Given an additive monoidal category $\mathcal{M}$ , a braided Lie algebra in $\mathcal{M}$ consists of a triple $\left(L,c,\left[-\right]:L\otimes L\rightarrow L\right)$ where $\left(L,c\right)$ is a braided object and the following equalities hold true:

	$$\displaystyle\left[-\right]=-\left[-\right]\circ c\text{ (skew-symmetry);}$$		(1.12)
	$$\displaystyle\left[-\right]\circ\left(L\otimes\left[-\right]\right)\circ\left[\mathrm{Id}_{L\otimes\left(L\otimes L\right)}+\left(L\otimes c\right)a_{L,L,L}\left(c\otimes L\right)a_{L,L,L}^{-1}+a_{L,L,L}\left(c\otimes L\right)a_{L,L,L}^{-1}\left(L\otimes c\right)\right]=0$$		(1.13)
	(Jacobi condition);
	$$\displaystyle c\circ\left(L\otimes\left[-\right]\right)a_{L,L,L}=\left(\left[-\right]\otimes L\right)a_{L,L,L}^{-1}\left(L\otimes c\right)a_{L,L,L}\left(c\otimes L\right);$$		(1.14)
	$$\displaystyle c\circ\left(\left[-\right]\otimes L\right)a_{L,L,L}^{-1}=\left(L\otimes\left[-\right]\right)a_{L,L,L}\left(c\otimes L\right)a_{L,L,L}^{-1}\left(L\otimes c\right).$$		(1.15)

Let $\mathcal{M}$ be an additive braided monoidal category. A Lie algebra in $\mathcal{M}$ consists of a pair $\left(L,\left[-\right]:L\otimes L\rightarrow L\right)$ such that $\left(L,c_{L,L},\left[-\right]\right)$ is a braided Lie algebra in the additive monoidal category $\mathcal{M}$ , where $c_{L,L}$ is the braiding $c$ of $\mathcal{M}$ evaluated on $L$ (note that in this case the conditions ( 1.14 ) and ( 1.15 ) are automatically satisfied).

Let ${\Bbbk}$ be a field. A BiHom-associative algebra over $\Bbbk$ is a 4-tuple $\left(A,\mu,\alpha,\beta\right)$ , where $A$ is a $\Bbbk$ -linear space, $\alpha:A\rightarrow A$ , $\beta:A\rightarrow A$ and $\mu:A\otimes A\rightarrow A$ are linear maps, with notation $\mu\left(a\otimes a^{\prime}\right)=aa^{\prime}$ , satisfying the following conditions, for all $a,a^{\prime},a^{\prime\prime}\in A:$

	$$\displaystyle\alpha\circ\beta=\beta\circ\alpha,$$		(2.1)
	$$\displaystyle\alpha\left(aa^{\prime}\right)=\alpha\left(a\right)\alpha\left(a^{\prime}\right)\text{ and }\beta\left(aa^{\prime}\right)=\beta\left(a\right)\beta\left(a^{\prime}\right),\quad\text{(multiplicativity)}$$		(2.2)
	$$\displaystyle\alpha\left(a\right)\left(a^{\prime}a^{\prime\prime}\right)=\left(aa^{\prime}\right)\beta\left(a^{\prime\prime}\right).\quad\text{(BiHom-associativity)}$$		(2.3)

We call $\alpha$ and $\beta$ (in this order) the structure maps of $A$ .

A morphism $f:(A,\mu_{A},\alpha_{A},\beta_{A})\rightarrow(B,\mu_{B},\alpha_{B},\beta_{B})$ of BiHom-associative algebras is a linear map $f:A\rightarrow B$ such that $\alpha_{B}\circ f=f\circ\alpha_{A}$ , $\beta_{B}\circ f=f\circ\beta_{A}$ and $f\circ\mu_{A}=\mu_{B}\circ(f\otimes f)$ .

