We prove the Bernoulli property for a class of counter-twisting linked twist maps. These compose orthogonal linear shears on the torus, orientated in the opposite sense to their co-twisting counterparts (where the shears reinforce one another). Compared to previous studies we focus on the parameter space corresponding to weak shears, near the critical parameter below which hyperbolicity is lost and the map is non-mixing. The approach developed to deal with this situation appears applicable to a broad range of non-uniformly hyperbolic examples.

JMH supported by EPSRC under Grant Ref. EP/W524372/1.

§ INTRODUCTION

A well known example of non-uniform hyperbolicity, linked twist maps <cit.> (hereafter LTMs) arise in a variety of applications. As models of chaotic advection in the presence of boundaries, their dynamics are central to problems in laminar mixing (see <cit.> and the references therein) and other physical phenomena <cit.>. More recently <cit.> drew a connection between LTMs and certain contact flows <cit.>.

We adopt the general form of a linear toral LTM from <cit.>. Fix four constants $0\leq x_1<x_2<1$, $0 \leq y_1<y_2<1$ and parameterise the torus by $(x,y)\in S^1 \times S^1$. Let $f:[y_0,y_1] \to S^1$ and $g:[x_0,x_1] \to S^1$ be given by
$f(y) = (y-y_0)/(y_1-y_0)$ and $g(x) = (x-x_0)(x_1-x_0)$. Defining horizontal and vertical annuli $P = \{ (x,y) \, | \, y_0 \leq y \leq y_1 \}$ and $Q =\{ (x,y) \, | \, x_0 \leq x\leq x_1 \}$, union $R= P \cup Q$, let $F,\tilde{G}:R \to R$ be given by

\[F(x,y) = \begin{cases} (x+f(y),y) & (x,y) \in P, \\ (x,y) & \text{otherwise,} \end{cases} \quad \tilde{G}(x,y) = \begin{cases} (x,y+g(x)) & (x,y) \in Q, \\ (x,y) & \text{otherwise,} \end{cases} \]

with the coordinates calculated modulo 1. For integers $k,l \neq 0$, the composition $H_{k,l} = \tilde{G}^l \circ F^k$ forms a continuous piecewise-linear Lebesgue measure preserving transformation on $R$.

We focus on the case of counter-twisting LTMs where $k$ and $l$ have opposite signs. Proving mixing properties of co-twisting LTMs (with $kl>0$) is more straightforward; see <cit.>. Defining constants $\alpha=kf' = k/(y_0-y_1)$ and $\beta = lg' = l/(x_1-x_0)$, <cit.> showed that if $|k|,|l| \geq 2$ and $\alpha\beta<-C \approx -17.244$ then $H_{k,l}$ is Bernoulli. Recently <cit.> revisited the problem, removing the constraint $|k|,|l| \geq 2$ and proving mixing properties up to ergodicity for $\alpha\beta<-C \approx -12.04$. Here we continue this effort, focusing on the case of single twists $|k| = |l| = 1$ of weak strength $|\alpha|,|\beta| \approx 2$. As in <cit.> we rescale so that $|\alpha| = |\beta|$. Without loss of generality we take $k=1$, $l=-1$, and shift $R$ so that $x_0=y_0= 0$, giving $x_1=y_1=1/\alpha$. Writing $G = \tilde{G}^{-1}$, the map $H = G \circ F$ is then parameterised by a single positive parameter $\alpha$. A sketch is given in Figure <ref>. Our main theorem is as follows:

= 2.4;
at (-5.2,0)
(0,0) – (0,3) – (3/,3) – (3/,3/) – (3,3/) – (3,0) – (0,0);
(0,3/) – (3/,3/) – (3/,0);

at (3/(2*),(3 + 3/)/2) $R \setminus P$;
at (3/(2*),3/(2*)) $P \cap Q$;
at ((3 + 3/)/2,3/(2*)) $R \setminus Q$;
[very thick,dashed] (3/,0) – (3,0);
[very thick,dashed] (3/,3/) – (3,3/);

[very thick,dashed] (3/,3/(2*)) – (1.5,3/(2*));
[very thick,dashed] (3,3/(2*)) – (1.5+3/,3/(2*));

at (  (3/+  1.5 )/2  , 3/(2*) + 0.15 ) $L_2^\star$;
at (  (3 +  1.5+3/)/2  , 3/(2*) + 0.15 ) $L_2$;

at ((3 + 3/)/2,0.15) $L_1$;
at ((3 + 3/)/2,3/+0.15) $L_1^\star$;

at (0,0)
(0,0) – (0,3) – (3/,3) – (3/,3/) – (3,3/) – (3,0) – (0,0);
(0,3/) – (3,3/) – (0,0);
in 1,...,5
[->] (0,3*/(6*) – (3*/6-0.15,3*/(6*);

[scale=2] at ((3 + 3/)/2,(3 + 3/)/2) $F$;

at (5.2,0)
(0,0) – (0,3) – (3/,3) – (3/,3/) – (3,3/) – (3,0) – (0,0);
(0,3) – (3/,0) – (3/,3);

in 1,...,5
[->] (3*/(6*,3) – (3*/(6*,3-3*/6+0.15);

[scale=2] at ((3 + 3/)/2,(3 + 3/)/2) $G$;

A linear counter-twisting toral linked twist map $H=G \circ F$ on the region $R = P \cup Q$. Dashed lines denote periodic segments in $R \setminus Q$; case illustrated $\alpha  = 2.4$.

Let $3 > \alpha > \alpha_0 \approx 2.1319$. Over this parameter range $H$ has the Bernoulli property.

For comparison with <cit.>, Theorem <ref> covers the range $-4.545 \approx -\alpha_0^2 > \alpha\beta > -9$. This brings us close to, yet still bounded away from, the `optimal' shear parameter of $\alpha=2$, below which $H$ is non-ergodic (see e.g. <cit.>, Figure 6.12). We claim that $\alpha_0$ is essentially the lowest achievable bound on the mixing window when relying on a single iterate of the canonical induced map for expansion. We discuss this further in section <ref>, outlining the likely necessary method for dealing with the remaining parameter space.

The fundamental obstacle in the counter-twisting setting is outlined in <cit.>. While hyperbolicity provides the expansion needed to prove mixing properties, it is tempered by the folding effect of the singularities, where the Jacobian of $H$ or its higher powers are undefined. Showing that hyperbolicity dominates this interaction (establishing so called complexity estimates) is a key step in proving statistical properties of various chaotic systems, e.g. billiards. <cit.> give detailed examinations of these estimates for maps very similar to $H$, using them to prove results on mixing rates using the schemes of <cit.>.

In slow mixing systems such as LTMs, laborious calculations are often necessary to verify the estimates directly. Particularly, as is the case with $H$ for small $\alpha$, when hyperbolicity is weak and one must consider higher powers of the map. An indirect approach is often more practical, relying on other features of the map to simplify the problem. Whereas <cit.> relies on the specific structure of periodic points under $F$, here we employ basic complexity estimates and exploit the self-similar structure of the induced map. This approach, developed in <cit.>, appears applicable to a broad range of non-uniformly hyperbolic examples, whose induced return maps typically admit this self-similar structure. For example in <cit.> it is applied to a non-monotonic LTM, possessing both co-twisting and counter-twisting dynamics. The present work demonstrates the method for a well known map, over a broad parameter range. It is organised as follows. In section <ref> we recall some classical results, in particular the scheme of <cit.> used to show the Bernoulli property. Sections <ref>, <ref> deal with the structure of the induced map and the images of certain line segments interacting with it. Section <ref> proves the growth lemma, the key step in the of proof Theorem <ref>, given in section <ref>. We conclude with some remarks on extension to the full expected mixing parameter range $\alpha \geq 2$.

§ BACKGROUND RESULTS

By <cit.> $H$ is hyperbolic, possessing non-zero Lyapunov exponents almost everywhere, for all $\alpha >2$. This, together with mild conditions on the singularity set (see <cit.> for a detailed treatment), implies the existence of local unstable and stable manifolds $\gamma_u(z)$ and $\gamma_s(z)$ at almost every $z \in R$. This result is due to <cit.>, a generalisation of Pesin theory <cit.> for `smooth maps with singularities'. The Bernoulli property (and by extension the lower rungs of the ergodic hierarchy: strong mixing, ergodicity etc.) follows from establishing:

* For almost any $z,\zeta \in R$, there exist $M,N$ such that for all $m\geq M$ and $n\geq N$, $H^m\gamma_u(z) \cap H^{-n}\gamma_s(\zeta) \neq \varnothing$.

As $H$ is piecewise-linear and non-uniformly hyperbolic, local manifolds are line segments whose diameters may be arbitrarily small. A key step in showing (MR) is establishing exponential expansion in the diameters of $H^m\gamma_u(z)$ and  $H^{-n}\gamma_s(\zeta)$, growing these images up to some tangible size where intersections may be inferred.

A uniformly hyperbolic induced map forms the basis for this expansion. The canonical choice is the return map $H_S: S \to S$, $z \mapsto H^{r}(z)$, where $r = r(z;H,S) = \min \{  i > 0 \, | \, H^i(z) \in S \} $ denotes the return time of $z$ to $S= P \cap Q$ under $H$. It decomposes as the composition $H_S = G_S \circ F_S$ of returns under $F$ then $G$ so $H_S(z) = G^l \circ F^k(z)$ for some naturals $k,l \geq 1$ depending on $z$. The return time of $z$ to $S$ is then given by $r = k + l -1$. Recall from <cit.> the cone $\mathcal{C}$, defined by vectors $(v_1,v_2) \in \mathbb{R}^2 \setminus \{ \mathbf{0} \}$ with $ L \leq v_1/v_2 \leq 0$ where $L = \frac{1}{2} \left( -\alpha + \sqrt{\alpha^2 -4} \right)$. Similarly defining $\mathcal{C}'$ by $v_1/v_2 \geq L + \alpha$, one has $DF^k \mathcal{C} \subset \mathcal{C}'$ and $DG^l \mathcal{C}' \subset \mathcal{C}$ for all $k,l \geq 1$. Hence $\mathcal{C}$ is invariant under the Jacobian $DH_S$; it further provides bounds on the gradients of local manifolds mapped into $S$ by $H$. In particular at almost every $z \in R$, we can find $i>0$ such that $H^i \gamma_u(z)$ contains a linear segment $\Gamma_0 \subset S$, aligned with some $v \in \mathcal{C}$ <cit.>. The stable cone $\mathcal{C}^s$, defined by the inequality $ L \leq v_2/v_1 \leq 0$, similarly bounds the backwards images of stable manifolds $H^{-i}\gamma_s(z) \subset S$.

