2$. Analogously to the case of square-free words, we introduce also the notions of descendant, ancestor, and closed word for minimally repetitive words, and denote by ${\cal L}_m$ the set of all words from $\Sigma^*_3$ which do not contain closed words from ${\cal F}(m)$ as factors. Denote also by ${\cal F}_m$ the set of all minimally repetitive words from ${\cal L}_m$, by ${\cal F}'(m)$ the set of all words~$w$ from ${\cal F}(m)$ such that $w[1]=0$ and $w[2]=1$, and by ${\cal F}''(m)$ the set of all words from ${\cal F}'(m)$ which are not closed. As in the case of square-free words, we introduce the notions of quasi-descendant and quasi-ancestor, define for ${\cal F}''(m)$ the matrix $\Delta_m$ of size $s\times s$ where $s=|{\cal F}''(m)|$, and compute the maximal in modulus eigenvalue~$r$ of this matrix. If $r>1$ and all components of the eigenvector $\tilde x=(x_1;\ldots; x_s)$ corresponding to~$r$ are positive, then we denote by $\mu$ the ratio $\max_i x_i/\min_i x_i$, and for $n\ge m$ consider $S^{\langle\mbox{lf}\rangle}_m(n)=\sum_{i=1}^s x_i\cdot |{\cal F}_m^{(w_i)}(n)|$ where $w_i$ is $i$-th word of the set ${\cal F}''(m)$, $i=1,\ldots ,s$. Analogously to equality~(\ref{Fwin}), for $i=1,\ldots ,s$ we have \begin{equation} |{\cal F}_m^{(w_i)}(n+1)|=|{\cal G}^{(w_i)}(n+1)|-|{\cal H}^{(w_i)}(n+1)| \label{Fwin-1} \end{equation} where ${\cal G}^{(w_i)}(n+1)$ is the set of all words~$w$ from ${\cal L}_m^{(w_i)}(n+1)$ such that the words $w[1:n]$ and $w[n-m+1:n+1]$ are minimally repetitive, and ${\cal H}^{(w_i)}(n+1)$ is the set of all words from ${\cal G}^{(w_i)}(n+1)$ which contain some prohibited repetition as a suffix. Analogously to equality~(\ref{Lwin}), we can obtain $$ |{\cal G}^{(w_i)}(n+1)|=\sum_{w\in\pi (i)} |{\cal F}_m^{(w)}(n)| $$ where $\pi (i)$ is the set of all quasi-ancestors of $w_i$. For any word~$w$ from ${\cal H}^{(w_i)}(n+1)$ denote by~$\lambda (w)$ the minimal period of the shortest prohibited repetition which is a suffix of~$w$. Then, analogously to equality~(\ref{Lwin2}), $$ |{\cal H}^{(w_i)}(n+1)|=\sum_{\lfloor (4m+3)/7\rfloor\lfloor 7p/4\rfloor$. Let $w$ be an arbitrary word from ${\cal H}_j^{(w_i)}(n+1)$ where $j\le p$. Then the suffix $w[n-\lfloor 7p/4\rfloor +1:n+1]$ is a prohibited repetition which contains neither closed words from ${\cal F}(m)$ nor other prohibited repetitions as factors and contains the word $w_i$ as a suffix. Let $v_1,\ldots,v_t$ be all possible prohibited repetitions with minimal period~$j$ which satisfy the given conditions. Denote by ${\cal H}_{j,k}^{(w_i)}(n+1)$ the set of all words from ${\cal