A BiHom-associative algebra $(A,\mu,\alpha,\beta)$ is called unital if there exists an element $1_{A}\in A$ (called a unit ) such that $\alpha(1_{A})=1_{A}$ , $\beta(1_{A})=1_{A}$ and

$\displaystyle a1_{A}=\alpha(a)\;\;\;and\;\;\;1_{A}a=\beta(a),\;\;\;\forall\;a\in A.$

A morphism of unital BiHom-associative algebras $f:A\rightarrow B$ is called unital if $f(1_{A})=1_{B}$ .

A BiHom-Lie algebra over a field $\Bbbk$ is a 4-tuple $\left(L,\left[-\right],\alpha,\beta\right)$ , where $L$ is a $\Bbbk$ -linear space, $\alpha:L\rightarrow L$ , $\beta:L\rightarrow L$ and $\left[-\right]:L\otimes L\rightarrow L$ are linear maps, with notation $\left[-\right]\left(a\otimes a^{\prime}\right)=\left[a,a^{\prime}\right]$ , satisfying the following conditions, for all $a,a^{\prime},a^{\prime\prime}\in L:$

	$$\displaystyle\alpha\circ\beta=\beta\circ\alpha,$$
	$$\displaystyle\alpha(\left[a^{\prime},a^{\prime\prime}\right])=\left[\alpha\left(a^{\prime}\right),\alpha\left(a^{\prime\prime}\right)\right]\;\;\text{ and }\;\;\beta(\left[a^{\prime},a^{\prime\prime}\right])=\left[\beta\left(a^{\prime}\right),\beta\left(a^{\prime\prime}\right)\right],$$
	$$\displaystyle\left[\beta\left(a\right),\alpha\left(a^{\prime}\right)\right]=-\left[\beta\left(a^{\prime}\right),\alpha\left(a\right)\right],\;\;\;\;\text{ (skew-symmetry)}$$
	$$\displaystyle\left[\beta^{2}\left(a\right),\left[\beta\left(a^{\prime}\right),\alpha\left(a^{\prime\prime}\right)\right]\right]+\left[\beta^{2}\left(a^{\prime}\right),\left[\beta\left(a^{\prime\prime}\right),\alpha\left(a\right)\right]\right]+\left[\beta^{2}\left(a^{\prime\prime}\right),\left[\beta\left(a\right),\alpha\left(a^{\prime}\right)\right]\right]=0.$$
	(BiHom-Jacobi condition).

We call $\alpha$ and $\beta$ (in this order) the structure maps of $L$ .

A morphism $f:\left(L,\left[-\right],\alpha,\beta\right)\rightarrow\left(L^{\prime},\left[-\right]^{\prime},\alpha^{\prime},\beta^{\prime}\right)$ of BiHom-Lie algebras is a linear map $f:L\rightarrow L^{\prime}$ such that $\alpha^{\prime}\circ f=f\circ\alpha$ , $\beta^{\prime}\circ f=f\circ\beta$ and $f(\left[x,y\right])=[f(x),f(y)]^{\prime}$ , for all $x,y\in L$ .

A BiHom-coassociative coalgebra is a 4-tuple $(C,\Delta,\psi,\omega)$ , in which $C$ is a linear space, $\psi,\omega:C\rightarrow C$ and $\Delta:C\rightarrow C\otimes C$ are linear maps, such that:

	$\displaystyle\psi\circ\omega=\omega\circ\psi,$
	$\displaystyle(\psi\otimes\psi)\circ\Delta=\Delta\circ\psi,$
	$\displaystyle(\omega\otimes\omega)\circ\Delta=\Delta\circ\omega,$
	$\displaystyle(\Delta\otimes\psi)\circ\Delta=(\omega\otimes\Delta)\circ\Delta.$

We call $\psi$ and $\omega$ (in this order) the structure maps of $C$ .

A morphism $g:(C,\Delta_{C},\psi_{C},\omega_{C})\rightarrow(D,\Delta_{D},\psi_{D},\omega_{D})$ of BiHom-coassociative coalgebras is a linear map $g:C\rightarrow D$ such that $\psi_{D}\circ g=g\circ\psi_{C}$ , $\omega_{D}\circ g=g\circ\omega_{C}$ and $(g\otimes g)\circ\Delta_{C}=\Delta_{D}\circ g$ .