§ STRUCTURE OF RETURN TIMES

[scale = 0.9] at (0,0)

[scale = 2.5]
= 2.8;

(0,0) rectangle (10/,10/);

at (10/,0) $\bullet$;
at (0,10/) $\bullet$;
at (0,5/) $\bullet$;
at (10/,5/) $\bullet$;

at (10/+0.1,0) $p_1$;
at (-0.1,10/) $p_1^\star$;
at (-0.1,5/) $p_2$;
at (10/+0.1,5/) $p_2^\star$;

at  (10/+0.1,  10*(1/- 1/^2)/2 ) $y_2$;
at  (10/+0.1,  10*(1/- 1/^2)/3 ) $y_3$;
at  (10/+0.1,  10*(1/- 1/^2)/4 ) $y_4$;

(0,0) rectangle (10/,10/);

at (5/,5/) $A_2$;
at (1.5/,1.5/) $A_1$;
at (8.5/,8.5/) $A_1^\star$;
at (3.5/,3.5/) $A_3$;
at (6.5/,6.5/) $A_3^\star$;
at (5/,2.4/) $A_4$;
[scale=0.8] at (2.7/,4.35/) 5;
at (5/,7.6/) $A_4^\star$;
[scale=0.8] at (7.3/,5.65/) 5;
[scale=0.8] at (5.5/,1.8/) 5;
[scale=0.7] at (6.45/,1.4/) 6;

[scale=0.8] at (4.5/,8.2/) 5;
[scale=0.7] at (3.55/,8.6/) 6;

[scale=1.2] at  (3.3/,2.1/) $\mathscr{L}$;

[scale=1] at  (9/,3.7/) $\mathscr{L}_2$;
[scale=1] at  (1/,6.3/) $\mathscr{L}_2^\star$;

[very thick] (0,10/^2) – (10*( 3/- 1)/2, 10*(1/- 1/^2)/2  );
at (0.4/, 10/^2 - 0.17) $\tilde{\mathscr{L}}$;

(0, 10/(*)  ) – (10/,0);

įn 2,...,120

( 10*(-̨)/( *-̨ )   , 10*(-1)/( **-̨ *)  ) – (10,0);

(10,0) – ( 10*(2 -)/( *2 - )   , 10*(-1)/( **2 - *)  ) – (0,0) – (10,0);
in 1,...,20

( 0 , 10*(+ 1/)/( (2*+1 )*)) – ( 10/, 10*/((2*+1)*)  );

at (0,(5*+ 5)/(*))

[scale = 2.5,rotate=180]
(0, 10/(*)  ) – (10/,0);

įn 2,...,120

( 10*(-̨)/( *-̨ )   , 10*(-1)/( **-̨ *)  ) – (10,0);

(10,0) – ( 10*(2 -)/( *2 - )   , 10*(-1)/( **2 - *)  ) – (0,0) – (10,0);
in 1,...,20

( 0 , 10*(+ 1/)/( (2*+1 )*)) – ( 10/, 10*/((2*+1)*)  );

[dashed] (-4.2/,5.8/) circle (2.8/);
[dashed] (-4.2/+ 2.8*cos(300)/,5.8/+ 2.8*sin(300)/) – (0.2/,4.9/);
[dashed] (-4.2/+ 2.8*cos(40)/,5.8/+ 2.8*sin(40)/) – (0.2/,4.9/);
(-4.2/,5.8/) circle (2.8/);
(-6/,10/) – (-6/,0);
(-6/,12/) – (0,0);

(-6/,7/) – (-3.15/,6.3/);
(-6/,6.3/) – (-2.6/,5.2/);
(-6/,5.2/) – (-1.8/,3.6/);

[scale=0.8] at (-6.2/,7/) $Y_k$;
[scale=0.8] at (-6.35/,6.3/) $Y_{k-2}$;
[scale=0.8] at (-6.35/,5.2/) $Y_{k-4}$;

[scale=0.8] at (-2.7/,6.3/) $Y_{k-2}$;
[scale=0.8] at (-2.15/,5.2/) $Y_{k-4}$;

at (-4/,3.8/) $\dots$;
at (-5/,7.5/) $\dots$;
[scale=0.9] at (-4.5/, 6.2/) $A_k^2$;
[scale=0.9] at (-4.1/, 5/) $A_{k-2}^2$;
[scale=1.2] at (-2.9/, 7.4/) $A_2$;

Partition of $S$ into sets $A_k^{(\dots)}$ of return time $k$ under $F$. Each are bounded by preimages of $\partial S$; for example near $p_1$ each $A_k$ is bounded between $\mathscr{L}_{k-1}$ and $\mathscr{L}_k$, meeting $\partial S$ at $(1/\alpha, y_{k-1})$ and $(1/\alpha, y_k)$ respectively. Near $p_2$ the accumulating sets have odd return times, bounded between $\mathscr{L}_{k-2}^2$ and $\mathscr{L}_k^2$ which meet $\partial S$ at $(0,Y_{k-2})$, $(0,Y_k)$ respectively. Case illustrated $\alpha = 2.8$.

We begin by describing the structure of hitting times $h(z;F,S) = \min \{  i > 0 \, | \, F^i(z) \in S \}$ over $z \in P$. For $2 < \alpha < 3$ such integers exist outside of the four line segments in $P \setminus S$:

* $L_1$: $y=0$, $\frac{1}{\alpha}< x < 1$,

* $L_1^\star$: $y=1$, $\frac{1}{\alpha}< x < 1$,

* $L_2$: $y = \frac{1}{2\alpha}$, $\frac{1}{2} + \frac{1}{\alpha} < x <1$,

* $L_2^\star$: $y = \frac{1}{2\alpha}$, $\frac{1}{\alpha} < x < \frac{1}{2}$,

each periodic under $F$ with period given by the subscript. These segments are sketched as the dashed lines in Figure <ref>. Across the rest of the parent circles $y \in \{0, \frac{1}{2\alpha}, \frac{1}{\alpha} \}$, points hit $S$ in just one or two iterates. The structure of return times to $S$ is plotted in Figure <ref>; the subscripts of the labelled regions correspond to the return time $r(\cdot;F,S)$. Near the circles $y \in \{0, \frac{1}{2\alpha}, \frac{1}{\alpha} \}$ we see either very fast returns, $r \in \{ 1,2\}$, or very slow returns, with $r$ diverging as we approach the accumulation points $p_i^{(\star)}$. In particular each $p_i^{(\star)}$ lies in the closure of $L_i^{(\star)}$ on $\partial S$; we label the four segments $\partial S_j$ which make up this boundary as shown in Figure <ref>. For example near $p_1$ we have the large immediately returning set $A_1 \subset F^{-1}(S) \cap S$ of points which shift no further than $\partial S_2$ under $F$, and a self similar family of sets $A_k$. For $k \geq 4$, the orbit of a point $(x,y) \in A_k$ maps into $P \setminus S$ under $F$ then shifts horizontally by $k-1$ further increments of $\alpha y$, hitting $S$ just beyond the left boundary $\partial S_1$. Each $A_k$ is then bounded by the segment $\mathscr{L} \subset F^{-1}(\partial S_2)$, $\partial S_2$, $\mathscr{L}_{k-1}$ and $\mathscr{L}_k$, where $\mathscr{L}_k \subset F^{-k}(\partial S_1)$. For future reference, $\mathscr{L}$ lies on the line

\begin{equation}
\label{eq:L}
y = -\frac{1}{\alpha} \left( x-\frac{1}{\alpha} \right)
\end{equation}

and the $\mathscr{L}_k$ lie on the lines

\begin{equation}
\label{eq:Lk}
y =  -\frac{1}{k\alpha} \left(x -1 \right).
\end{equation}

Near $p_2$, points $(x,y)$ either return under $F^2$ (the set $A_2$, bounded by the segment $\mathscr{L}_2 \subset F^{-2}(\partial S_1) $) or $F(x,y)$ falls just short of the line $x=1/2$. Letting $(x_n,y) = F^{2n+1}(x,y)$, the sequence $x_n = x_{n-1} + 2\alpha y \text{ mod }1 $ is strictly decreasing, giving $N$ such that $x_N < 1/\alpha$, so that $F^{2N+1}(x,y)$ lies left of $\partial S_2$. This gives odd first return times to $S$ as the even iterates up to $n=2N$ lie further right near the segment $L_2$. The number $N$ diverges as we approach $p_2$ ($y \to \frac{1}{2\alpha}$) giving, for $k \geq 5$, secondary accumulating sets of constant return time $A_k^2$ bounded by $\mathscr{L}_2$, $\partial S_1$, $\mathscr{L}_{k-2}^2$ and $\mathscr{L}_k^2$ where $\mathscr{L}_k^2 \subset F^{-k}(\partial S_2)$. For future reference, the $\mathscr{L}_k^2$ lie on the lines

\begin{equation}
\label{eq:Lk2}
y = \frac{1}{k\alpha} \left( \frac{k-1}{2} + \frac{1}{\alpha} - x \right).
\end{equation}

The remaining region between these two accumulating patterns forms the set $A_3$ of constant return time 3, completing the description of return times to $S$ over $y < \frac{1}{2\alpha}$. Noting that $F$ commutes with the involution $I_1(x,y) = \left( \frac{1}{\alpha} - x ,\frac{1}{\alpha} - y \right) \text{ mod 1}$, we may infer sets of constant return time above $y = \frac{1}{2\alpha}$ by mapping under $I_1$. Figure <ref> provides a plot at an example parameter, denoting the images under $I_1$ with a superscript $\star$.

Defining three further transformations
$I_2(x,y) = \left( x, \frac{1}{\alpha} - y \right) \text{ mod 1}$, $I_3(x,y) = \left( y , \frac{1}{\alpha} - x \right) \text{ mod 1}$, $I_4(x,y) = \left( y, x \right) $, and the map $\mathcal{H} = F \circ G$, the following relations are straightforward to verify:

* $I_1$ commutes with $F$, $G$ and by extension $H$.

* $I_2 \circ G = G^{-1} \circ I_2$.

* $I_3 \circ F = G \circ I_3$.

* $I_3 \circ H = \mathcal{H} \circ I_3$.

* $I_4 \circ H = H^{-1} \circ I_4$.

The same relations hold for all powers of the maps $F,G,H$, and by extension the return maps $F_S,G_S,H_S$.

The transformation $I_3$ allows us to deduce the structure of $G_S$ from that of $F_S$, sketched in Figure <ref>. The labelling scheme follows directly from Figure <ref> with e.g. the set $B_4$ having return time 4 under $G$, bounded by the segments $\mathscr{I} = I_3(\mathscr{L})$, $\mathscr{I}_3 = I_3(\mathscr{L}_3)$, $\mathscr{I}_4 = I_3(\mathscr{L}_4)$, $\partial S_3 = I_3(\partial S_2)$. For future reference $\mathscr{I}$ lies on the line $y= \alpha x$ and the $\mathscr{I}_k$ lie on the lines

\begin{equation}
\label{eq:Ik}
y = k \alpha x - 1 +\frac{1}{\alpha}.
\end{equation}

[rotate=-90] at (0,0)

[scale = 2]
= 2.8;

(0,0) rectangle (10/,10/);

[rotate=90] at (10/,0) $\bullet$;
[rotate=90] at (0,10/) $\bullet$;
[rotate=90] at (0,5/) $\bullet$;
[rotate=90] at (10/,5/) $\bullet$;

[rotate=90] at (10/+0.15,0) $q_1$;
[rotate=90] at (-0.15,10/) $q_1^\star$;
[rotate=90] at (-0.15,5/) $q_2$;
[rotate=90] at (10/+0.15,5/) $q_2^\star$;

(0,0) rectangle (10/,10/);

[rotate=90] at (5/,5/) $B_2$;
[rotate=90] at (1.5/,1.5/) $B_1$;
[rotate=90] at (8.5/,8.5/) $B_1^\star$;
[rotate=90] at (3.5/,3.5/) $B_3$;
[rotate=90] at (6.5/,6.5/) $B_3^\star$;
[rotate=90] at (5/,2.4/) $B_4$;
[rotate=90] at (5/,7.6/) $B_4^\star$;

[rotate=90][scale=1.2] at  (3.3/,2.1/) $\mathscr{I}$;

[rotate=90] at  (9/,3.7/) $\mathscr{I}_2$;
[rotate=90] at  (1/,6.3/) $\mathscr{I}_2^\star$;

(0, 10/(*)  ) – (10/,0);

įn 2,...,120

( 10*(-̨)/( *-̨ )   , 10*(-1)/( **-̨ *)  ) – (10,0);

(10,0) – ( 10*(2 -)/( *2 - )   , 10*(-1)/( **2 - *)  ) – (0,0) – (10,0);
in 1,...,20

( 0 , 10*(+ 1/)/( (2*+1 )*)) – ( 10/, 10*/((2*+1)*)  );

at (0,(5*+ 5)/(*))

[scale = 2,rotate=180]
(0, 10/(*)  ) – (10/,0);

įn 2,...,120

( 10*(-̨)/( *-̨ )   , 10*(-1)/( **-̨ *)  ) – (10,0);

(10,0) – ( 10*(2 -)/( *2 - )   , 10*(-1)/( **2 - *)  ) – (0,0) – (10,0);
in 1,...,20

( 0 , 10*(+ 1/)/( (2*+1 )*)) – ( 10/, 10*/((2*+1)*)  );

[scale = 2]
= 2.8;

[fill=gray!50] (5/, 0) – ( 10/- 10*(1/+ 1/^2)/2 ,10/) – (5/, 10/) – ( 10*(1/+ 1/^2)/2 ,0) – (5/, 0);

[very thick] (0,0) rectangle (10/,10/);
[white] at (5/, -0.21) $q_2^\star$;
[scale=1.2] at (0.2,6.5/) $\partial S_1$;
[scale=1.2] at (10/- 0.2,6.5/) $\partial S_2$;
[scale=1.2] at (5/,-0.15) $\partial S_3$;
[scale=1.2] at (5/, 10/+ 0.15) $\partial S_4$;

[very thick] ( 6.4/, 0 ) – (4/,10/);

(5/, 10/) – ( 10*(1/+ 1/^2)/2 ,0);
(5/, 0) – ( 10/- 10*(1/+ 1/^2)/2 ,10/);

[scale=0.9] at (2.8/,7.5/) $I_2(\mathscr{I}_2^\star)$;
[scale=0.9] at (6.2/,7.5/) $I_2(\mathscr{I}_2)$;

[scale=1.2] at (5.6/,1/) $\Gamma''$;

(10*( 3/- 1)/2, 10*(1/- 1/^2)/2  ) – (10/, 10*(1/- 1/^2)/3);
(0,5/) – (10/, 10*(1/- 1/^2)/2);

at (10*( 3/- 1)/2, 10*(1/- 1/^2)/2  ) $\bullet$;
[scale=0.9] at (10*( 3/- 1)/2, 10*(1/- 1/^2)/2 - 0.15  ) $\mathscr{L}_3 \cap \mathscr{L}$;

at (9/,3.7/) $\mathscr{L}_2$;
at (9/,1.85/) $\mathscr{L}_3$;

Part (a) shows the distribution of return times to $S$ under $G$, using a similar labelling scheme to Figure <ref>. Part (b) gives a labelling of $\partial S$ and sketches a $v$-segment $\Gamma'' \subset G^2(B_2)$ (shaded).

