned} \textrm{ind}\left( P^k|_{{\tilde{U}}_{W[c]}}\right) =-L\left( \left( {\overline{\mu }}_{W}-I_{H(W_0^{--})}\right) ^{c_W}\right) , \end{aligned}$$

(4)

where $c_W:=\textrm{card}\, \{s\in \{0,\ldots , k-1\}: c_s=W\}\ge 1$.

Proof

It follows by Theorem 3.1 in [16] that

$$\begin{aligned} \textrm{ind}\left( P^k|_{{\tilde{U}}_{W[c]}}\right) =L\left( \left( {\tilde{\mu }}_{W}-I_{H(W_0, W_0^{--})}\right) ^{c_W}\right) . \end{aligned}$$

For $n\ge 1$ we have

$$\begin{aligned} L({\tilde{\mu }}^n)=L(m_W^n)-L({\overline{\mu }}_W^n)=\chi (W_0)-L({\overline{\mu }}_W^n), \end{aligned}$$

because $m_W\simeq \textrm{id}_{W_0}$ by Lemma 6. We put ${\tilde{\mu }}^0:=I_{H(W_0,W_0^{--})}$ and ${\overline{\mu }}_W^0:=I_{H(W_0^{--})}$. Consequently,

$$\begin{aligned} \textrm{ind}(P^k|_{{\tilde{U}}_{W[c]}})= & {} \sum _{l=0}^{c_W}(-1)^{c_W-l}\left( {\begin{array}{c}c_W\\ l\end{array}}\right) L({\tilde{\mu }}_W^l)\\= & {} \sum _{l=0}^{c_W}(-1)^{c_W-l}\left( {\begin{array}{c}c_W\\ l\end{array}}\right) (\chi (W_0)-L({\overline{\mu }}_W^l))= \sum _{l=0}^{c_W}(-1)^{c_W-l}\left( {\begin{array}{c}c_W\\ l\end{array}}\right) \chi (W_0)\\{} & {} -\sum _{l=0}^{c_W}(-1)^{c_W-l}\left( {\begin{array}{c}c_W\\ l\end{array}}\right) L({\overline{\mu }}_W^l))= L((I_{H(W_0)}-I_{H(W_0)})^{c_W})\\{} & {} -L(({\overline{\mu }}_{W} -I_{H(W_0^{--})})^{c_W})=-L(({\overline{\mu }}_{W}-I_{H(W_0^{--})})^{c_W}). \end{aligned}$$

$\square$

Corollary 17

Assume that (W, Z) is a strong pair of periodic segments over [0, T] for the local semi process $\Phi$. Let

$$\begin{aligned} D=\{c\in \{0,1\}^k: L(({\overline{m}}_W-I_{H(W_0^{--})})^{c_W})\ne 0\}. \end{aligned}$$

If $D\subset {\mathbb {P}}$ is dense in $\Sigma _2$ then the Poincaré map P has a $\Sigma _2$-weak chaotic dynamics.

Proof

Let $c\in D$. It follows by Theorem 16 that $g^{-1}(c)$ contains a fixed point of $P^k$, so (ch2) holds. $\square$

Example 18

Let (W, Z) be a strong pair of periodic segments over [0, T] such that $L({\overline{m}}_W)\ne \chi (W_0^{--})$ and ${\overline{m}}_W^2=\textrm{id}_{W_0^{--}}$. Then

$$\begin{aligned} L(({\overline{m}}_W-I_{H(W_0^{--})})^{c_W})= & {} \sum _{i=0}^{c_W} (-1)^{c_W-i}\left( {\begin{array}{c}c_W\\ i\end{array}}\right) L({\overline{m}}_W^i) \\= & {} (-1)^{c_W} 2^{c_W-1}(\chi (W_0^{--})-L({\overline{m}}_W))\ne 0. \end{aligned}$$

Example 19

For a given $l\in {\mathbb {N}}$ we consider the basic sequence $\textrm{reg}_l$ defined as

$$\begin{aligned} \textrm{reg}_l (n)= {\left\{ \begin{array}{ll} l &{} \text {if} \; l\equiv 0 \pmod {l},\\ 0 &{} \text {if} \; l\not \equiv 0 \pmod {l}. \end{array}\right. } \end{aligned}$$

Assume that $L({\overline{m}}_W^n)=\textrm{reg}_l (n)$ for $n\ge 1$. It follows by Remark 3.2 in [17] that $L(({\overline{m}}_W-I_{H(W_0^{--})})^{m})=0$ iff l is odd and m is odd multiplicity of l.

In order to shed more light on behavior of the sequence $L(({\overline{\mu }}_{W}-I_{H(W_0^{--})})^n)$, we will focus on the sequence $\textrm{tr}(A-I_k)^n$, where A is $k\times k$-integer matrix. If $\lambda \in \sigma (A)$ is the eigenvalue of A, then by $m_A(\lambda )$ we denote its algebraic multiplicity. Let ${\mathcal {R}}\subset {\mathbb {C}}$ be the set of all complex roots of unity i.e., $\lambda \in {\mathcal {R}}$ iff $\lambda ^n=1$ for some $n\in {\mathbb {N}}$. If A is an integer matrix then the following conditions are equivalent ([5]):

• (a) the sequence of traces $\textrm{tr}\, A^n$ is bounded,

• (b) the sequence $\textrm{tr}\, A^n$ is periodic,

• (c) $\sigma (A)\subset \{0\}\cup {\mathcal {R}}$.