A BiHom-coassociative coalgebra $(C,\Delta,\psi,\omega)$ is called counital if there exists a linear map $\varepsilon:C\rightarrow\Bbbk$ (called a counit ) such that

	$\displaystyle\varepsilon\circ\psi=\varepsilon,\quad\varepsilon\circ\omega=\varepsilon,$
	$\displaystyle(id_{C}\otimes\varepsilon)\circ\Delta=\omega\quad and\quad(\varepsilon\otimes id_{C})\circ\Delta=\psi.$

A morphism of counital BiHom-coassociative coalgebras $g:C\rightarrow D$ is called counital if $\varepsilon_{D}\circ g=\varepsilon_{C}$ , where $\varepsilon_{C}$ and $\varepsilon_{D}$ are the counits of $C$ and $D$ , respectively.

A surface $M$ is a translation surface if it can be parametrized by a patch

(2.1)

$$x(u,v)=\left(u,v,f(u)+g(v)\right).$$

A surface $M$ is a factorable surface if it can be parametrized by a patch

(2.2)

$$x(u,v)=\left(u,v,f(u)g(v)\right).$$

Let $D$ be normed ring. Element $a\in D$ is called limit of a sequence $\{a_{n}\}$

$$a=$$

if for every $\epsilon\in R$ , $\epsilon>0$ , there exists positive integer $n_{0}$ depending on $\epsilon$ and such, that

$$|a_{n}-a|<\epsilon$$

for every $n>n_{0}$ . ∎

Let $a,b\in\mathbb{R}$ be two real numbers, one of which is positive and another negative. We consider two closed intervals $I_{\eta}=[a,0]$ and $I_{\xi}=[b,0]$ . A commuting pair of class $C^{r}$ , $r\geq 3$ ( $C^{\infty}$ , analytic) acting on the intervals $I_{\eta}$ , $I_{\xi}$ is a pair of maps $\zeta=(\eta,\xi)$ with the corresponding smoothness, such that the following properties hold:

(i) $\eta$ is an orientation-preserving homeomorphism of the closed intervals $I_{\eta}\mapsto\eta(I_{\eta})\subset I_{\eta}\cup I_{\xi}$ ; $\xi$ is an orientation-preserving homeomorphism $I_{\xi}\mapsto\xi(I_{\xi})\subset I_{\eta}\cup I_{\xi}$ ; $a=\xi(0)$ and $b=\eta(0)$ ;

(ii) $\eta^{\prime}(x)\neq 0$ for all $x\in\hat{I}_{\eta}\setminus\{0\}$ , and $\xi^{\prime}(y)\neq 0$ for all $y\in\hat{I}_{\xi}\setminus\{0\}$ ; and $0$ is a cubic critical point for both maps;

(iii) there exist homeomorphic extensions of $\eta$ and $\xi$ of the same smoothness to interval neighborhoods $\hat{I}_{\eta}\Supset I_{\eta}$ and $\hat{I}_{\xi}\Supset I_{\xi}$ which commute:

$$\eta\circ\xi=\xi\circ\eta$$

where defined;

(iv) $\xi\circ\eta(0)\in I_{\eta}$ .

A Smale space $(X,\varphi)$ consists of a compact metric space $X$ with metric $d$ along with a homeomorphism $\varphi:X\to X$ such that there exist constants $\varepsilon_{X}>0,0<\lambda<1$ and a continuous bracket map

$$\{(x,y)\in X\times X\mid d(x,y)\leq\varepsilon_{X}\}\mapsto[x,y]\in X$$

satisfying the bracket axioms:

• B1

  $\left[x,x\right]=x$ ,

• B2

  $\left[x,[y,z]\right]=[x,z]$ ,

• B3

  $\left[[x,y],z\right]=[x,z]$ , and

• B4

  $\varphi[x,y]=[\varphi(x),\varphi(y)]$ ;

for any $x,y,z$ in $X$ when both sides are defined. In addition, $(X,\varphi)$ is required to satisfy the contraction axioms:

• C1

  For $x,y\in X$ such that $[x,y]=y$ , we have $d(\varphi(x),\varphi(y))\leq\lambda d(x,y)$ and

• C2

  For $x,y\in X$ such that $[x,y]=x$ , we have $d(\varphi^{-1}(x),\varphi^{-1}(y))\leq\lambda d(x,y)$ .