§ MAPPING INTO $V$-SEGMENTS

Given a line segment $\Gamma \subset S$, we say that $\Gamma$ is a $h$-segment if it connects $\partial S_1$ to $\partial S_2$. Similarly we call $\Gamma$ a $v$-segment if it connects $\partial S_3$ to $\partial S_4$ (see Figure <ref> for an example). We begin by showing how particular line segments map into $v$-segments.

Let $\alpha \geq \alpha_1 \approx 2.125$, the largest root of the cubic equation $2\alpha^3 -4 \alpha^2 - \alpha +1 =0$. Given a line segment $\Gamma \subset S$ aligned with some $v \in \mathcal{C}$:

* If $\Gamma$ connects $\mathscr{L}_{k-1}$ to $\mathscr{L}_k$ for some $k \geq 4$, $F_S(\Gamma)$ contains a segment connecting $\mathscr{I}_{k-2}$ to $\mathscr{I}_{k-1}$.

* If $\Gamma$ connects $\mathscr{L}_2$ to $\mathscr{L}_3$, $H^4(\Gamma)$ contains a $v$-segment.

Starting with the first statement, restrict $\Gamma$ to $A_k$ and denote its endpoints by $z_k \in \mathscr{L}_k$ and $z_{k-1} \in \mathscr{L}_{k-1}$. We note that the $y$-coordinate of $z_{k-1}$ is bounded below by that of $\mathscr{L}_{k-1} \cap \partial S_2$, which by (<ref>) is

\begin{equation}
\label{eq:ybound}
y \geq y_{k-1} := \frac{1}{k-1} \left( \frac{1}{\alpha} - \frac{1}{\alpha^2} \right),
\end{equation}

and above by that of $\mathscr{L}_{k-1} \cap \mathscr{L}$, equal to $y_{k-2}$.
With $\Gamma \subset A_k$ we have that $F_S(\Gamma) = F^k(\Gamma)$ and by definition $F^k(\mathscr{L}_k) \subset \partial S_1$ and $F^k(\mathscr{L}_{k-1}) \subset F(\partial S_1)$. The segment $F_S(\Gamma)$ thus connects $F^k(z_k) \in \partial S_1$ to $(\alpha y, y) : = F^k(z_{k-1}) \in F(\partial S_1)$. Given that $F_S(\Gamma)$ has non-negative gradient (aligned with $DF^kv \in \mathcal{C}'$), for $F_S(\Gamma)$ to intersect $\mathscr{I}_{k-2}$ and $\mathscr{I}_{k-1}$ it is sufficient to show that $(\alpha y, y)$ lies right of the line segment $\mathscr{I}_{k-2}$, with $y$ bounded above by the $y$-coordinate of $\mathscr{I}_{k-1} \cap \mathscr{I}$. Using the inequality $y_{k-1} \leq y \leq y_{k-2}$ and (<ref>), this holds over the given parameter range for all $k \geq 4$ as required.

The second statement follows by a similar argument. If $(\alpha y, y)$ lies right of the segment $\mathscr{I}_2^\star$ then $F^3(\Gamma \cap A_3) \subset F_S(\Gamma)$ contains a segment $\Gamma' \subset B_2$, connecting $\mathscr{I}_2$ to $\mathscr{I}_2^\star$. The image of such a segment under $G_S = G^2$ is a $v$-segment, so that $H^4(\Gamma)$ contains a $v$-segment. The segment $\mathscr{I}_{2}^\star$ lies on the line $y = 2\alpha x - 1$; since $y \geq y_2$ it is sufficient to verify:

\begin{equation}
\label{eq:2inequality}
y_2 \leq  2 \alpha^2 y_2 - 1
\end{equation}

which reduces to

\[ 2\alpha^3 - 4 \alpha^2 - \alpha + 1 \geq 0,\]

valid for all $\alpha \geq \alpha_1$ as required.

Consider the point $z_p = (x_p,y_p) \in A_3$, where

\begin{equation}
\label{eq:P4point}
(x_p,y_p) = \left(  \frac{2\alpha-4}{3\alpha^3 -8\alpha} ,\frac{\alpha^2 + \alpha -4}{3\alpha^3 -8\alpha} \right).
\end{equation}

It is periodic, of period 4 under $H$ with $F^3(z_p) \in B_1^\star$, $GF^3(z_p) \in A_1^\star$, $FGF^3(z_p) \in B_1$, giving

\[ DH_{z_p}^4 = DG \, DF \, DG \, DF^3 = \begin{pmatrix} -\alpha^2 + 1 &  3\alpha^3 + 4\alpha \\ \alpha^3-2 \alpha & 3\alpha^4 - 7\alpha^2 + 1\end{pmatrix}.  \]

For $\alpha > \sqrt{8/5} \approx 1.633$ this matrix is hyperbolic, possessing expanding and contracting eigenvectors $(1,g_+)^T$ and $(1,g_-)^T$ where

\[ g_{\pm} = \frac{4-2\alpha^{2}}{3\alpha^{3}-6\alpha \mp \sqrt{9\alpha^{6}-48\alpha^{4}+76\alpha^{2}-32}}.\]

The region $M = F^{-3}( G^{-1} (  F^{-1} (  B_1 ) \cap A_1^\star ) \cap B_1^\star ) \cap A_3$ of points $z$ around $z_p$ similarly satisfying $DH_z^4 = DH_{z_p}^4$ is shaded in Figure <ref>, bounded by $\partial A_3$ and the preimages $\mathscr{M}_1 \subset (FGF^3)^{-1}(\partial S_1)$, $\mathscr{M}_2 = (FGF^3)^{-1}(\mathscr{I})$. The line segment passing through $z_p$ with gradient $g_-$ and endpoints on $\partial M$ forms the stable manifold $\gamma_s$ at $z_p$. Defining the relative interior of a line segment $\Gamma$ with endpoints $z_1,z_2$ as $\Gamma^\circ = \Gamma \setminus \{ z_1, z_2 \}$, we have the following:

Let $\alpha > \alpha_2 \approx 2.127$. Let $\Gamma$ be a line segment, aligned with $v \in \mathcal{C}$, which intersects $\gamma_s$ at some point $z_0 \in \Gamma^\circ$. Then there exists $k$ such that $H^k(\Gamma)$ contains a $v$-segment.

We essentially apply the inclination or $\lambda$-lemma. Let $\Gamma_0 = \Gamma \cap M$ and iteratively define $\Gamma_i := H^4(\Gamma_{i-1} \cap M)$. This generates sequence of line segments $\Gamma_i \subset H^{4i}(\Gamma)$, aligned with $v_i := DH_{z_p}^{4i} v$, which pass through $z_i := H^{4i}(z_0) \in \gamma_s$. In effect, $\Gamma_i$ limits exponentially fast onto the unstable manifold $\gamma_u$ through $z_p$, with gradient $g_+$ and endpoints on $H^4(\mathscr{M}_1) \subset \partial S_1$ and $H^4(\mathscr{M}_2) \subset \partial S_3$ (plotted as the dashed line in Figure <ref>).

For all $\alpha > \alpha_2$ we claim that either $F_S(\gamma_u)$ contains a segment $\gamma_u'$ whose relative interior intersects
$\mathscr{I}_2$ and $\mathscr{I}_2^\star$, or $F_S \circ H_S(\gamma_u)$ satisfies this intersection property. The lemma then follows, noting we can find finite $i$ such that $F_S(\Gamma_i)$ or $F_S \circ H_S(\Gamma_i)$ similarly intersects $\mathscr{I}_2$ and $\mathscr{I}_2^\star$, so that $H_S(\Gamma_i)$ or $H_S^2(\Gamma_i)$ contains a $v$-segment. In particular, letting $j$ denote the $j$th image $H^j(\Gamma_i)$ of $\Gamma_i$ containing this $v$-segment, $k$ in the lemma statement is given by $k=4i+j$.

For all $\alpha>2$ the segment $\gamma_u$ intersects $\mathscr{L}_3^2$ and $\mathscr{L}$; write this latter intersection as $(x_u,y_u)$. If $\alpha > \alpha_3 \approx 2.694$, the parameter value for which $y_u = y_2$, the manifold $\gamma_u$ also intersects $\mathscr{L}_3$. The image $\gamma_u' = F^3(\gamma_u \cap A_3) \subset F_S(\gamma_u)$ is then a $h$-segment, intersecting $\mathscr{I}_2$ and $\mathscr{I}_2^\star$ in the desired fashion. Otherwise $\gamma_u'$ connects $(x_u',y_u)$ to $\partial S_2$, where $x_u' = 2\alpha(y-y_u)$ (as the image $F^3(\mathscr{L})$ has gradient $\frac{1}{2\alpha}$). The segment $\gamma_u'$ then intersects $\mathscr{I}^\star$ (with parent line $y= 1/\alpha + \alpha x -1$) provided that

\[  y_u \geq 1/\alpha + \alpha x_u' -1 . \]

This holds for $\alpha \geq \alpha_4 \approx 2.1239$, the precise parameter for which $(x_u',y_u) \in \mathscr{I}^\star$. Writing $(x,y) = \gamma_u' \cap \mathscr{I}^\star$,

\[ x =  \frac{y - \frac{1}{\alpha} + 1}{\alpha},\]

by cone alignment $\gamma_u'$ has non-negative gradient so $y \geq y_u$. The segment $\mathscr{I}^\star$ maps into $\partial S_4$ under $G$, so $\gamma_u'' = G(\gamma_u' \cap B_1^\star) \subset H_S(\gamma_u)$ joins $\partial S_2$ to $(x, 1/\alpha) \in \partial S_4$. In particular this endpoint lies to the right of $\mathscr{I}_2^\star$ provided

\[ \frac{y - \frac{1}{\alpha} + 1}{\alpha} > \frac{1}{2}\left( \frac{1}{\alpha} + \frac{1}{\alpha^2} \right), \]

the $x$-coordinate of $\mathscr{I}_2^\star \cap \partial S_4$. By $y \geq y_u$ it is sufficient to check this bound for $y=y_u$. Indeed it holds for $\alpha > \alpha_2 \approx 2.127$. As it lies in $F^3(\mathscr{L}_3^2)$, the other endpoint of $\gamma_u''$ on $\partial S_2$ lies below $\mathscr{L}^\star \cap \partial S_2$ so that $\gamma_u''$ intersects $\mathscr{L}^\star$. The image $F(\gamma_u'' \cap A_1^\star) \subset F_S\circ H_S(\gamma_u)$ then connects $\partial S_1$ to $(x , 1/\alpha)$, intersecting $\mathscr{I}_2$ and $\mathscr{I}_2^\star$ in the desired fashion.