Lemma 20

Assume that A is a $k\times k$-integer matrix and the sequence of traces $\textrm{tr}\, A^n$ is bounded. The following conditions are equivalent:

• (i) the sequence $\textrm{tr} (A-I_k)^n$ is bounded,

• (ii) $\sigma (A)\subset \{0,1,e^{\pi i/3}, e^{-\pi i /3}\}$.

Proof

Obviously, (ii) implies (i). We show that (i) implies (ii). If (i) holds then $\sigma (A), \sigma (A-I)\subset \{0\}\cup {\mathcal {R}}$. Let $\lambda \in \sigma (A)\setminus \{0,1\}$. It follows that $\lambda , \lambda -1\in {\mathcal {R}}\setminus \{1\}$. In particular, $1=|\lambda |=|\lambda -1|$ and consequently, $\lambda =e^{\pm \pi i/3}$. $\square$

5 Using the Connected Components of $Z^{--}$

Assume that (W, Z) is a strong pair of periodic segments over [0, T] for a local semi-process. Let E be the connected component of $Z^{--}$. We say that $x\in W_0$ leaves Zthrough E iff $\Phi _{(0,t)}(x)\in E$ for some $t\in [0,T]$. Let

$$\begin{aligned} F=\{x\in W_0:\text {x leaves Z through} \; E\}. \end{aligned}$$

We define $g_E:W_0\longrightarrow W_0$ by

$$\begin{aligned} g_E(x)={\left\{ \begin{array}{ll} \Phi _{(0,\sigma _Z (0,x))}(x) &{} \text {if} \; x\in W_0\setminus F,\\ m_{\sigma _W(0,x)}(\Phi _{(0,\sigma _W(0,x))}(x)) &{} \text {if} \; x\in F, \end{array}\right. } \end{aligned}$$

where

$$\begin{aligned} m_s:W_s\ni x\longrightarrow \pi _2h(T,\pi _2h^{-1}(s,x))\in W_0, \quad s\in [0,T]. \end{aligned}$$

Observe that $m_0=m_W$ and $m_T=\textrm{id}_{W_0}$.

Lemma 21

Assume that (W, Z) is a strong pair of periodic segments over [0, T] and E is a connected component of $Z^{--}$. The map $g_E:W_0\longrightarrow W_0$ has the following properties:

• (1) $g_E$ is continuous,

• (2) $g_E\simeq \textrm{id}_{W_0}$,

• (3) $g_E (W_0^{--})=W_0^{--}$,

• (4) $g_E|_{E_0}={\overline{m}}_W|_{E_0}$,

• (5) $g_E|_{W_0^{--}{\setminus } E_0}=\textrm{id}_{W_0^{--}{\setminus } E_0}$.

In particular, if ${\overline{m}}_W(E_0)\subset W_0^{--}\setminus E_0$ then $L(g_E|_{W_0^{--}})=\chi (W_0^{--})-\chi (E_0)$.

Proof

We have

$$\begin{aligned} \textrm{int} (F)= & {} \{x\in W_0:\text {x leaves Z through} \; E,\, \sigma _Z(0,x)<T\}, \\ F\setminus \textrm{int}(F)= & {} \{x\in W_0:\text {x leaves Z through} \; E,\,\sigma _Z(0,x)=T \}. \end{aligned}$$

It is sufficient to show that $g_E$ is continuous in $x_0\in F{\setminus } \textrm{int}(F)$. For $x_0\in F{\setminus } \textrm{int}(F)$ we have $\sigma _Z(0,x_0)=T$ and $P(x_0)\in E$. Since $Z\subset W$ and $Z_0^{--}=W_0^{--}$, so $\sigma _W(0,x_0)=T$. Let $x_n\in W_0{\setminus } F$ and $x_n\rightarrow x_0$. It follows by the continuity of $\sigma _Z$ that $\sigma _Z(0,x_n)\rightarrow T$ and

$$\begin{aligned} \Phi _{(0,\sigma _Z(0,x_n))}(x_n)\in (Z^{--}\setminus E)\cup (W_T\setminus W_T^{--}). \end{aligned}$$

Suppose that $\Phi _{(0,\sigma _Z(0,x_n))}(x_n)\in Z^{--}\setminus E$. Then $\Phi _{(0,\sigma _Z(0,x_n))}(x_n)\rightarrow P(x_0)\in E$. This leads to a contradiction because $Z^{--}\setminus E$ is closed. It follows that $\Phi _{(0,\sigma _Z(0,x_n))}(x_n)\in W_T{\setminus } W_T^{--}$, so $x_n\in U_Z$. In particular, $\sigma _Z(0,x_n)=T$ and $g_E(x_n)=P(x_n)$. On the other hand, we have $g_E(x_0)=P(x_0)$. Consequently, $g_E(x_n)\rightarrow g_E(x_0)$ and $g_E$ is continuous.