A physical bit is the bit physically transfered over the channel. A physical bit represents some type of coded realization of a payload bit. The number of physical bits, denoted $\#\text{physical bits}$ , is

$$\#\text{payload bits}=\#\text{physical bits}\cdot\text{code rate}.$$

When measuring signal-to-noise ratio (SNR), it will be done with respect to physical bits.

$(L,\vee,\wedge)$ is distributive if $\forall a,b,c\in L$ :

$$a\vee(b\wedge c)=(a\vee b)\wedge(a\vee c)$$

(or equivalently: $a\wedge(b\vee c)=(a\wedge b)\vee(a\wedge c)$ )

For a fixed monomial order, a basis $\mathcal{G}=\left\{g_{1},g_{2},\ldots,g_{s}\right\}$ of a polynomial ideal $J\subset\mathbb{R}\left[\mathbf{x}\right]$ is a Gröbner basis (or standard basis) if for all $f\in\mathbb{R}\left[\mathbf{x}\right]$ there exist a unique $r\in\mathbb{R}\left[\mathbf{x}\right]$ and $g\in J$ such that

$$f=g+r$$

and no monomial of $r$ is divisible by any of the leading monomials in $\mathcal{G}$ , i.e., by any of the monomials $\operatorname{LM}\left(g_{1}\right),\operatorname{LM}\left(g_{2}\right),\ldots,\operatorname{LM}\left(g_{s}\right)$ .

Let $\mathfrak{h}$ be a Lie algebra, $G$ a group, $\varphi:G\to{\rm Aut}_{\rm Lie}(\mathfrak{h})$ a morphism of groups, $\gamma\in G$ and $\mathfrak{h}_{\gamma}:=\{y-\gamma\triangleright y\,|\,y\in\mathfrak{h}\}$ . The action $\varphi$ is called $\gamma$ -abelian if:

$$[g\triangleright z,\,g^{\prime}\triangleright z^{\prime}]=0$$

(24)

for all $g\neq g^{\prime}\in G$ and $z$ , $z^{\prime}\in\mathfrak{h}_{\gamma}$ .

(I) For any two chromosomes $x$ and $y$ in $\mathcal{A}$ , the path from $x$ to $y$ is the function

$\displaystyle{{[x\leftrightarrow y]}}(a,\ell)=\begin{cases}1&\text{if $x$ or $y$, but not both, inherits from $a$ at $\ell$}\\ 0&\text{otherwise}.\end{cases}$

(10)

(II) For any two collections of chromosomes $X$ and $Y$ , having $|X|$ and $|Y|$ elements each, the path from $X$ to $Y$ is the function

$\displaystyle{{[X\leftrightarrow Y]}}(a,\ell)=\frac{1}{C(X,Y)}\sum_{x,y}{{[x\leftrightarrow y]}}(a,\ell),$

(11)

where the sum is over distinct pairs $x\in X$ and $y\in Y$ with $x\neq y$ , and $C(X,Y)$ is the number of such pairs. These are depicted in figure 1 .

The Heisenberg group $\mathbb{H}^{n}$ is $\mathbb{R}^{2n+1}$ equipped with the non commutative group law:

$$(a,b,c)(a^{\prime},b^{\prime},c^{\prime})=(a+a^{\prime},\,b+b^{\prime},\,c+c^{\prime}-2(\langle a,b^{\prime}\rangle-\langle b,a^{\prime}\rangle)).$$

If $(B;\cdot)$ is a group, then the bijection, $\alpha:B\rightarrow B$ , is called a holomorphism of $(B;\cdot)$ if

$$\alpha(x\cdot y^{-1}\cdot z)=\alpha x\cdot(\alpha y)^{-1}\cdot\alpha z,$$

for every $x,y,z\in B$ .