[scale = 8]
= 2.15;

(0,  10/(2*) + 0.2 ) – (0,  10*(-1)/(2**- ) -0.2   );

(0,  10*(-1)/(2**- ) -0.2   ) rectangle (  10*(-*+ 3*-1)/(2**- ) +0.8   , 10/(2*) + 0.2 );

(0,5/) – (0,10/(^2)) – ( 10*(3/- 1)/2 , 10*(1/- 1/(^2))/2  ) – (10/, 10*(1/- 1/(^2))/3) – (10/, 10*(1/- 1/(^2))/2) – (0,5/);
[thick, dotted, fill = gray!50] (  0,  10*(^3 + ^2 -3*-1)/( 1-7*^2 + 3*^4 )    ) – (  10,  10*(^2 -- 1)/( 1-7*^2 + 3*^4 )    ) – (  10,  10*(1-)/(-3*^3 + 4*)    ) –   (  0,  10*(1-)*(+2)/(-3*^3 + 4*)    ) – (  0,  10*(^3 + ^2 -3*-1)/( 1-7*^2 + 3*^4 )    );
[ red, domain=0:4/, smooth, variable=,  thick] plot (,  10*(^2 + -4)/( 3*^3 - 8*) + (   (4-2*^2)/(3*^3-6*+ (9*^6-48*^4+76*^2-32 )^0.5 ) )*(- 10*(2*- 4)/( 3*^3 - 8*))    );

[red, domain=0:2/, smooth, variable=, dashed,  thick] plot (,  10*(^2 + -4)/( 3*^3 - 8*) + (   (4-2*^2)/(3*^3-6*- (9*^6-48*^4+76*^2-32 )^0.5 ) )*(- 10*(2*- 4)/( 3*^3 - 8*))    );

[scale=0.7] at ( 10*(2*- 4)/( 3*^3 - 8*)  , 10*(^2 + -4)/( 3*^3 - 8*)   ) $\bullet$;

[very thick] ( 0, 10/(*) ) – (   10*(-*+ 3*-1)/(2**- )   ,    10*(-1)/(2**- )   );
[dashed] (   10*(-*+ 3*-1)/(2**- )   ,    10*(-1)/(2**- )   ) – (   10*(-*+ 3*-1)/(2**- )   ,    3.85/   );
at (   10*(-*+ 3*-1)/(2**- )   ,    3.9/   ) $\bar{x}$;

[dashed] (    ( 10 - 2**10*(^2 + -4)/( 3*^3 - 8*) + 2**(   (4-2*^2)/(3*^3-6*+ (9*^6-48*^4+76*^2-32 )^0.5 ) )*10*(2*- 4)/( 3*^3 - 8*)  )/(  1 + 2**    (4-2*^2)/(3*^3-6*+ (9*^6-48*^4+76*^2-32 )^0.5 ) )   ,   (10 - ( 10 - 2**10*(^2 + -4)/( 3*^3 - 8*) + 2**(   (4-2*^2)/(3*^3-6*+ (9*^6-48*^4+76*^2-32 )^0.5 ) )*10*(2*- 4)/( 3*^3 - 8*)  )/(  1 + 2**    (4-2*^2)/(3*^3-6*+ (9*^6-48*^4+76*^2-32 )^0.5 ) ))/(2*)       ) – (    ( 10 - 2**10*(^2 + -4)/( 3*^3 - 8*) + 2**(   (4-2*^2)/(3*^3-6*+ (9*^6-48*^4+76*^2-32 )^0.5 ) )*10*(2*- 4)/( 3*^3 - 8*)  )/(  1 + 2**    (4-2*^2)/(3*^3-6*+ (9*^6-48*^4+76*^2-32 )^0.5 ) )     ,    3.95/   );

at (    ( 10 - 2**10*(^2 + -4)/( 3*^3 - 8*) + 2**(   (4-2*^2)/(3*^3-6*+ (9*^6-48*^4+76*^2-32 )^0.5 ) )*10*(2*- 4)/( 3*^3 - 8*)  )/(  1 + 2**    (4-2*^2)/(3*^3-6*+ (9*^6-48*^4+76*^2-32 )^0.5 ) )     ,    3.9/   ) $x_s$;

[red] at (0.3/,5.2/) $\gamma_u$;
[red] at (1/,4.58/) $\gamma_s$;

at (0.47/,4.66/) $z_4$;

[scale=1.2] at (2.1/,3.58/) $\bar{\mathscr{L}}$;
[scale=1.2] at (3.3/,4.3/) $\mathscr{L}_2$;

at (1.6/, 4.61/) $\mathscr{M}_1$;
at (1.6/, 4.33/) $\mathscr{M}_2$;

(0, 10/(*)  ) – (10/,0);

įn 2,...,120

( 10*(-̨)/( *-̨ )   , 10*(-1)/( **-̨ *)  ) – (10,0);

(10,0) – ( 10*(2 -)/( *2 - )   , 10*(-1)/( **2 - *)  ) – (0,0) – (10,0);
in 1,...,50

( 0 , 10*(+ 1/)/( (2*+1 )*)) – ( 10/, 10*/((2*+1)*)  );

Close up of the singularity set for $F_S$ near $p_2$, parameter value $\alpha=2.15$.

Let $\alpha > \alpha_2 \approx 2.127$. Let $\Gamma \subset S$ be a line segment aligned with some $v \in \mathcal{C}$. If $\Gamma$ connects $\mathscr{L}$ to $\mathscr{L}_2$, then there exists $k$ such that $H^k(\Gamma)$ contains a $v$-segment.

Restrict $\Gamma$ to the open region bounded by $\mathscr{L},\mathscr{L}_2$. Observing Figure <ref>, points in $\Gamma$ return to $S$ over three or more iterates of $F$. Since $\alpha \geq \alpha_1$, if $\Gamma$ intersects $\mathscr{L}_3$ then the result holds with $k=4$ by Lemma <ref>. Otherwise $\Gamma$ intersects the subset $\tilde{\mathscr{L}} \subset \mathscr{L}$ at a point $(1/\alpha - \alpha y, y)$ with $y_2 < y \leq 1/\alpha^2 $ (shown in bold in Figure <ref>). The endpoints of $\tilde{\mathscr{L}}$ map to $(0,y_2)$ and $(  3/\alpha -1 ,1/\alpha^2)$ under $F^3$, so that $F^3( \tilde{\mathscr{L}})$ lies entirely left of $\mathscr{I}_2$, see (<ref>), if

\[ \frac{1}{\alpha^2} \geq 2 \alpha \left(  \frac{3}{\alpha} -1  \right) -1 + \frac{1}{\alpha}, \]

i.e. $\alpha > \left(3 + \sqrt{5} \right)/2 \approx 2.618$. In such a case $F_S(\Gamma)$ contains a segment joining $\mathscr{I}_2$ to $\mathscr{I}_2^\star$ and $H^4(\Gamma)$ contains a $v$-segment. For $\alpha \leq \left(3 + \sqrt{5} \right)/2$ the segment $F^3(\tilde{\mathscr{L}})$ intersects $\mathscr{I}_2$ at $(\bar{y},\bar{y})$,

\begin{equation}
\label{eq:ybar}
\bar{y} = \frac{\alpha -1}{2\alpha^2 - \alpha},
\end{equation}

and $F^3(\Gamma)$ joins $\mathscr{I}_2$ to $\mathscr{I}_2^\star$ provided that $y< \bar{y}$. Analogous to before, $H^4(\Gamma)$ then contains a $v$-segment.

It remains to consider the case where $\Gamma$ intersects the subset $\bar{\mathscr{L}} \subset \mathscr{L}$ at $(1/\alpha - \alpha y, y)$ satisfying $y \geq \bar{y}$. This segment is plotted in Figure <ref>, with endpoints $(0,1/\alpha^2)$ and $(\bar{x},\bar{y})$ where

\[ \bar{x} =  \frac{1}{\alpha} - \alpha \bar{y} = \frac{-\alpha^2 + 3 \alpha -1}{2\alpha^2 - \alpha}.  \]

By cone alignment $\Gamma$ can then only intersect $\mathscr{L}_2$ at some point $(x, (1-x)/(2\alpha))$ with $0\leq x \leq \bar{x}$. Comparing the equation for the parent line $y-y_p = g_-(x-x_p)$ of $\gamma_s$ with those of $\partial M$ (for reference $\mathscr{M}_1$ and $\mathscr{M}_2$ lie on the lines

\[  y= \frac{\alpha^2 -1}{-3 \alpha^3 + 4 \alpha} \left( x - \frac{1}{\alpha + 1} -1 \right) \text{ and } y= \frac{2 \alpha - \alpha^3}{1 - y \alpha^2 + 3 \alpha^4} \left( x - \frac{-\alpha^2 + \alpha + 1}{2 \alpha - \alpha^3} -1 \right)  \]

respectively), one can verify that over the remaining parameter range $2 < \alpha \leq \left(3 + \sqrt{5} \right)/2$ the stable manifold intersects $\partial M$ on $\partial S_1$ and $\mathscr{L}_2$. Writing $(x_s,y_s) = \gamma_s \cap \mathscr{L}_2$, if $\bar{x} < x_s$ then $\Gamma$ intersects $\gamma_s$ at some point in $\Gamma^\circ$ and the result follows over $\alpha > \alpha_2$ by Lemma <ref>. Indeed the inequality holds for all $\alpha > \alpha_5 \approx 2.124$, the parameter value for which $\bar{x} = x_s$.

§ GROWTH LEMMA

Given a line segment $\Gamma \subset S$, we define its height as $\ell_v(\Gamma) = \nu \left( \{y \,|\, (x,y) \in \Gamma \} \right)$ and width as $\ell_h(\Gamma) = \nu \left( \{x \,|\, (x,y) \in \Gamma \} \right)$, where $\nu$ is the Lebesgue measure on $\mathbb{R}$.

Let $\alpha > \alpha_0 \approx 2.1319$. Let $\Gamma \subset S$ be a line segment aligned with some $v \in \mathcal{C}$. Either:

* There exists $\delta >0$ such that $F_S(\Gamma)$ contains a segment $\Gamma'$ with $\ell_h(\Gamma') > (1+ \delta) \, \ell_v(\Gamma)$, or

* $\Gamma$ connects $\mathscr{L}_{k-1}$ to $\mathscr{L}_k$ or $\mathscr{L}_{k-1}^\star$ to $\mathscr{L}_k^\star$ for some $k \geq 3$, or

* There exists $k$ such that $H^k(\Gamma)$ contains a $v$-segment.

Suppose first that $\Gamma$ intersects $\mathscr{L}_2$ and $\mathscr{L}_2^\star$. Then $\Gamma' = F^2(\Gamma \cap A_2) \subset F_S(\Gamma)$ is a $h$-segment and $\Gamma'' = G^2(\Gamma' \cap B_2) \subset G_S(\Gamma')$ is a $v$-segment, so that (C3) follows with $k=3$. Otherwise $\Gamma$ lies entirely below the segment $\mathscr{L}_2^\star$ or entirely above $\mathscr{L}_2$. We consider the first case now, addressing the latter case at the end.

Suppose $\Gamma$ lies entirely within some set of constant return time $k$ under $F$. The image $\Gamma' = F_S(\Gamma)$ then satisfies $\ell_h(\Gamma) \geq E_k \ell_v(\Gamma)$ where

\begin{equation}
\label{eq:expansionFactors}
E_k(\alpha) := \inf_{v \in C} \frac{\|DF^k v\|_\infty}{\| v \|_\infty} = k \alpha + L
\end{equation}

denote minimum expansion factors under the $\| \cdot \|_\infty$ norm. For all $\alpha >2$ and $k \in \mathbb{N}$ this factor is strictly greater than 1 so (C1) holds. More generally, suppose $\Gamma$ splits into multiple components $\Gamma_i$, $i \in I$, of constant return time under $F$. Denoting the list of these return times by $K = [r(\Gamma_i;,F,S) \, | \, i \in I]$, if

\begin{equation}
\label{eq:growthCondition}
\sum_{k \in K} \frac{1}{E_k(\alpha)} < 1
\end{equation}

then there exists $i \in I$ such that $\Gamma' = F_S(\Gamma_i)$ similarly satisfies $\ell_h(\Gamma') > \ell_v(\Gamma)$[This is a basic complexity estimate; a detailed proof is found in <cit.>.].