We show that $g_E$ is homotopic to $\textrm{id}_{W_0}$. Let $H:[0,1]\times W_0\longrightarrow W_0$ be given by

$$\begin{aligned} H(t,x)={\left\{ \begin{array}{ll} m_{tT}(\Phi _{(0,t\sigma _Z (0,x))}(x)) &{} \text {if} \; x\in W_0\setminus F,\\ m_{t\sigma _W(0,x)}(\Phi _{(0,t\sigma _W(0,x))}(x)) &{} \text {if} \; x\in F. \end{array}\right. } \end{aligned}$$

If $x\in F\setminus \textrm{int}(F)$ then $\sigma _Z(0,x)=\sigma _W(0,x)=T$, hence $H(t,x)=m_{tT}(\Phi _{(0,tT)}(x))$. Let $x_n\in W_0{\setminus } F$, $x_n\rightarrow x_0$ and $t_n\rightarrow t$. Then $\sigma _Z(0,x_n)\rightarrow T$, hence

$$\begin{aligned} H(t_n,x_n)=m_{t_nT}(\Phi _{(0,t_n\sigma _Z(0,x_n))}(x_n)\longrightarrow m_{tT}(\Phi _{(0,tT)}(x))=h(t,x). \end{aligned}$$

Consequently, H is continuous. We have $m_0=m_W$ and $m_1=g_E$, hence $g_E\simeq m_W$ and consequently, $g_E\simeq \textrm{id}_{W_0}$ by Lemma 6. $\square$

Example 22

Assume that $Z^{--}$ has a three connected components E(1), E(2), E(3), ${\overline{m}}_W^3|_{W_0^{--}}=\textrm{id}_{W_0^{--}}$ and

$$\begin{aligned} {\overline{m}}_W(E(1)_0)=E(2)_0, \quad {\overline{m}}_W(E(2)_0)=E(3)_0,\quad {\overline{m}}_W(E(3)_0)=E(1)_0. \end{aligned}$$

We claim that $g_{E(1)}\circ g_{E(2)}|_{W_0^{--}}\ne g_{E(2)}\circ g_{E(1)}|_{W_0^{--}}$. Both sides restricted to $E(3)_0$ are the identity maps. We have

$$\begin{aligned}{} & {} g_{E(1)}\circ g_{E(2)}(E(1)_0)=g_{E(1)}(E(1)_0)={\overline{m}}_W(E(1)_0)=E(2)_0, \\{} & {} g_{E(2)}\circ g_{E(1)}(E(1)_0)=g_{E(2)}({\overline{m}}_W(E(1)_0))=g_{E(2)}(E(2)_0)=E(3)_0. \end{aligned}$$

Nevertheless, $L(g_{E(1)}\circ g_{E(2)}|_{W_0^{--}})=L(g_{E(2)}\circ g_{E(1)}|_{W_0^{--}})=\chi (E(3)_0)$. On the other hand,

$$\begin{aligned} L(g_{(E(3)}\circ g_{E(2)}\circ g_{E(1)}|_{W_0^{--}})=2\chi (E(1)_0), \quad L(g_{(E(2)}\circ g_{E(3)}\circ g_{E(1)}|_{W_0^{--}})=0. \end{aligned}$$

For a connected component E of $Z^{--}$ we define

$$\begin{aligned} \eta (E_0)={\left\{ \begin{array}{ll} 0 &{} \text {if} \; E_0\; \text {is not invariant for}\; {\overline{m}}_W,\\ \chi (E_0) &{} \text {if} \; E_0\; \text {is invariant for}\; {\overline{m}}_W. \end{array}\right. }. \end{aligned}$$

Corollary 23

Let E be a connected component of $Z^{--}$. The following conditions hold:

• (i) if $E_0$ is invariant for ${\overline{m}}_W$ then $g_{E}=\textrm{id}_{W_0^{--}}$,

• (ii) $g_{E}^l|_{{W_0}^{--}}=g_{E}|_{{W_0}^{--}}$ ($l\ge 1$) and

  $$\begin{aligned} L(g_{E}|_{{W_0}^{--}})=\chi (W_0^{--})-\chi (E_0)+\eta (E_0). \end{aligned}$$

Theorem 24

Assume that (W, Z) is a strong pair of periodic segments and E is a component of $Z^{--}$. If $\chi (E_0)-\eta (E_0)\ne 0$ then the Poincaré map P has a fixed point $x\in W_0\setminus W_0^{--}$ such that

• (1) x leaves Z through E in time less then T,

• (2) $\Phi _{(0,t)}(x)\in W_t\setminus W_t^{--}$ for $t\in [0,T]$.

We do not provide a proof of the above result here since it is a corollary of a more general Theorem 31.