H}_j^{(w_i)}(n+1)$ which contain $v_k$ as a suffix, $k=1,\ldots, t$. Let $w\in {\cal H}_{j,k}^{(w_i)}(n+1)$. Analogously to the case of square-free words, the symbol $w[n-\lfloor 7j/4\rfloor ]$ is determined uniquely by $v_k$ as the symbol from $\Sigma_3$ which is different from the two distinct symbols $v_k[1]$ and $v_k[j]$. Denoting this symbol by $b_k$, we conclude that the factor $w[n-\lfloor 7j/4\rfloor :n]$ is determined uniquely as the word $b_k v_k[1:\lfloor 7j/4\rfloor ]$. Let this word belong to ${\cal F}_m$. Then we denote by $u'_k$ the word from ${\cal F}'(m)$ which is isomorphic to the word $b_k v_k[1:m-1]$. Analogously to inequality~(\ref{Ljwin}), one can obtain the inequality $$ |{\cal H}_j^{(w_i)}(n+1)|\le\sum_{u\in U_j(w_i)} |{\cal F}_m^{(u)}(n+m-\lfloor 7j/4\rfloor -1)| $$ where $U_j(w_i)$ is the set of all words\footnote{Note that, as in the case of square-free words, the same word can be counted several times in $U_j(w_i)$.} $u'_k$. Thus, \begin{equation} |{\cal H}^{(w_i)}(n+1)|\le A_p^{(w_i)}(n+1)+ \sum_{p p$, i.e., $j\ge 4m/3$. Note that the sets ${\cal H}_j^{(w_i)}(n+1)$ are non-overlapping. So we have the obvious inequality \begin{equation} \sum_{i=1}^s x_i\cdot |{\cal H}_j^{(w_i)}(n+1)|\le |{\cal M}_j|\cdot \max_{i=1,\ldots,s} x_i \label{sumHj} \end{equation} where ${\cal M}_j=\bigcup_{i=1}^s {\cal H}_j^{(w_i)}(n+1)$. Note also that any word $w$ from ${\cal M}_j$ is determined uniquely by the prefix $w[1:n-\lfloor 3j/4\rfloor ]$ and satisfies the conditions $w[n+2-m-j]=w[n+2-m]=0$ and $w[n+3-m-j]=w[n+3-m]=1$. Thus $|{\cal M}_j|\le |{\cal M}'_j|$ where ${\cal M}'_j$ is the set of all words $w$ from ${\cal F}_m(n-\lfloor 3j/4\rfloor )$ such that $w[n+2-m-j]=0$ and $w[n+3-m-j]=1$. Consider also the set ${\cal M}''_j$ of all words $w$ from ${\cal F}_m(n-\lfloor 3j/4\rfloor )$ such that $w[n+1-\lfloor 3j/4\rfloor -m]=0$ and $w[n+2-\lfloor 3j/4\rfloor -m]=1$. There is an evident bijection between the sets ${\cal M}'_j$ and ${\cal M}''_j$, so $|{\cal M}'_j|=|{\cal M}''_j|$. Note also that the set ${\cal M}''_j$ is the union of the non-overlapping sets ${\cal H}_j^{(w_i)}(n-\lfloor 3j/4\rfloor )$ for $i=1,\ldots, s$, i.e., $$ |{\cal M}''_j|\le \sum_{i=1}^s |{\cal H}_j^{(w_i)}(n-\lfloor 3j/4\rfloor )| \le S^{\langle\mbox{lf}\rangle}_m(n-\lfloor 3j/4\rfloor )/ (\min_{i=1,\ldots,s} x_i). $$ Therefore, it follows from~(\ref{sumHj}) that $$ \sum_{i=1}^s x_i\cdot |{\cal H}_j^{(w_i)}(n+1)|\le |{\cal M}'_j|\cdot \max_{i=1,\ldots,s} x_i = |{\cal M}''_j|\cdot \max_{i=1,\ldots,s} x_i \le \mu\cdot S^{\langle\mbox{lf}\rangle}_m(n-\lfloor 3j/4\rfloor ). $$ Thus, \begin{equation} \sum_{p 1$ and each $i=m, m+1,\ldots , n-1$ the inequality $S^{\langle\mbox{lf}\rangle}_m(i+1)\ge \alpha S^{\langle\mbox{lf}\rangle}_m(i)$ be valid, i.e., $S^{\langle\mbox{lf}\rangle}_m(i)\le S^{\langle\mbox{lf}\rangle}_m(n)/\alpha^{n-i}$. Then \begin{equation} \sum_{i=1}^s x_i\cdot A_p^{(w_i)}(n+1)\le \sum_{d=d_0}^q \rho_d\cdot (S^{\langle\mbox{lf}\rangle}_m(n)/\alpha^d)= {\cal P}_m^{(p,q)}(1/\alpha)\cdot S^{\langle\mbox{lf}\rangle}_m(n) \label{sumA_p} \end{equation} where ${\cal P}_m^{(p,q)}(z)=\sum_{d=d_0}^q \rho_d\cdot z^d$. Moreover, it follows from~(\ref{sumplej}) that $$ \begin{array}{c} \displaystyle \sum_{p &S^{\langle\mbox{lf}\rangle}_m(n) \cdot\Biggl(r-{\cal P}_m^{(p,q)}(1/\alpha)-\\ &&\mu\cdot\Bigl(1/\bigl(\alpha^{\lfloor (3p-1)/4\rfloor} (\alpha -1)\bigr) +1/\bigl(\alpha^{3\lfloor p/4\rfloor}(\alpha^3-1)\bigr)\Bigr)\Biggr). \end{eqnarray*} Therefore, if $$ r-{\cal P}_m^{(p,q)}(1/\alpha)-\mu\cdot\Bigl( 1/\bigl(\alpha^{\lfloor (3p-1)/4\rfloor} (\alpha -1)\bigr)+ 1/\bigl(\alpha^{3\lfloor p/4\rfloor}(\alpha^3-1)\bigr)\Bigr) \ge\alpha $$ then $S^{\langle\mbox{lf}\rangle}_m(n+1)\ge \alpha S^{\langle\mbox{lf}\rangle}_m(n)$ for any~$n$, i.e., $S^{\langle\mbox{lf}\rangle}_m(n)=\Omega (\alpha^n)$. Since the order of growth of $S^{\langle\mbox{lf}\rangle}(n)$ is not less than $S^{\langle\mbox{lf}\rangle}_m(n)$, in this case we have $S^{\langle\mbox{lf}\rangle}(n)=\Omega (\alpha^n)$, i.e., $\gamma^{\langle\mbox{lf}\rangle}\ge\alpha$. Using computer computations with the parameters $m=42$, $p=72$, $q=85$, we obtained that $|{\cal F}''(42)|=36141$, $r=1.247500$, all components of the eigenvector corresponding to~$r$ were positive, and ${\cal P}_{42}^{(72,85)}(z)$ was $$ \begin{array}{l} 1.976268\cdot z^{42} + 1.148062\cdot z^{44} + 3.519576\cdot z^{45} + 1.741046\cdot z^{47} + \\ 9.687624\cdot z^{49} + 0.126312\cdot z^{50}+ 31.479339\cdot z^{52} + 12.284335\cdot z^{53} + \\ 21.010557\cdot z^{54} + 24.183001\cdot z^{56} + 96.529327\cdot z^{61} + 129.216325\cdot z^{64} + \\ 256.213310\cdot z^{66} + 14.826731\cdot z^{67} + 64.163103\cdot z^{68} + 6.862805\cdot z^{69} + \\ 84.819931\cdot z^{70} + 2.337610\cdot z^{72} + 175.026144\cdot z^{73} + 41.068102\cdot z^{74} + \\ 335.714818\cdot z^{75} + 341.576384\cdot z^{78} + 329.970329\cdot z^{80} + 693.282157\cdot z^{81} + \\ 763.104210\cdot z^{82} + 303.272754\cdot z^{83} + 583.157071\cdot z^{84} + 10510.070498\cdot z^{85}. \end{array} $$ Let $\alpha =1.245$. It is immediately checked that $$ r-{\cal P}_{42}^{(72,85)}(1/\alpha)-\frac{1}{\alpha^{53}(\alpha-1)}- \frac{1}{\alpha^{54}(\alpha^3-1)}\ge\alpha. $$ Moreover, we estimate $S^{\langle\mbox{lf}\rangle}_{42}(n+1)\ge \alpha S^{\langle\mbox{lf}\rangle}_{42}(n)$ for each $n=42, 43,\ldots , q+m-1=126$ in the same inductive way with evident modifications following from the restriction $n 2$ we can introduce the notions of descendant, ancestor, and closed word for cube-free words. Denote also by ${\cal L}_m$ the set of all words from $\Sigma^*_2$ which do not contain closed words from ${\cal F}(m)$ as factors, and by ${\cal F}_m$ the set of all cube-free words from ${\cal L}_m$. By ${\cal F}'(m)$ we denote the set of all words~$w$ from ${\cal F}(m)$ such that $w[1]=0$. Note that for any word~$w$ from ${\cal F}(m)$ there exists a single word from ${\cal F}'(m)$ which is isomorphic to~$w$. By ${\cal F}''(m)$ we denote the set of all words from ${\cal F}'(m)$ which are not closed. We introduce also the notions of quasi-descendant and quasi-ancestor, define for ${\cal F}''(m)$ the matrix $\Delta_m$ of size $s\times s$ where $s=|{\cal F}''(m)|$, and compute the maximal in modulus eigenvalue~$r$ of this matrix. If $r>1$ and all components of the eigenvector $\tilde x=(x_1;\ldots; x_s)$ corresponding to~$r$ are positive, then for $n\ge m$ we consider $S^{\langle\mbox{cf}\rangle}_m(n)=\sum_{i=1}^s x_i\cdot |{\cal F}_m^{(w_i)}(n)|$ where $w_i$ is $i$-th word of the set ${\cal F}''(m)$, $i=1,\ldots ,s$. As in the case of square-free words, for $i=1,\ldots ,s$ we have \begin{equation} |{\cal F}_m^{(w_i)}(n+1)|=|{\cal G}^{(w_i)}(n+1)|-|{\cal H}^{(w_i)}(n+1)| \label{Fwin-3} \end{equation} where ${\cal G}^{(w_i)}(n+1)$ is the set of all words~$w$ from ${\cal L}_m^{(w_i)}(n+1)$ such that the words $w[1:n]$ and $w[n-m+1:n+1]$ are cube-free, and ${\cal H}^{(w_i)}(n+1)$ is the set of all words from ${\cal G}^{(w_i)}(n+1)$ which contain some cube as a suffix. Analogously to equality~(\ref{Lwin}), we obtain $$ |{\cal G}^{(w_i)}(n+1)|=\sum_{w\in\pi (i)} |{\cal F}_m^{(w)}(n)| $$ where $\pi (i)$ is the set of all quasi-ancestors of $w_i$. For any word~$w$ from ${\cal H}^{(w_i)}(n+1)$ denote by~$\lambda (w)$ the period of the minimal cube which is a suffix of~$w$. Then, analogously to equality~(\ref{Lwin2}), \begin{equation} |{\cal H}^{(w_i)}(n+1)|=\sum_{\lfloor (m+1)/3\rfloorp$, analogously to inequality~(\ref{Ljwin1}), we have \begin{equation} |{\cal H}_j^{(w_i)}(n+1)|\le |{\cal F}_m^{(w_i)}(n-2j+1)|. \label{Ljwin-1} \end{equation} Thus, from (\ref{Lwin-2}), (\ref{Ljwin0}) and~(\ref{Ljwin-1}) we obtain that \begin{equation} |{\cal H}^{(w_i)}(n+1)|\le A_p^{(w_i)}(n+1)+ B_p^{(w_i)}(n+1) \label{Ljwin3} \end{equation} where \begin{eqnarray*} A_p^{(w_i)}(n+1)&=&\sum_{\lfloor (m+1)/3\rfloor 1$ and each $i=m, m+1,\ldots , n-1$ the inequalities $S^{\langle\mbox{cf}\rangle}_m(i+1)\ge \alpha S^{\langle\mbox{cf}\rangle}_m(i)$ be valid. Then $$ \sum_{i=1}^s x_i\cdot A_p^{(w_i)}(n+1)\le \sum_{d=d_0}^q \rho_d\cdot (S^{\langle\mbox{cf}\rangle}_m(n)/\alpha^d)={\cal P}_m^{(p,q)}(1/\alpha)\cdot S^{\langle\mbox{cf}\rangle}_m(n), $$ where ${\cal P}_m^{(p,q)}(z)=\sum_{d=d_0}^q \rho_d\cdot z^d$. Moreover, it follows from~(\ref{sum_pjl1}) that $$ \sum_{i=1}^s x_i\cdot B_p^{(w_i)}(n+1)\le \sum_{p &S^{\langle\mbox{cf}\rangle}_m(n)\cdot\left(r-{\cal P}_m^{(p,q)}(1/\alpha)- \frac{1}{\alpha^{2p-1}(\alpha^2-1)}\right). \end{eqnarray*} Therefore, if $$ r-{\cal P}_m^{(p,q)}(1/\alpha)-\frac{1}{\alpha^{2p-1}(\alpha^2-1)} \ge\alpha, $$ then $S^{\langle\mbox{cf}\rangle}_m({n+1})\ge \alpha S^{\langle\mbox{cf}\rangle}_m(n)$ for any~$n$, i.~e. $S^{\langle\mbox{cf}\rangle}_m(n)=\Omega (\alpha^n)$. Since the order of growth of $S^{\langle\mbox{cf}\rangle}(n)$ is not less than $S^{\langle\mbox{cf}\rangle}_m(n)$, we obtain in this case that $S^{\langle\mbox{cf}\rangle}(n)= \Omega (\alpha^n)$, i.~e. $\gamma^{\langle\mbox{cf}\rangle}\ge\alpha$. Using computer computations with the parameters $m=35$, $p=35$, $q=70$, we obtained that $|{\cal F}''(35)|=732274$, $r=1.457599$, all components of the eigenvector corresponding to~$r$ were positive, and ${\cal P}_{35}^{(35,70)}(z)$ was $$ \begin{array}{l} 0.890340\cdot z^{35} + 1.398382\cdot z^{37} + 1.096456\cdot z^{38} + 30.292784\cdot z^{40} + \\ 2.533687\cdot z^{41} + 1.296919\cdot z^{42} + 28.893958\cdot z^{43} + 22.780262\cdot z^{44} + \\ 10.699704\cdot z^{45} + 64.314464\cdot z^{47} + 92.853910\cdot z^{49} + 91.743094\cdot z^{50} + \\ 67.688387\cdot z^{51} + 48.613345\cdot z^{52} + 68.285930\cdot z^{53} + 113.239316\cdot z^{54} + \\ 144.612325\cdot z^{56} + 346.136318\cdot z^{58} + 173.468149\cdot z^{59} + 465.000388\cdot z^{60} + \\ 134.993653\cdot z^{61} + 224.831969\cdot z^{62} + 585.928351\cdot z^{63} + 355.591901\cdot z^{65} + \\ 1335.518621\cdot z^{67} + 343.074473\cdot z^{68} + 2202.468159\cdot z^{69} + 11098.126369\cdot z^{70}. \end{array} $$ It is immediately checked that for $\alpha =1.457567$ the inequality $$ r-{\cal P}_{35}^{(35,70)}(1/\alpha)-\frac{1}{\alpha^{69}(\alpha^2-1)} \ge\alpha $$ is valid. Moreover, the inequalities $S^{\langle\mbox{cf}\rangle}_{35}(n+1) \ge \alpha S^{\langle\mbox{cf}\rangle}_{35}(n)$ for $n=35, 36,\ldots ,q+m-1=104$ are also verified in the same inductive way with evident modifications following from the restriction $n