Suppose $ \# K<4$. If $\Gamma$ intersects $A_1$ and $A_2$, it must connect $\mathscr{L}$ to $\mathscr{L}_2$. By Lemma <ref>, (C3) follows. Otherwise, noting that $E_k$ is strictly increasing in $k$, the summation (<ref>) is bounded above by that on $K = [ 1, 3, 4]$. Letting $\alpha_0 \approx 2.1319$ denote the parameter value for which

\[ \frac{1}{E_1(\alpha_0)} + \frac{1}{E_3(\alpha_0)} + \frac{1}{E_4(\alpha_0)} = 1,  \]

in the case $\# K <4$, the lemma then follows for $\alpha > \alpha_0$.

Suppose, then, that $\# K \geq 4$. If $\Gamma$ avoids the secondary accumulation sets $A_k^2$ then, noting Figure <ref>, there exists $k \geq 3$ such that $\Gamma$ intersects $A_{k-1}$, $A_k$, and $A_{k+1}$. In doing so it connects $\mathscr{L}_{k-1}$ to $\mathscr{L}_k$ so that (C2) is satisfied. Assume, then, that $\Gamma$ intersects at least one of the secondary accumulation sets. Again noting Figure <ref>, $\Gamma$ either

* Connects $\mathscr{L}_3$ to $\mathscr{L}_3^2$, or

* Intersects some trio $A_{k-2}^2$, $A_k^2$, $A_{k+2}^2$ for some $k \geq 5$ (defining $A_3^2 = A_3$).

In the first case $\Gamma' = F^3(\Gamma \cap A_3) \subset F_S(\Gamma)$ is a $h$-segment and $\Gamma'' = G^2(\Gamma' \cap B_2) \subset G_S(\Gamma')$ is a $v$-segment. Moving onto the second case, it follows that

* $\Gamma$ traverses $A_k^2$, connecting $\mathscr{L}_{k-2}^2$ to $\mathscr{L}_{k}^2$, for some $k \geq 5$.

By (<ref>) each $\mathscr{L}_k^2$ intersects $\partial S_1$ at the point

\begin{equation}
\label{eq:Yk}
(0,Y_k) = \left ( 0 , \frac{(k-1)\alpha + 2}{2k \alpha^2} \right)
\end{equation}

and $\mathscr{L}_2$ at the point

\begin{equation}
\label{eq:YkL2}
\left( \frac{\alpha-2}{(k-2)\alpha} , \frac{(k-3)\alpha + 2}{2 (k-2) \alpha^2} \right) = \left( \frac{\alpha-2}{(k-2)\alpha} , Y_{k-2} \right) ;
\end{equation}

see the magnified part of Figure <ref>.

Suppose that $\Gamma$ satisfying ($\dag$) violates the lemma; we will show that this leads to a contradiction by an inductive argument. To avoid satisfying (C1) the restriction $\Gamma_2 = \Gamma \cap A_2$ must satisfy

\[ \frac{\ell_v(\Gamma_2)}{\ell_v(\Gamma)} \leq \frac{1}{E_2},  \]

\begin{equation}
\label{eq:proportionEstimate}
\frac{\ell_v(\tilde{\Gamma})}{\ell_v(\Gamma)} \geq 1 - \frac{1}{E_2}
\end{equation}

where $\tilde{\Gamma} = \Gamma \setminus \Gamma_2$. As the base stage of the induction suppose ($\dag$) holds with $k=5$, intersecting $\mathscr{L}_3^2$ and $\mathscr{L}_5^2$. Noting Lemma <ref>, to violate the lemma we must have $\Gamma^\circ \cap \gamma_s = \varnothing$. This gives an upper bound

\begin{equation}
\label{eq:baseCaseUpper}
\ell_v(\tilde{\Gamma}) \leq \frac{1}{2\alpha} - y_m
\end{equation}

where $(x_m,y_m)$ denotes the intersection of $\gamma_s$ with the line passing through $\mathscr{L}_5^2 \cap \mathscr{L}_2$, gradient $1/L$, the lowest point along $\gamma_s$ that a segment intersecting $\mathscr{L}_5^2$ and aligned with some $v \in \mathcal{C}$ can hit.
Using (<ref>), $y_m$ is given by

\[ y_m = \frac{ g_-( 1-2 \alpha Y_3 - x_p - L Y_3 ) + y_p  }{1-Lg_-}. \]

The shortest height of any segment aligned with $v \in \mathcal{C}$ connecting $\mathscr{L}_{k-2}^2$ to $\mathscr{L}_k^2$ is that which lies on $\partial S_1$, given by $Y_k - Y_{k-2}$. Denoting $\Gamma_5 = \Gamma \cap A_5^2$, by (<ref>) then (<ref>) the image $\Gamma' = F^5(\Gamma_5) \subset F_S(\Gamma)$ satisfies

\begin{equation*}
\begin{split}
\ell_h(\Gamma') - \ell_v(\Gamma) & \geq \ell_h(\Gamma') - \frac{1}{1 - \frac{1}{E_2}} \ell_v(\tilde{\Gamma}) \\
& \geq  (Y_5-Y_3) E_5 - \frac{1}{1 - \frac{1}{E_2}} \left( \frac{1}{2\alpha} - y_m \right) \\
\end{split}
\end{equation*}

which is positive for all $\alpha > \alpha_7 \approx 2.072$. Noting $\alpha_0 > \alpha_7 $, if $\Gamma$ violates the lemma it cannot traverse $A_5^2$.

For the inductive step assume $\Gamma$ traverses $A_k^2$, but does not traverse $A_{k-2}^2$. It therefore intersects $\mathscr{L}_k^2$ and $\mathscr{L}_{k-2}^2$, but does not intersect $\mathscr{L}_{k-4}^2$. Analogous to (<ref>) this gives an upper bound

\[ \ell_v(\tilde{\Gamma}) < \frac{1}{2\alpha} - y_l  \]

where $(x_l,y_l)$ denotes the intersection of $\mathscr{L}_{k-4}^2$ with the line passing through $\mathscr{L}_k^2 \cap \mathscr{L}_2$, gradient $1/L$. Again using (<ref>), $y_l$ is given by

\begin{equation}
\label{eq:yl}
y_l = \frac{\left(k-4\right)\alpha Y_{k-4}-\frac{\alpha-2}{\left(k-2\right)\alpha}+LY_{k-2}}{L + \left(k-4\right)\alpha}.
\end{equation}

Analogous to the base case $\Gamma' = F^k(\Gamma \cap A_k^2)$ then satisfies

\begin{equation}
\label{eq:inductiveEstimate}
\ell_h(\Gamma') - \ell_v(\Gamma) > (Y_k-Y_{k-2}) E_k - \frac{1}{1 - \frac{1}{E_2}} \left( \frac{1}{2\alpha} - y_l \right) := f_\alpha(k).
\end{equation}

The function $f_\alpha(k)$ is positive[This is formally verified by noting that $f_\alpha(k)$ is continuous on $k > 9/2 \geq 4 - L/\alpha$ and shares its roots with $k(k-1)[L + (k-4)\alpha]f_\alpha(k)$, a quadratic with roots $k_1,k_2<7$ for $\alpha>\alpha_8$. This gives $\mathrm{sgn}f_\alpha(k) = \mathrm{sgn}f_\alpha(7)$ for all $k \geq 7$, with $f_\alpha(7)$ positive for all $\alpha>\alpha_8$.] for all $k \geq 7$ provided that $\alpha > \alpha_8 \approx 2.012$, the parameter value for which $f_{\alpha_8}(7) = 0$. It follows by induction that for $\Gamma$ to violate the lemma it must not traverse $A_7^2$, nor $A_9^2$, and so on. But this directly contradicts ($\dag$), so no such $\Gamma$ exists, verifying the lemma for the case where $\Gamma$ lies entirely below $\mathscr{L}_2^\star$.

The case where $\Gamma$ lies entirely above $\mathscr{L}_2$ follows similarly. The image $\Gamma^\star = I_1(\Gamma)$ lies entirely below the line $\mathscr{L}_2^\star$ and is aligned with $DI_1 v = -v \in \mathcal{C}$. By the above, $\Gamma^\star$ satisfies one of (C1-3). Noting that $I_1$ commutes with $F_S$ and preserves $\ell_h$, if $\Gamma^\star$ satisfies (C1), so does $\Gamma$. It similarly inherits (C2) or (C3) from $\Gamma^\star$, noting that $I_1$ interchanges $\mathscr{L}_k \leftrightarrow \mathscr{L}_k^\star$, commutes with $H$, and maps $v$-segments to $v$-segments.

Recalling $\mathcal{H} = F \circ G$, we have an analogous result for growth under $G_S$:

Let $\alpha > \alpha_0 \approx 2.1319$. Let $\Lambda \subset S$ be a line segment aligned with some $v' \in \mathcal{C}'$ Either:

* There exists $\delta >0$ such that $G_S(\Lambda)$ contains a segment $\Lambda'$ with $\ell_v(\Lambda') > (1+ \delta) \, \ell_h(\Lambda)$, or

* $\Lambda$ connects $\mathscr{I}_{k-1}$ to $\mathscr{I}_k$ or $\mathscr{I}_{k-1}^\star$ to $\mathscr{I}_k^\star$ for some $k \geq 3$, or

* There exists $k$ such that $\mathcal{H}^k(\Lambda$) contains a $h$-segment.

Let $\Gamma = I_3^{-1}(\Lambda)$, a line segment in $S$ aligned with $v = DI_3^{-1}v' \in \mathcal{C}$. By Lemma <ref>, one of (C1-3) follows. In case (C1) the segment $\Lambda' = I_3(\Gamma') \subset G_S(\Lambda)$ has height $\ell_v(\Lambda') = \ell_h(\Gamma') > (1+\delta) \, \ell_v(\Gamma) =  (1+\delta) \,\ell_h(\Lambda)$, satisfying (C1'). Cases (C2') and (C3') similarly follow from (C2) and (C3), noting that $I_3$ maps the $\mathscr{L}_k^{(\star)}$ onto the $\mathscr{I}_k^{(\star)}$, satisfies $I_3 \circ H^k = \mathcal{H}^k \circ I_3$, and maps $v$-segments into $h$-segments.

§ PROOF OF THE MAIN THEOREM

As noted in section <ref>, it is sufficient to establish (MR). Given $\gamma_u(z)$, we iteratively apply Lemmas <ref>, <ref> to $\Gamma_0 \subset H^i\gamma_u(z) \subset S$, aligned with some $v \in \mathcal{C}$. This generates two sequences of line segments $(\Gamma_m)$, $(\Lambda_m)$ with $\Lambda_m = \Gamma_m'$ from (C1) and $\Gamma_m = \Lambda_{m-1}'$ from (C1'). Each $\Gamma_m$ lies in $H^{m'}(\Gamma_0)$ for some integer $m'$ and $\Lambda_m$ lies in $F \circ H^{m''}(\Gamma_0)$ for some integer $m'' \geq m'$. Their diameters $\ell(\cdot) = \max \{ \ell_v(\cdot), \ell_h(\cdot) \}$ grow exponentially, so that after some finite number $m_1$ steps either $\Gamma_{m_1}$ satisfies (C2) or (C3), or $\Lambda_{m_1}$ satisfies (C2') or (C3').

Starting with case (C3), we can find $k$ such that the image $H^k(\Gamma_{m_1}) \subset H^{m_1' + k}(\Gamma_0)$ contains a $v$-segment. Similarly in case (C3') the image $\mathcal{H}^k(\Lambda_{m_1}) \subset \mathcal{H}^k \circ F \circ H^{m_1''}(\Gamma_0) = F \circ H^{m_1'' +k}(\Gamma_0)$ contains a $h$-segment, connecting $\mathscr{I}_2$ to $\mathscr{I}_2^\star$. As seen previously, the image $H^{m_1'' +k + 2}(\Gamma_0)$ then contains a $v$-segment.