Let E, F be some two connected components of $Z^{--}$. We have the following four possibilities:

• (c1) ${\overline{m}}_W(E_0)=F_0$ and ${\overline{m}}_W(F_0)=E_0$,

• (c2) ${\overline{m}}_W(E_0)\ne F_0$ and ${\overline{m}}_W(F_0)\ne E_0$,

• (c3) ${\overline{m}}_W(E_0)= F_0$ and ${\overline{m}}_W(F_0)\ne E_0$,

• (c4) ${\overline{m}}_W(E_0)\ne F_0$ and ${\overline{m}}_W(F_0)= E_0$.

Corollary 25

Let E, F be some two connected components of $Z^{--}$. Then (c2) holds iff $g_{E}\circ g_{F}|_{W_0^{--}} = g_{F}\circ g_{E}|_{W_0^{--}}$

Proof

It follows that the both $g_{E}\circ g_{F}|_{W_0^{--}}$ and $g_{F}\circ g_{E}|_{W_0^{--}}$ restricted to $W_0^{--}{\setminus } (E_0\cup F_0)$ are the identity maps. We have

$$\begin{aligned} g_{E}\circ g_{F}|_{E_0}= & {} g_{E}|_{E_0}={\overline{m}}_W|_{E_0}, \\ g_{F}\circ g_{E}|_{E_0}= & {} g_{F}\circ {\overline{m}}_W|_{E_0}, \end{aligned}$$

so $g_{E}\circ g_{F}|_{E(i)_0}=g_{F}\circ g_{E}|_{E_0}$ iff ${\overline{m}}_W(E_0)\ne F_0$. By the symmetry, $g_{E}\circ g_{F}|_{F_0}=g_{F}\circ g_{E}|_{F_0}$ iff ${\overline{m}}_W(F_0)\ne E_0$, hence the proof is finished. $\square$

Lemma 26

Assume that E, F are connected components of $Z^{--}$ and $l\ge 1$. Then

$$\begin{aligned} L((g_{F}\circ g_{E})^l|_{W_0^{--}})={\left\{ \begin{array}{ll} \chi (W_0^{--})-\chi (E_0) &{} \text {if} \;\, (c1)\; holds,\\ \chi (W_0^{--})-\chi (E_0)-\chi (F_0)+\eta (E_0)+\eta (F_0) &{} \text {if} \;\, (c2)\; holds,\\ \chi (W_0^{--})-\chi (E_0)-\chi (F_0)+\eta (F_0) &{} \text {if} \;\, (c3)\; holds,\\ \chi (W_0^{--})-\chi (E_0)-\chi (F_0)+\eta (E_0) &{} \text {if} \;\, (c4)\; holds. \end{array}\right. } \end{aligned}$$

Proof

Obviously, $(g_F\circ g_E)^l|_{W_0^{--}}$ restricted to $W_0^{--}\setminus (E_0\cup F_0)$ is the identity map on $W_0^{--}\setminus (E_0\cup F_0)$. We have to inspect the action of $(g_F\circ g_E)^l$ on $E_0\cup F_0$.

Assume (c1).:

Then $\eta (E_0)=\eta (F_0)=0$, $\chi (E_0)=\chi (F_0)$, $(g_{F}\circ g_{E})^l|_{F_0}$ has no fixed points and $(g_{F}\circ g_{E})^l|_{E_0}=\textrm{id}_{E_0}$, hence the result follows.

Assume (c2).:

Then $(g_{F}\circ g_{E})^l|_{E_0\cup F_0}={\overline{m}}_W|_{E_0\cup F_0}$, so the formula holds.

Assume (c3).:

$\eta (E_0)=0$, $\chi (E_0)=\chi (F_0)$, $(g_{F}\circ g_{E})^l|_{E_0}$ has no fixed points and $(g_{F}\circ g_{E})^l|_{F_0}={\overline{m}}_W|_{F_0}$, hence the result follows.

Assume (c4).:

Then $\eta (F_0)=0$, $\chi (E_0)=\chi (F_0)$, $(g_{F}\circ g_{E})^l|_{E_0}$ has no fixed points and $(g_{F}\circ g_{E})^l|_{E_0}={\overline{m}}_W|_{E_0}$, so the proof is finished.

$\square$

Lemma 27

Assume that E, F are connected components of $Z^{--}$ and $l\ge 1$. Then

$$\begin{aligned} L\left( g_E \circ (g_{F}\circ g_{E})^l|_{W_0^{--}}\right) =L\left( (g_{F}\circ g_{E})^l|_{W_0^{--}}\right) . \end{aligned}$$

Proof

It follows by the same arguments like the ones used in the proof of Lemma 26. $\square$

Assume that $E(1),\ldots , E(n)$ are the connected components of $Z^{--}$. We put $g_Z:=m_Z=\textrm{id}_{W_0}$. Let ${\tilde{c}}=({\tilde{c}}_0, \ldots , {\tilde{c}}_{k-1})\in \{Z,E(1),\ldots , E(n)\}^k$. We consider a continuous map

$$\begin{aligned} g_{{\tilde{c}}}:=g_{{\tilde{c}}_{k-1}}\circ \ldots \circ g_{{\tilde{c}}_{0}}:W_0\longrightarrow W_0 \end{aligned}$$

