In this paper, we study the following max-type system of difference equations of higher order: $$ \textstyle\begin{cases} x_{n} = \max \{A ,\frac{y_{n-t}}{x_{n-s}} \}, \\ y_{n} = \max \{B ,\frac{x_{n-t}}{y_{n-s}} \},\end{cases}\displaystyle \quad n\in \{0,1,2,\ldots \}, $${xn=max{A,yn−txn−s},yn=max{B,xn−tyn−s},n∈{0,1,2,…}, where $A,B\in (0, +\infty )$A,B∈(0,+∞) , $t,s\in \{1,2,\ldots \}$t,s∈{1,2,…} with $\gcd (s,t)=1$gcd(s,t)=1 , the initial values $x_{-d},y_{-d},x_{-d+1},y_{-d+1}, \ldots , x_{-1}, y_{-1}\in (0,+ \infty )$x−d,y−d,x−d+1,y−d+1,…,x−1,y−1∈(0,+∞) and $d=\max \{t,s\}$d=max{t,s} .

In this paper, we study global behavior of the following max-type system of difference equations of the second order with four variables and period-two parameters $ \left\{\begin{array}{ll}x_{n} = \max\Big\{A_n , \frac{z_{n-1}}{y_{n-2}}\Big\}, \ y_{n} = \max \Big\{B_n, \frac{w_{n-1}}{x_{n-2}}\Big\}, \ z_{n} = \max\Big\{C_n , \frac{x_{n-1}}{w_{n-2}}\Big\}, \ w_{n} = \max \Big\{D_n, \frac{y_{n-1}}{z_{n-2}}\Big\}, \ \end{array}\right. \ \ n\in \{0, 1, 2, \cdots\}, $ where $ A_n, B_n, C_n, D_n\in (0, +\infty) $ are periodic sequences with period 2 and the initial values $ x_{-i}, y_{-i}, z_{-i}, w_{-i}\in (0, +\infty)\ (1\leq i\leq 2) $. We show that if $ \min\{A_0C_1, B_0D_1, A_1C_0, B_1D_0\} < 1 $, then this system has unbounded solutions. Also, if $ \min\{A_0C_1, B_0D_1, A_1C_0, B_1D_0\}\geq 1 $, then every solution of this system is eventually periodic with period $ 4 $.

Let f be a continuous map on a dendrite D with f (D) = D. Denote by R(f ) and AP (f ) the set of recurrent points and the set of almost periodic points of f , respectively, and denote by ω(x, f ), Λ(x, f ), Γ(x, f ) and Ω(x, f ) the set of ω-limit points, the set of α-limit points, the set of γ-limit points and the set of weak ω-limit points of x under f , respectively. In this paper, we show that the following statements are equivalent: (1) D = R(f ). (2) D = AP (f ). (3) Ω(x, f ) = ω(x, f ) for any x ∈ D. (4) Ω(x, f ) = Γ(x, f ) for any x ∈ D. (5) f is equicontinuous. (6) [c, d] ⊄ Ω(x, f ) for any c, d,x ∈ D with c ≠ (7) Ω(x, f ) is minimal for any x ∈ D. (8) Card(Λ ^{− 1}(x, f ) ∩ (D−End(D))) < ∞ for any x ∈ D, where Λ ^{− 1}(x, f ) = {y : x ∈ Λ(y, f )}, End(D) is the set of endpoints of D and Card(A) is the cardinal number of set A. (9) If x ∈ Λ(y, f ) with x, y ∈ D, then y ∈ ω(x, f ). (10) Map h : x → ω(x, f ) (x ∈ D) is continuous and for any x, y ∈ D with x ∉ ω(y, f ), ω(x, f ) ≠ ω(y, f ). Besides, we also study characteristic of pointwise-recurrent maps on a dendrite with the number of branch points being finite.

Let $I$ be a fuzzy metric interval and $f$ be a continuous map from $I$ to $I$. Denote by $R(f)$, $\Omega(f)$ and $\omega (x,f)$ the set of recurrent points of $f$, the set of non-wandering points of $f$ and the set of $\omega$- limit points of $x$ under $f$, respectively. Write $\omega(f)=\cup_{x\in I}\omega(x,f)$, $\omega^{n+1}(f)=\cup_{x\in \omega^n(f)}\omega (x,f)$ and $\Omega_{n+1} (f)=\Omega (f|_{\Omega_{n} (f)})$ for any positive integer $n$. In this paper, we show that $\Omega_2(f)=\overline{R(f)}$ and the depth of $f$ is at most 2, and $\omega^3(f)=\omega^2 (f)$ and the depth of the attracting centre of $f$ is at most 2.