For case (C2) write $\Gamma = \Gamma_{m_1}$; connecting $\mathscr{L}_{k-1}$ to $\mathscr{L}_k$ or $\mathscr{L}_{k-1}^\star$ to $\mathscr{L}_k^\star$ for some $k \geq 3$. If $k$ is odd, making use of the transformations $I_1$ and $I_3$, we may apply Lemma <ref> some $k-3$ times until $H_S^{\frac{k-3}{2}}(\Gamma)$ contains a segment joining $\mathscr{L}_2$ to $\mathscr{L}_3$ or $\mathscr{L}_2^\star$ to $\mathscr{L}_3^\star$. The second part of Lemma <ref>, together with the transformation $I_1$, ensures that we then map into a $v$-segment under $H^4$. Similarly if $k$ is even then $F_S \circ H_S^{\frac{k-4}{2}}(\Gamma)$ contains a segment $\Gamma'$ connecting $\mathscr{I}_2$ to $\mathscr{I}_3$ or $\mathscr{I}_2^\star$ to $\mathscr{I}_3^\star$ and lies in $F\circ H^{m_2}(\Gamma)$ for some integer $m_2$. Now using $I_1$, $I_3$, and Lemma <ref>, the image $\mathcal{H}^4(\Gamma')$ contains a $h$-segment, connecting $\mathscr{I}_2$ to $\mathscr{I}_2^\star$. Hence $H \circ G \circ \mathcal{H}^4(\Gamma') \subset H^{6+m_2}(\Gamma)$ contains a $v$-segment. Case (C2') can then be reduced to case (C2), again making use of the transformation $I_3$.

In any case, then, we can find $M_0$ such that $H^{M_0}(\Gamma_0)$ contains a $v$-segment $\Gamma$. Letting $\Gamma' = F^2(\Gamma \cap A_2)$, the image $\Gamma'' = G^2( \Gamma' \cap B_2) \subset H^3(\Gamma)$ is similarly a $v$-segment. Hence $H^{M_0+3}(\Gamma_0)$ contains a $v$-segment, as does $H^{M_0+3k}(\Gamma_0)$ for all $k \geq 1$ by induction. In particular $\Gamma''$ lies in $G^2(B_2)$, with endpoints on $\partial S_3$ and $\partial S_4$, bounded to the right by $I_2(\mathscr{I}_2)$ and to the left by $I_2(\mathscr{I}_2^\star)$. An example sketch is given in Figure <ref>. Such a segment intersects $\mathscr{L}_3$ provided $\mathscr{L}_3 \cap \mathscr{L} = ( 1/\alpha - \alpha y_2, y_2 )$ lies left of $I_2(\mathscr{I}_2^\star)$. Noting that $I_2(\mathscr{I}_2^\star)$ lies on the line $y = 1 - 2\alpha x$, this amounts to checking that

\[ y_2 \leq 1-2\alpha \left( \frac{1}{\alpha} - \alpha y_2 \right).\]

This is equivalent to (<ref>) so holds for all $\alpha \geq \alpha_1$. Noting $\alpha_0 > \alpha_1$, the segment $\Gamma''$ intersects $\mathscr{L}_2$ and $\mathscr{L}_3$. By Lemma <ref>, $H^4(\Gamma'')$ contains a $v$-segment, similarly in $G^2(B_2)$. The integer combinations $3k+4l$ with $k\geq 1$ and $l \geq 0$ cover all integers greater than 8, so $H^m\gamma_u(z)$ contains a $v$-segment for all $m \geq M = i + M_0 + 9$.

Given $\gamma_s(\zeta)$, we may find $i$ such that $H^{-i}\gamma_s(\zeta)$ contains a segment $\Gamma^s \subset S$, aligned with $v^s \in \mathcal{C}^s$. Now $\Gamma_0 := I_4(\Gamma^s)$ is a line segment in $S$, aligned with $v \in \mathcal{C}$. By the above we can find some integer $M_0$ such that $H^m(\Gamma_0)$ contains a $v$-segment for all $m \geq M_0 + 9$. Since $I_4^{-1}$ maps $v$-segments to $h$-segments, $H^{-n}\gamma_s(\zeta)$ = $(I_4^{-1} \circ H^n \circ I_4) \gamma_s(\zeta)$ contains a $h$-segment for all $n \geq N = i + M_0 + 9$.

§ FINAL REMARKS

As alluded to in the introduction, the lower bound $\alpha_0$ forms a natural barrier to analysis when relying on the canonical induced map $H_S$ (or rather its components $F_S, G_S$) for growth. It is the parameter value below which the `one-step expansion condition' of <cit.> fails for the map $F_S$ over unstable manifolds bounded away from the accumulation points $p_i^{(\star)}$ (the growth near which we ensure using the inductive argument, else[While unnecessary here, the inductive argument may similarly be applied to establish growth near $p_1^{(\star)}$.] map into $v$-segments by repeatedly applying Lemma <ref>). Considering expansion under the full composition $H_S$ or its higher powers may widen the mixing window to some $\alpha_0'<\alpha_0$. This is no simple task, however, owing to the increased complexity of the singularity set. Further $\alpha_0'$ would always be bounded some distance away from the optimal shear parameter $\alpha = 2$, where $H_S$ and all its powers lose uniform hyperbolicity. At this parameter, the problematic region is $T = H^{-1}(S) \cap S $ on which $DH_S = DG \, DF = \left( \begin{smallmatrix}
1 & 2 \\ -2 & -3
\end{smallmatrix} \right) $ is parabolic. Analogous to the map in <cit.>, $T$ contains a pair of periodic line segments on which Lyapunov exponents are zero and nearby points may remain trapped for arbitrary long periods, introducing a new source of intermittent behaviour. Following <cit.>, an appropriate induced map for establishing growth is the return map $H_{\sigma}$, where $\sigma = S \setminus H(T)$. While it is uniformly hyperbolic over $\alpha \geq 2$, the complexity of the singularity set likely precludes a concise analysis.

The other bound $\alpha<3$ of Theorem <ref> is however one of convenience, allowing for a more compact argument. While new accumulation points arise each time $\alpha$ surpasses an integer $k$, sets of constant return time $k$ under $F$ become (like $A_2$ in the present work) quadrilaterals with sides on $\partial S_1, F^{-k}(\partial S_1),\partial S_2, F^{-k}(\partial S_2)$. Any $\Gamma$ traversing these sets maps into a $v$-segment, so these sets divide up the analysis into cases analogous to those encountered here. Growth estimates are no more difficult to show, as expansion factors (<ref>) are universally larger with increasing $\alpha$.

Missing 'biblatex' package
The bibliography requires the 'biblatex' package.

Berlin, Heidelberg

booktitleGlobal Theory of Dynamical Systems
seriesLecture Notes in Mathematics
titleErgodicity of linked twist maps
Full Measure,Invariant Curf,Measure Zero,Orbit Structure,Unstable Manifold

abstractWhile many dynamical systems of mechanical origin, in particular billiards, are strongly chaotic—enjoy exponential mixing, the rates of mixing in many other models are slow (algebraic, or polynomial). The dynamics in the latter are intermittent between regular and chaotic, which makes them particularly interesting in physical studies. However, mathematical methods for the analysis of systems with slow mixing rates were developed just recently and are still difficult to apply to realistic models. Here, we reduce those methods to a practical scheme that allows us to obtain a nearly optimal bound on mixing rates. We demonstrate how the method works by applying it to several classes of chaotic billiards with slow mixing as well as discuss a few examples where the method, in its present form, fails.
titleBilliards with polynomial mixing rates

abstractWe study hyperbolic systems with singularities and prove the coupling lemma and exponential decay of correlations under weaker assumptions than previously adopted in similar studies. Our new approach allows us to study the mixing rates of the reduced map for certain billiard models that could not be handled by the traditional techniques. These models include modified Bunimovich stadia, which are bounded by minor arcs, and flower-type regions that are bounded by major arcs.
journaltitleJournal of Statistical Physics
titleOn Statistical Properties of Hyperbolic Systems with Singularities

abstractFor each pair of nonzero integers $j, k$, we define a homeomorphism $f_{j, k}$ of the two-disk minus three holes. We show that there exists a compact, invariant, hyperbolic set for each $f_{j, k}$ on which $f_{j, k}$ is conjugate to a subshift of finite type. This implies that the topological entropy of $f_{j, k}$ is bounded below by $4|j| |k| - 2|j| - 2|k|$.
journaltitleProceedings of the American Mathematical Society
titleSubshifts of Finite Type in Linked Twist Mappings

journaltitleAnnales Henri Lebesgue
titleOrbit growth of contact structures after surgery

journaltitleMathematics Research Reports
titleHyperbolicity from contact surgery

Berlin Heidelberg

abstractInvariant Manifolds, Entropy and Billiards. Smooth Maps with Singularities...
seriesLecture Notes in Mathematics
titleInvariant Manifolds, Entropy and Billiards. Smooth Maps with Singularities

abstractNon-monotonic velocity profiles are an inherent feature of mixing flows obeying non-slip boundary conditions. There are, however, few known models of laminar mixing which incorporate this feature and have proven mixing properties. Here we present such a model, alternating between two non-monotonic shear flows which act in orthogonal (i.e. perpendicular) directions. Each shear is defined by an independent variable, giving a two-dimensional parameter space within which we prove the mixing property over open subsets. Within these mixing windows, we use results from the billiards literature to establish exponential mixing rates. Outside of these windows, we find large parameter regions where elliptic islands persist, leading to poor mixing. Finally, we comment on the challenges of extending these mixing windows and the potential for a non-exponential mixing rate at particular parameter values.
journaltitlePhysica D: Nonlinear Phenomena
titleExponential mixing by orthogonal non-monotonic shears
Deterministic chaos,Laminar mixing,Non-uniform hyperbolicity

abstractWe consider a Lebesgue measure preserving map of the 2-torus, given by the composition of orthogonal tent shaped shears. We establish strong mixing properties with respect to the invariant measure and polynomial decay of correlations for Holder observables, making use of results from the chaotic billiards literature. The system serves as a prototype example of piecewise linear maps which sit on the boundary of ergodicity, possessing null measure sets around which mixing is slowed and which birth elliptic islands under certain perturbations.
notearXiv:2303.08515 [math]
titleLoss of Exponential Mixing in a Non-Monotonic Toral Map
37A25,Mathematics - Dynamical Systems

University of Leeds

titleMixing in the presence of non-monotonicity
typePhD Thesis

abstractWe reduce the earlier known optimal shear parameter for which ergodicity is established in the linked twist map with two linear shears in opposite sense, in the most general setting. Further, here we obtain ergodicity with possibly only one-fold twists in either lobe, while earlier results only applied for twist parameters at least 2. Almost hyperbolicity is easily established for shear parameters greater than 2, while in the most general setting of the linked twist map with both shears of equal magnitude, ergodicity was earlier established in the most general setting by Przytycki for shear parameters greater than 4.15 with at least two-fold twists in each lobe. Here we reduce this optimal shear parameter to 3.47 in the general setting. These techniques can be effected to make further improvements when additional assumptions are made on the dimensions of the strips or when the linked twist map is modified in other natural ways.
notearXiv:2307.08124 [math]
titleErgodicity in the linked twist map with oppositely oriented shears
Mathematics - Dynamical Systems

journaltitleRussian Mathematical Surveys
titleCharacteristic Lyapunov Exponents and Smooth Ergodic Theory

journaltitleAnnales scientifiques de l'École Normale Supérieure
titleErgodicity of toral linked twist mappings

abstractLinked twist maps of ergodic theory are modified to provide a model for chaotic orbits in ‘double bowl’ potentials in conservative systems, e.g. direct ‘exchange’ orbits in the planar restricted problem of three bodies. The maps are typically ergodic (and mixing), and provide a clue to the nature of relaxation or diffusion in chaotic Hamiltonian systems.
journaltitleCelestial Mechanics and Dynamical Astronomy
titleLinked twist maps in hamiltonian systems
Chaotic Orbit,Conservative System,Ergodic Theory,Hamiltonian System,Restricted Problem