Let $c\in \{Z,W\}^k$ be obtained from ${\tilde{c}}$ by replacing the symbols $E(1), \ldots , E(n)$ by W. We say that $x\in {\tilde{U}}_{W[c]}$ follows the sequence ${\tilde{c}}$ if ${\tilde{c}}_l\ne Z$ ($l=0,\ldots , k-1$) implies that $P^l(x)$ leaves Z through $c_l$. We put

$$\begin{aligned} {\tilde{U}}_{W[{\tilde{c}}]}=\{ x\in {\tilde{U}}_{W[c]}: x \; \text {follows}\; {\tilde{c}}\}. \end{aligned}$$

Let ${\tilde{b}}\in \{Z,E(1),\ldots , E(n)\}^k$. We say that ${\tilde{b}}<{\tilde{c}}$ if the following conditions hold:

• ${\tilde{c}}_i=Z$ implies ${\tilde{b}}_i=Z$,

• ${\tilde{c}}_i=E(s)$ implies ${\tilde{b}}_i\in \{Z, E(s)\}$,

• $1\le b_W<c_W$.

Definition 28

We say that a sequence ${\tilde{c}} \in \{Z,E(1),\ldots , E(n)\}^k$ is hereditary if $L(g_{{\tilde{b}}}|_{W_0^{--}})=L(g_{{\tilde{c}}}|_{W_0^{--}})$ for every ${\tilde{b}}<{\tilde{c}}$.

Corollary 29

If ${\tilde{c}}$ is a hereditary sequence and ${\tilde{b}}<{\tilde{c}}$ then ${\tilde{b}}$ is hereditary.

Corollary 30

Assume that E, F are connected components of $Z^{--}$.

• (1) If ${\tilde{c}}\in \{Z,E\}^k$ and $c_W>1$ then

  $$\begin{aligned} L(g_{{\tilde{c}}}|_{W_0^{--}})=\chi (W_0^{--})-\chi (E_0)+\eta (E_0). \end{aligned}$$

• (2) If (c1) holds, ${\tilde{c}}\in \{Z,E,F\}^k$ and $c_W>1$ then

  $$\begin{aligned} L(g_{{\tilde{c}}}|_{W_0^{--}})=\chi (W_0^{--})-\chi (E_0). \end{aligned}$$

  In particular, ${\tilde{c}}$ is a hereditary sequence.

Proof

The condition (1) is a consequence of Corollary 23. Assume that (c1) holds. Then $\eta (E_0)=\eta (F_0)=0$. By (1), we can assume that ${\tilde{c}}\in \{Z,E, F\}^k{\setminus } (\{Z,E\}^k\cup \{Z,F\}^k)$ with $c_W\ge 2$. Without loss of generality we assume that ${\tilde{c}}_0=E$. It follows by Corollary 23 that $g_{{\tilde{c}}}$ is one of the following form:

• (i) $g_{{\tilde{c}}}=(g_{F}\circ g_{E})^l$ for some $l\ge 1$,

• (ii) $g_{{\tilde{c}}}=g_{E}\circ (g_{F}\circ g_{E})^l$ for some $l\ge 0$.

Consequently, the result follows by Lemmas 26 and 27. $\square$

Theorem 31

Let ${\tilde{c}}\in \{Z, E(1),\ldots ,E(n)\}^k$. The set ${\tilde{U}}_{W[{\tilde{c}}]}$ is open in $W_0\setminus W_0^{--}$, the set of fixed point of $P^k$ restricted to ${\tilde{U}}_{W[{\tilde{c}}]}$ is compact and

$$\begin{aligned} \textrm{ind}(P^k|_{{\tilde{U}}_{W[{\tilde{c}}]}})=\chi (W_0^{--})-L(g_{{\tilde{c}}}|_{W_0^{--}})-\sum _{{\tilde{b}}<{\tilde{c}}} \textrm{ind}(P^k|_{{\tilde{U}}_{W[{\tilde{b}}]}}). \end{aligned}$$

(5)

In particular, if $c_W=1$ then

$$\begin{aligned} \textrm{ind}(P|_{{\tilde{U}}_{W[{\tilde{c}}]}})=\chi (W_0^{--})-L(g_{{\tilde{c}}}|_{W_0^{--}}). \end{aligned}$$