Cambridge University Press

abstractMixing processes occur in many technological and natural applications, with length and time scales ranging from the very small to the very large. The diversity of problems can give rise to a diversity of approaches. Are there concepts that are central to all of them? Are there tools that allow for prediction and quantification? The authors show how a variety of flows in very different settings possess the characteristic of streamline crossing. This notion can be placed on firm mathematical footing via Linked Twist Maps (LTMs), which is the central organizing principle of this book. The authors discuss the definition and construction of LTMs, provide examples of specific mixers that can be analyzed in the LTM framework and introduce a number of mathematical techniques which are then brought to bear on the problem of fluid mixing. In a final chapter, they present a number of open problems and new directions.
shorttitleThe Mathematical Foundations of Mixing
titleThe Mathematical Foundations of Mixing: The Linked Twist Map as a Paradigm in Applications: Micro to Macro, Fluids to Solids
Mathematics / Applied,Mathematics / General,Mathematics / Linear & Nonlinear Programming,Science / Mechanics / Fluids

University of Bristol

titleErgodic properties of linked-twist maps
typePhD thesis

abstractLinked-twist maps are area-preserving, piecewise diffeomorphisms, defined on a subset of the torus. They are non-uniformly hyperbolic generalizations of the well-known Arnold cat map. We show that a class of canonical examples have polynomial decay of correlations for $\alpha $ -Hölder observables, of order $1/ n$ .
issn0143-3857, 1469-4417
journaltitleErgodic Theory and Dynamical Systems
titlePolynomial decay of correlations in linked-twist maps

journaltitleAnnals of the New York Academy of Sciences
titleLinked Twist Mappings Have the K-Property


# UVL Sentinel: a tool for parsing and syntactic correction of UVL datasets

David Romero-Organvidez  Dept. of Computer Languages and SystemsUniversity of
SevilleAv. Reina Mercedes S/NSevilleSpain<EMAIL_ADDRESS>,  Jose A.
Galindo  Dept. of Computer Languages and SystemsUniversity of
SevilleSevillaSpain<EMAIL_ADDRESS>and  David Benavides  Dept. of Computer
Languages and SystemsUniversity of SevilleAv. Reina Mercedes S/NSevilleSpain
<EMAIL_ADDRESS>
(2024)

###### Abstract.

Feature models have become a _de facto_ standard for representing variability
in software product lines. UVL (Universal Variability Language) is a language
which expresses the features, dependencies, and constraints between them. This
language is written in plain text and follows a syntactic structure that needs
to be processed by a parser. This parser is software with specific syntactic
rules that the language must comply with to be processed correctly.
Researchers have datasets with numerous feature models. The language
description form of these feature models is tied to a version of the parser
language. When the parser is updated to support new features or correct
previous ones, these feature models are often no longer compatible, generating
incompatibilities and inconsistency within the dataset. In this paper, we
present UVL Sentinel. This tool analyzes a dataset of feature models in UVL
format, generating error analysis reports, describing those errors and,
eventually, a syntactic processing that applies the most common solutions.
This tool can detect the incompatibilities of the feature models of a dataset
when the parser is updated and tries to correct the most common syntactic
errors, facilitating the management of the dataset and the adaptation of their
models to the new version of the parser. Our tool was evaluated using a
dataset of 1,479 UVL models from different sources and helped semi-
automatically fix 185 warnings and syntax errors.

feature model, uvl, dataset, parser, syntactic error

††copyright: acmlicensed††journalyear: 2024††doi: XXXXXXX.XXXXXXX††conference:
6th International Workshop on Languages for Modelling Variability (MODEVAR
2024); February 6, 2024; Bern, Switzerland††isbn:
978-1-4503-XXXX-X/18/06††ccs: Software and its engineering Software product
lines††ccs: Software and its engineering Requirements analysis††ccs: Software
and its engineering Software design engineering††ccs: Software and its
engineering Software implementation planning

##  1\. Introduction

UVL (Universal Variability Language) is a community proposal to establish a
universal textual language for feature modelling (Sundermann et al., 2022).
This language describes feature models and facilitates the exchange of
scientific knowledge. It is also the basis for feature model analysis tools
such as FeatureIDE (Sundermann et al., 2021) or Flama (Galindo et al., 2023).

There are numerous datasets containing an arbitrary number of feature models.
These feature models are extracted from various sources and transformed into
the UVL language. The feature models in UVL are processed by a parser, a
software that parses the model description. The parser validates that specific
defined syntactic rules are met and reports possible errors in the model
syntax.

Listing 1 shows a UVL file representing a mobile phone feature model. This
feature model represents the constraints and dependencies between features,
such as _Camera_ , _Audio_Formats_ and _Camera_Resolution_.

Listing 1: Example of feature model in UVL format for a specific parser
version (mobile phone).

⬇

1features

2 MOBILE_PHONE {abstract}

3 or

4 MP3_Recording

5 Camera_Resolution

6

7 alternative

8 2.1MP

9 5 MP

10 3.1MP

11 Camera

12 Audio_Formats

13 or

14 WAV

15 MP3

16

17constraints

18 MP3_Recording => MP3

Figure 1. UVL Sentinel main components

Within the feature _Camera_Resolution_ , there are three alternatives for the
megapixel resolution of the camera: _2.1 MP_ , _5 MP_ and _3.1 MP_ (lines
8-10). However, the feature names contain characters that are not allowed: a
dot, ”.”, a blank space ” ” , and beginning with a digit instead of a
character (5 MP). In addition, a blank line (line 6) is not allowed in the
syntax of the current parser version.

Syntactic errors can occur in the UVL transformation process. These errors
cause the syntax to be incorrect for the parser. Furthermore, even if the
transformation does not cause errors, updating the parser to support new
features or syntax changes causes these models to return errors. If a dataset
has fewer feature models, less than 20 models, it may be feasible to solve
them by hand. However, it is impossible to solve massive datasets with more
than 500 models without using some systematic process.

In this paper, we present UVL Sentinel, a tool that analyzes a dataset
composed of several feature models in UVL format. This tool generates valuable
reports for researchers about possible syntactic errors in the models and
helps to fix these errors semi-automatically, thanks to common error patterns.

##  2\. A tool for syntactic checking of datasets

Our tool can analyze a dataset with any logical organization structure,
generate a report of the previous analysis with a UVL parser and, eventually,
try to fix errors by applying regular patterns. The four characteristics in
detail are:

  * •

_Search for files in the dataset_. Researchers often have a dataset that
follows a specific logical structure, such as grouping the feature models
according to characteristics, number of valid products, configurations, and
subject matter. Therefore, the tool goes through the directory tree in depth
until it finds all the files with extension _.uvl_. Each file is an input for
the parser, which will dump the parsed information either by console or plain
text.

  * •

_Analysis report generation_. The tool generates a report file with a
fundamental analysis of each _.uvl_ file found, indicating (1) the dataset to
which it belongs, (2) the name of the file with extension _uvl_ , (3) whether
a warning, an unexpected exception or both have been raised and (4) a
description of the error.

  * •

_Error reporting analysis_. From this text report, the tool counts the number
of warnings, exceptions, and errors, among others, found in the report in the
previous phase and generates a new report with the error summary.

  * •

_Correction of common formatting errors_. Many of the syntactic errors and
incompatibilities are due to, but not limited to, (1) characters not allowed
in the name of features or constraints, (2) blank lines, (3) tabs on a blank
line, (4) encoding not allowed in the language. In the case of the UVL
language, using tabulators, as in the Python language, is indispensable to
distinguish which elements belong to the same block. Tabulators on blank
lines, being non-printable characters, are not easy to detect with just a
visual glance at the development environment. Our tool detects and corrects
these anomalies using regular patterns and performing matching between these
patterns and the syntactic errors found.

##  3\. Implementation

Figure 1 shows the main components of the tool, the inputs, the outputs and
the configurations we can make to adapt a dataset to the needs of a specific
version of a parser.

  1. (1)

_Original dataset_. We start with a dataset of an arbitrary number of feature
models in UVL format. The organization of these files in directories or
subdirectories is unknown to the tool; this organization responds to the
logical structure defined by the researchers, and that makes sense within its
scope. The tool performs an in-depth search for all files with UVL extension
and stores the path for reporting purposes.

  2. (2)

_UVL parser version_. If the parser is available through a package manager,
the version we want to test can be defined in a dependency file. The tool is
abstracted from the version, allowing a report to be generated for different
versions of the same parser.

  3. (3)

_UVL parser_. After discovering all the UVL files in this component, the tool
calls the parser, passing a single UVL file as a parameter. The output, either
by console (desirable) or dumped to a file, is processed by the tool and
stored in the Syntactic analysis report.

  4. (4)

_Syntactic analysis report_. The output of the parser upon receiving each UVL
file can generate (1) the plain text representation of the feature model, a
symptom that the parser has worked, (2) a warning of possible
incompatibilities between the parser version and the current syntax of the UVL
file or (3) an exception because it has been impossible to parse the UVL file.
The tool generates a report of the full path to each UVL file, whether a
warning or exception was raised, and the output generated by the parser to
help identify the error.

  5. (5)

_Summary of parsing report_. Sometimes, the _Syntactic analysis report_ needs
to be shorter to be studied at a glance. Therefore, the tool summarizes (1)
correctly parsed UVL files, (2) files with warnings and (3) files where an
unhandled exception was caught.

  6. (6)

_Common patterns of syntactic errors_. Often, many syntactic errors arising
from incompatibilities between different versions of a parser are due to
disallowed characters, blank lines or problems associated with the file’s
encoding. For this reason, the tool has a base of regular expressions as error
detection patterns.

  7. (7)

_Syntax error correction_. Thanks to the _syntactic analysis report_ and the
_common patterns of syntactic errors_ , the tool helps to match the error
obtained when reading a UVL file with the possible solution. For example, the
middle hyphen character ”-” is not allowed as part of a feature name, and the
solution is to replace it with an underscore ”_”. The tool processes each UVL
individually and generates a new UVL with solutions applied based on the error
patterns mentioned in Stage 6.

  8. (8)

_Fixed dataset_. The tool starts from the original dataset structure. It makes
an exact copy of the files, replacing those original UVL files with the ones
generated and fixed in the previous component. This new dataset serves, again,
as input for the tool to generate new reports, allowing to compare the number
of errors before using the tool and after and to study if there are
significant differences.

##  4\. Validation

For the validation of the tool, we used 1,479 feature models in UVL format
organized in 20 datasets. We have performed a massive analysis of all models,
generated reports before and after the syntactic correction and analyzed them.

###  4.1. Warning, errors and exceptions

For a correct understanding of the validation, the following terms are
defined:

  * •

_Warning_. A warning is a warning from the parser that there are
inconsistencies in the syntax but that it is possible to build the model.

  * •

_Error_. An error is a message that there is a syntactically incorrect element
and must be corrected.

  * •

_Exception_. When an exception occurs, the parser cannot read the model
because of a formatting problem.

###  4.2. Analysis of the dataset before syntax error correction

The data from the syntactic analysis report before running the syntactic
correction have been:

  * •

_Syntactic analysis report_. We have analyzed the 1,479 and found descriptive
syntactic errors such as (1) invalid characters (_extraneous input_), (2)
blank lines (_mismatched input_), and (3) characters not allowed as beginnings
of feature names (_token recognition_).

  * •

_Summary of parsing report_. The total number of UVL files with warnings is 0
(0.00% of total files), and the total number of UVL files with exceptions is
185 (12.51%).

###  4.3. Analysis of the dataset after syntax error correction

The data from the syntactic analysis report after running the syntactic
correction have been:

  * •

_Syntactic analysis report_. Visually, the full errors report reflects that
there has been an improvement in the number of syntactic problems. However, we
still see errors such as _extraneous input_ in some UVL files.

  * •

_Summary of parsing report_. The total number of UVL files with warnings is 0
(0.00% of total files), and the total number of UVL files with exceptions is
25 (1.69%).

We note that UVL Sentinel has helped to detect and correct several syntactic
errors by 86.49%. However, there are errors that we have not contemplated in
the set of defined patterns because they are more difficult to define or do
not contemplate all the possible options of syntactic variants. Our tool is
limited to defining this set of patterns and their associated solutions. A
detailed study covering more syntactic error detection patterns is needed.