Proof

It follows by Theorem (8.3) in [13] that the set ${\tilde{U}}_{W[{\tilde{c}}]}$ is open in $W_0\setminus W_0^{--}$ and the set of fixed point of $P^k$ restricted to ${\tilde{U}}_{W[{\tilde{c}}]}$ is compact, so the fixed point index $\textrm{ind}(P^k|_{{\tilde{U}}_{W[{\tilde{c}}]}})$ is well-defined. It is easy to check that the set of fixed points of $g_{{\tilde{c}}}$ splits into a sum of compact sets:

$$\begin{aligned} \textrm{Fix}(g_{{\tilde{c}}})=\textrm{Fix}(P^k|_{{\tilde{U}}_{W[{\tilde{c}}]}})\cup \bigcup _{{\tilde{b}}<{\tilde{c}}}\textrm{Fix}(P^k|_{{\tilde{U}}_{W[{\tilde{b}}]}})\cup \textrm{Fix}(P^k|_{U_{Z^k}})\cup \textrm{Fix}(g_{{\tilde{c}}}|_{W_0^{--}}). \end{aligned}$$

Moreover, there exists an open neighborhood V of $W_0^{--}$ such that $g(V)\subset W_0^{--}$. By the commutativity property of the fixed point index we have

$$\begin{aligned} \textrm{ind}(g_{{\tilde{c}}}|_V)=L(g_{{\tilde{c}}}|_{W_0^{--}}). \end{aligned}$$

Moreover, $\textrm{ind}(P^k|_{U_{Z^k}})=\chi (W_0)-\chi (W_0^{--})$ by Theorem 5 and $L(g_{{\tilde{c}}})=\chi (W_0)$ by Lemma 21. By the Lefschetz fixed point theorem and the additivity property of the fixed point index we have

$$\begin{aligned} \chi (W_0)=L(g_{{\tilde{c}}})= & {} \textrm{ind}(P^k|_{{\tilde{U}}_{W[{\tilde{c}}]}})+\sum _{{\tilde{b}}<{\tilde{c}}} \textrm{ind}(P^k|_{{\tilde{U}}_{W[{\tilde{b}}]}})\\{} & {} +\chi (W_0)-\chi (W_0^{--})+L(g_{{\tilde{c}}}|_{W_0^{--}}), \end{aligned}$$

hence the result follows. $\square$

Corollary 32

Assume that (W, Z) is a strong pair of periodic segments for two semi-processes $\Phi _0$ and $\Phi _1$ on X. Then

$$\begin{aligned} \textrm{ind}(P_{\Phi _0}^k|_{{\tilde{U}}_{W[{\tilde{c}}]}})=\textrm{ind}(P_{\Phi _1}^k|_{{\tilde{U}}_{W[{\tilde{c}}]}}), \end{aligned}$$

(6)

for every sequence ${\tilde{c}}\in \{Z, E(1),\ldots ,E(n)\}^k$.

Proof

Observe that $L(g_{{\tilde{c}}}|_{W_0^{--}})$ depends only on ${\overline{m}}_W|_{W_0^{--}}$ and is independent on the local semi-processes $\Phi _i$ ($i=1,2$). Consequently, the result follows by Theorem 31 and the induction on $c_W$. $\square$

Remark 33

Corollary 32 is a generalization of Theorem 10 in [18]. The equality (6) was proved there under the following assumptions:

• $\Phi _i$ ($i=0,1$) are local semi-processes on ${\mathbb {R}}^d$ joined by a continuous family of local semi-processes $\Phi _t$ ($t\in [0,1]$),

• (W, Z) is a strong pair of periodic sequences for each $\Phi _t$,

• there exists $\eta >0$ such that for every $\lambda \in [0,1]$ and for every $x\in W^-$ ($x\in Z^-$) there exists $t>0$ such that for $0<\tau \le t$ holds $\Phi _{\tau }(x)\notin W$ and $d(\Phi _t(x),W)>\eta$ (resp. $\Phi _{\tau }(x)\notin Z$ and $d(\Phi _t(x),Z)>\eta$).

Lemma 34

Assume that ${\tilde{c}}\in \{Z, E(1),\ldots ,E(n)\}^k$ ($c_W\ge 1$) is a hereditary sequence. Then

$$\begin{aligned} \textrm{ind}(P^k|_{{\tilde{U}}_{W[{\tilde{c}}]}})=(-1)^{c_W+1}(\chi (W_0^{--})-L(g_{{\tilde{c}}}|_{W_0^{--}})). \end{aligned}$$

Proof

We use the induction with respect to $n=c_W$. If $n=1$ then the result follows by Theorem 31. Assume that formula holds for all sequences ${\tilde{b}}$ with $b_W\le n$. We prove it for a sequence ${\tilde{c}}$ with $c_W=n+1$. It follows that every ${\tilde{b}}<{\tilde{c}}$ is a hereditary sequence with $L(g_{{\tilde{b}}})=L(g_{{\tilde{c}}})$. By Theorem 31 and the inductive step we get

$$\begin{aligned} \textrm{ind}(P^k|_{{\tilde{U}}_{W[{\tilde{c}}]}})=\chi (W_0^{--})-L(g_{{\tilde{c}}}|_{W_0^{--}}) +\sum _{l=1}^n(-1)^l\left( {\begin{array}{c}n+1\\ l\end{array}}\right) (\chi (W_0^{--})-L(g_{{\tilde{c}}}|_{W_0^{--}})), \end{aligned}$$

hence the result follows by the equality

$$\begin{aligned} \sum _{l=1}^n(-1)^l\left( {\begin{array}{c}n+1\\ l\end{array}}\right) ={\left\{ \begin{array}{ll} -2 &{} \text {if} \; n\; \text {is odd}\\ 0 &{} \text {if} \; n\; \text {is even}. \end{array}\right. } \end{aligned}$$

$\square$

Corollary 35

Assume that E is a connected component of $Z^{--}$ and ${\tilde{c}}\in \{Z,E\}^k$ with $c_W\ge 1$. Then

$$\begin{aligned} \textrm{ind}(P^k|_{{\tilde{U}}_{W[{\tilde{c}}]}})=(-1)^{c_W+1}(\chi (E_0)-\eta (E_0)). \end{aligned}$$

In particular, the following conditions are equivalent:

• (I) $\textrm{ind}(P^k|_{{\tilde{U}}_{W[{\tilde{c}}]}})\ne 0$,

• (II) $E_0$ is not invariant for ${\overline{m}}_W$ and $\chi (E_0)\ne 0$.

Proof

It follows by Corollary 30 and Lemma 34. $\square$

Corollary 36

Assume that (W, Z) is a strong pair of periodic segments over [0, T] and

$$\begin{aligned} m[W]:=\textrm{card}\, \{E: E\; \text {is a connected component of}\; Z^{--}\; \text {with}\; \eta (E_0)=0\ne \chi (E_0)\}. \end{aligned}$$

Then the Poincaré map has at least m[W] fixed points in the set I.

Corollary 37

Assume that E, F are two connected components of $Z^{--}$ and (c1) holds. Then

$$\begin{aligned} \textrm{ind}(P^k|_{{\tilde{U}}_{W[{\tilde{c}}]}})=(-1)^{c_W+1}\chi (E_0), \end{aligned}$$

for every ${\tilde{c}}\in \{Z,E,F\}^k$ with $c_W\ge 1$.

Proof

It follows by Corollary 30 and Lemma 34. $\square$

Lemma 38

Assume that E, F are two connected components $Z^{--}$ and ${\tilde{c}}=(E,F)\in \{E,F\}^2$. Then

$$\begin{aligned} \textrm{ind}\, (P^2|_{{\tilde{U}}_{W[{\tilde{c}}]}})= {\left\{ \begin{array}{ll} -\chi (E_0) &{} \text {if} \;\, (c1)\; \text {holds},\\ 0 &{} \text {otherwise}. \end{array}\right. }. \end{aligned}$$

Proof

The formula is true if (c1) holds by Corollary 37. By Theorem 31 and Corollary 35 we get

$$\begin{aligned} \textrm{ind}(P^2|_{{\tilde{U}}_{W[{\tilde{c}}]}})= & {} \chi (W_0^{--})-L(g_{{\tilde{c}}}|_{W_0^{--}})-\textrm{ind}(P^2|_{{\tilde{U}}_{W[(E,Z)]}})-\textrm{ind}(P^2|_{{\tilde{U}}_{W[(Z,F)]}}) \\= & {} \chi (W_0^{--})-L(g_{{\tilde{c}}}|_{W_0^{--}})-\chi (E_0)+\eta (E_0)-\chi (F_0)+\eta (F_0). \end{aligned}$$

If (c1) does not hold then the result follows by Lemma 26. $\square$

Proposition 39

Assume that E, F are two connected components $Z^{--}$ and ${\tilde{c}}\in \{Z,E, F\}^k\setminus (\{Z,E\}^k\cup \{Z,F\}^k)$ with $c_W\ge 2$.. Then

$$\begin{aligned} \textrm{ind}\, (P^k|_{{\tilde{U}}_{W[{\tilde{c}}]}})= {\left\{ \begin{array}{ll} (-1)^{c_W+1}\chi (E_0) &{} \text {if} \;\, (c1)\; \text {holds},\\ 0 &{} \text {otherwise}. \end{array}\right. }. \end{aligned}$$

Proof

By Corollary 37 the formula is true if (c1) holds. Assume that (c1) does not hold. We proceed by the induction with respect to $n=c_W$. For $n=2$ the result follows by Lemma 38. Assume that the result is true for every sequence ${\tilde{b}}\in \{Z,E, F\}^k{\setminus } (\{Z,E\}^k\cup \{Z,F\}^k)$ with $2\le c_W\le n$. We prove it for ${\tilde{c}}\in \{Z,E, F\}^k{\setminus } (\{Z,E\}^k\cup \{Z,F\}^k)$ with $c_W=n+1$. By Theorem 31 and the inductive step we get

$$\begin{aligned} \textrm{ind}(P^k|_{{\tilde{U}}_{W[{\tilde{c}}]}})= & {} \chi (W_0^{--})-L(g_{{\tilde{c}}}|_{W_0^{--}}) \\{} & {} \quad -\sum _{{\tilde{b}}<{\tilde{c}}, \, {\tilde{b}}\in \{Z,E\}^k} \textrm{ind}(P^k|_{{\tilde{U}}_{W[{\tilde{b}}]}}) -\sum _{{\tilde{b}}<{\tilde{c}}, \, {\tilde{b}}\in \{Z,F\}^k} \textrm{ind}(P^k|_{{\tilde{U}}_{W[{\tilde{b}}]}}). \end{aligned}$$

Let $S\in \{E,F\}$. We denote by $1\le |S|\le n$ the number of appearances of S in ${\tilde{c}}$. It follows by Corollary 35 and $\sum _{l=0}^{|S|}(-1)^i\left( {\begin{array}{c}|S|\\ l\end{array}}\right) =(1-1)^{|S|}=0$ that

$$\begin{aligned} \sum _{{\tilde{b}}<{\tilde{c}}, \, {\tilde{b}}\in \{Z,S\}^k} \textrm{ind}(P^k|_{{\tilde{U}}_{W[{\tilde{b}}]}})=(\chi (S_0)-\eta (S_0)) \sum _{l=1}^{|S|}(-1)^{l+1}\left( {\begin{array}{c}|S|\\ l\end{array}}\right) =\chi (S_0)-\eta (S_0), \end{aligned}$$

hence the result follows by Lemmas 26 and 27. $\square$

Theorem 40

Let (W, Z) be a strong pair of periodic segments over [0, T]. Assume that E is a connected component of $Z^{--}$ satisfying the following conditions:

• (a) $E_0$ is not invariant for ${\overline{m}}_W$,

• (b) $\chi (E_0)\ne 0$,

• (c) $\chi (W_0)\ne \chi (W_0^{--})$.

Then P has a $\Sigma _2$-chaotic dynamics. Moreover,

$$\begin{aligned} \textrm{card}\, (\textrm{Fix}\, (P^k)\cap {\tilde{U}}_{W[c]})\ge m[W]. \end{aligned}$$

Proof

It is sufficient to show that (ch2) holds with $D={\mathbb {P}}$. Let $c\in \{Z,W\}^k$. If $c_W=0$ then $W[c]=Z^k$ and by Theorem 5 we get

$$\begin{aligned} \textrm{ind}(P^k|_{U_{Z^k}})=\chi (W_0)-\chi (W_0^{--})\ne 0. \end{aligned}$$

For $c_W\ge 1$, we consider a sequence ${\tilde{c}}\in \{Z, E\}^k$ obtained from c by replacing all symbols W by E. It follows by Corollary 35 that

$$\begin{aligned} \textrm{ind}(P^k|_{{\tilde{U}}_{W[{\tilde{c}}]}})=(-1)^{c_W+1}\chi (E_0)\ne 0. \end{aligned}$$

The result follows because $\textrm{Fix}(P^k|_{{\tilde{U}}_{W[{\tilde{c}}]}})\ne \emptyset$ and ${\tilde{U}}_{W[{\tilde{c}}]}\subset {\tilde{U}}_{W[c]}$. $\square$

Example 41

Consider the following planar non-autonomous equation $(n\ge 1)$

$$\begin{aligned} {\dot{z}}=(1+e^{i\kappa t}|z|^2){\overline{z}}^n,\quad z\in {\mathbb {C}}, \end{aligned}$$

(7)

where $\kappa >0$ is a real parameter. The Eq. (7) is $T=2\pi /\kappa$ periodic. Let $\Phi$ be a generated T-periodic local process. It follows by Example (6.8) in [13] that for sufficiently small $\kappa$ the process $\Phi$ admits a strong periodic segment W(n) over [0, T]. We describe briefly how the segments W(n) look like. The segment W(n) is a twisted prism with a $2(n+1)$-gon base $W(n)_0$ centered at origin. Its time sections $W(n)_t$ are obtained by rotating $W(n)_0$ with the angular velocity $\frac{\kappa }{n+1}$ over the the time interval [0, T]. The essential exit set $W(n)^{--}$ consist of $n+1$ disjoint ribbons winding around the prism.

Proposition 42

Let P be a Poincaré map associated to Eq. (7). Then P has a $\Sigma _2$-chaotic dynamics.

Remark 43

It follows by Theorem 6.4 in [13] that P has $\Sigma _2$-chaotic dynamics if n is odd and P has $\Sigma _2$-weak chaotic dynamics if n is even.

Proposition 44

Assume that $Z^{--}$ has two connected components E(1), E(2), (c1) holds and $2\chi (E(1)_0)\notin \{0, \chi (W_0)\}$. Then the Poincaré map P has a $\Sigma _3$-chaotic dynamics.

Proof

Let the compact set I be defined by (3). For $x\in I$ we define $q(x)\in \Sigma _3$ by the following rule:

• if on the time interval $[iT,(i+1)T]$ the trajectory of x is contained in Z, then $q(x)_i=0$,

• if $P^i (x)$ leaves Z in time less than T through E(s), then $q(x)_i=s$ ($s=1,2$).

It follows that the map q is continuous and satisfies (ch1). It follows by Corollary 37 and Theorem 5 that (ch2) holds. $\square$