##  5\. Conclusion and future work

UVL Sentinel allows a comprehensive analysis of datasets with an arbitrary
structure of feature models in UVL format, generates valuable reports for
researchers and assists in the semi-automatic syntactic update of the model
set of one or several datasets to newer versions of a parser.

Our tool prevents a valuable dataset for the scientific community from being
tied to a particular parser version. It can no longer be used as a study
element because it is incompatible with new UVL parsers or the same parser but
with an updated version that generates incompatibilities in its syntax.
However, the success of syntax correction is limited to the set of regular
expression definitions and their possible solutions. A systematic and in-depth
study of all syntactic errors with a large volume of UVL models is needed to
determine the best strategy for correcting datasets and to keep them as part
of future studies on feature models and software product lines.

In future work, the idea is proposed to be extended to other UVL parsers or
feature model description languages with syntactic incompatibility problems.
It would also be helpful to have more error detection patterns and more
detailed reports, such as parse time or a cloud of the most common errors
encountered when reading models from a dataset.

## Material

The developed software, example datasets and deployment instructions can be
obtained from this repository:
https://github.com/diverso-lab/uvl-sentinel

## Acknowledgments

This work was partially supported by FEDER/Ministry of Science, Innovation and
Universities/Junta de Andalucía/State Research Agency/CDTI with the following
grants: _Data-pl_ (PID2022-138486OB-I00), TASOVA PLUS research network
(RED2022-134337-T) and MIDAS (IDI-20230256)

## References

  * (1)
  * Galindo et al. (2023) José A. Galindo, Jose-Miguel Horcas, Alexander Felferning, David Fernandez-Amoros, and David Benavides. 2023.  FLAMA: A Collaborative Effort to Build a New Framework for the Automated Analysis of Feature Models. In _Proceedings of the 27th ACM International Systems and Software Product Line Conference - Volume B_ (Tokyo, Japan) _(SPLC ’23)_. Association for Computing Machinery, New York, NY, USA, 16–19.  https://doi.org/10.1145/3579028.3609008
  * Sundermann et al. (2022) Chico Sundermann, Kevin Feichtinger, José Galindo, David Benavides, Rick Rabiser, Sebastian Krieter, and Thomas Thüm. 2022\.  Tutorial on the universal variability language. 260–260.  https://doi.org/10.1145/3546932.3547024
  * Sundermann et al. (2021) Chico Sundermann, Tobias Heß, Dominik Engelhardt, Rahel Arens, Johannes Herschel, Kevin Jedelhauser, Benedikt Jutz, Sebastian Krieter, and Ina Schaefer. 2021\.  Integration of UVL in FeatureIDE. In _25th SPLC_ , Vol. B. 73–79.  https://doi.org/10.1145/3461002.3473940


# The resonance absorption probability function for neutron and multiplicative
integral

V.D. Rusov111Corresponding author: Vitaliy D. Rusov, E-mail<EMAIL_ADDRESS>V.A. Tarasov, S.I. Kosenko, S.A. Chernegenko

###### Abstract

The analytical approximations for the moderating neutrons flux density like
Fermi spectra, widely used in reactor physics, involve the probability
function for moderating neutron to avoid the resonant absorption obtained
using some restrictive assumptions regarding the acceptable resonances width.
By means of multiplicative integral (Volterra integral) theory for a
commutative algebra an analytical expression for the probability function is
obtained rigorously without any restrictive assumptions.

Department of Theoretical and Experimental Nuclear Physics,
Odessa National Polytechnic University, Odessa, Ukraine

##  1 Introduction

The compound-nuclear reactions may be divided into resonant and non resonant.
It is known that the nucleus energy may possess only a discrete set of values
corresponding to its energy levels. However the idea of the levels with
strictly fixed energy is true only for the ground levels (non-excited levels)
of the stable nuclei. All the other (excited) levels do not possess the fixed
energy – they are more or less smeared over energy. A width $\Gamma_{i}$ ($i$
is the level index) of such smearing can be estimated using the uncertainty
relation for time and energy. According to this estimate, $\Delta
E_{i}=\Gamma_{i}=\hbar/\tau_{i}$, where $\tau_{i}$ is the lifetime for current
level. The nucleus may be excited through the interaction with some projectile
particle only if this excitation energy induced by interaction corresponds to
the energy levels spacing. Therefore a compound nucleus may form only in the
case that the projectile particle energy is such that the excitation energy of
the compound nucleus induced by this interaction fits into the corresponding
level’s interval of uncertainty $\Gamma_{i}$ (if the nucleus had been in its
ground state prior to interaction). If the width of compound nucleus energy
levels is less than the distance between them and the projectile particles
possess some fixed energy, the reaction may take place only trough some
$i^{th}$ single level. The reaction cross-section dependence on projectile
particle energy would display a resonant behavior. Correspondingly, this kind
of reactions is called resonant. If the levels are located so densely that the
distances between them is less than their width, they merge together and the
reaction may take place for any projectile particle energy. This kind of
reactions is called non-resonant. The form of resonances is known to be
described by the Breit–Wigner function.

As is known, the resonance region of the reactor neutron nuclear reactions of
fissile nuclides is located, depending on neutrons energy, in the
0.5$\div$1000 eV range (e.g. [3] and Fig. 1).

Figure 1: ${}^{235}U$ capture and fission cross-sections dependence on
neutrons energy. Dashed line represents the neutrons scattering cross-section
dependence.

Determining the neutrons spectrum for a moderating medium with resonant
absorption remains an important task for numerous problems of nuclear reactor
physics. Numerical methods of integro-differential equation solution for the
moderating neutrons flux density in the medium with resonant absorption are
used as well as the widespread analytical approximations for the flux density
like Fermi spectra. Fermi spectra involve the probability function for
moderating neutron to avoid the resonant absorption. The specific expression
for this probability function is usually obtained using some restrictive
assumptions regarding the acceptable resonances width. The narrow, broad
(infinite mass approximation) and intermediate resonance approximations are
used for that purpose. Therefore depending on the properties of nuclei
composing the moderating medium with absorption, one of the above
approximations is used.

We show in the present paper that the probability function for a moderating
neutron to avoid the resonant absorption (let us note that the neutron
absorption is the opposite event, and therefore the absorption probability
function is equal to one minus the probability function to avoid the
absorption) may be written down in a form of multiplicative integral (Volterra
integral) [1]. The analytical expression for this function was rigorously
obtained using the multiplicative integral theory for a commutative algebra
[2] without any restricting assumptions.

##  2 Probability function for avoiding the resonant capture with narrow
resonance approximation (standard approach)

Fermi spectrum for a moderating medium with resonant absorption, according to
[3, 4], has the form:

$\Phi_{\Phi}(E)=\dfrac{S}{\overline{\xi}\Sigma_{t}E}\cdot\varphi(E),$ (1)

where $\varphi(E)$ is the probability function for a moderating neutron to
avoid the resonant absorption; $S$ is the total volume neutron generation
rate;
$\overline{\xi}=\sum\limits_{i}\left(\xi_{i}\Sigma_{S}^{i}\right)/\Sigma_{S}$,
$\xi_{i}$ is the average logarithmic decrement of energy loss;
$\Sigma_{S}^{i}$ is the macroscopic scattering cross-section and
$\Sigma_{a}^{i}$ is the macroscopic absorption cross-section of the $i^{th}$
nuclide; $\Sigma_{t}=\sum\limits_{i}\Sigma_{S}^{i}+\Sigma_{a}^{i}$ is the
total macroscopic cross-section of the fissile material,
$\Sigma_{S}=\sum\limits_{i}\Sigma_{S}^{i}$ is the total macroscopic scattering
cross-section of the fissile material.

A following standard approach is used in reactor physics [3, 4] in order to
obtain the probability function $\varphi(E)$. According to the standard
slowing-down theory and Fig. 2, which represents the resonance region for
neutron absorption cross-section in a simplified visual manner, the
probability for neutron to avoid the absorption within the first resonance
region may be written in the form:

$\varphi_{1}=1-\dfrac{\Sigma_{a}}{\overline{\xi}\left(\Sigma_{S}+\Sigma_{a}\right)}\dfrac{\Delta
E_{1}}{E_{1}},$ (2)

where $\Sigma_{a}$ is the macroscopic absorption cross-section; $E_{1}$ is the
first resonance energy and $\Delta E_{1}$ is its width (see Fig. 2).

Figure 2: Two distant resonances $E_{1}$ and $E_{2}$ ($\Delta E_{1}$, $\Delta
E_{2}$ are the widths of the first and second resonances respectively).

The probability for neutron to avoid the absorption within the second
resonance region with energy $E_{2}$ and width $\Delta E_{2}$ has the
following form:

$\varphi_{2}=\left[1-\dfrac{\Sigma_{a}}{\overline{\xi}\left(\Sigma_{S}+\Sigma_{a}\right)}\dfrac{\Delta
E_{1}}{E_{1}}\right]\times\left[1-\dfrac{\Sigma_{a}}{\overline{\xi}\left(\Sigma_{S}+\Sigma_{a}\right)}\dfrac{\Delta
E_{2}}{E_{2}}\right].$ (3)

Thereby by analogy we have the probability function for a moderating neutron
to avoid the resonant absorption:

$\varphi(E)=\prod\limits_{i=1}^{N}\varphi_{i}=\prod\limits_{i=1}^{N}\varphi(E_{i})=\prod\limits_{i=1}^{N}\left(1-\dfrac{\Sigma_{a}}{\overline{\xi}\left(\Sigma_{S}+\Sigma_{a}\right)}\dfrac{\Delta
E_{i}}{E_{i}}\right),$ (4)

where $N$ is a number of resonances.

In order to pass from multiplication to summation, we take a logarithm of the
function $\varphi(E)$ defined in (4):

$\ln{\varphi(E)}=\prod\limits_{i=1}^{N}\ln\left(1-\dfrac{\Sigma_{a}}{\overline{\xi}\left(\Sigma_{S}+\Sigma_{a}\right)}\dfrac{\Delta
E_{i}}{E_{i}}\right)=\prod\limits_{i=1}^{N}\ln\left(1-x_{i}\right),$ (5)

where
$x_{i}=\dfrac{\Sigma_{a}}{\overline{\xi}\left(\Sigma_{S}+\Sigma_{a}\right)}\dfrac{\Delta
E_{i}}{E_{i}}$.

Next every multiplicand in expression (5) is expanded into series in terms of
a small quantity $x_{i}$ with an assumption that $\dfrac{\Delta
E_{i}}{E_{i}}\ll 1$. Limiting to the first expansion term, we obtain the
following expression:

$\ln\varphi(E)\approx-\sum\limits_{i=1}^{N}\dfrac{\Sigma_{a}}{\overline{\xi}\left(\Sigma_{S}+\Sigma_{a}\right)}\dfrac{\Delta
E_{i}}{E_{i}}.$ (6)

Let us divide the resonance region into $m$ energy intervals of width $\Delta
E_{j}$ and assume $\Sigma_{a}=0$ in the intervals between the resonances. Thus
for (6) we obtain:

$\ln\varphi(E)\approx-\sum\limits_{j=1}^{m}\dfrac{\Sigma_{a}}{\overline{\xi}\left(\Sigma_{S}+\Sigma_{a}\right)}\dfrac{\Delta
E_{j}}{E_{j}}.$ (7)

Passing from the logarithm to an exponent and making $m$ approach the
infinity, for entire resonance region from $E_{0}$ to $E_{f}$ we have:

$\varphi(E)\approx\lim\limits_{m\to\infty}\exp\left(-\sum\limits_{j=1}^{m}\dfrac{\Sigma_{a}}{\overline{\xi}\left(\Sigma_{S}+\Sigma_{a}\right)}\dfrac{\Delta
E_{j}}{E_{j}}\right)=\exp\left(-\int\limits_{E_{0}}^{E_{f}}\dfrac{\Sigma_{a}}{\overline{\xi}\left(\Sigma_{S}+\Sigma_{a}\right)}\dfrac{dE}{E}\right).$
(8)

The Fermi spectrum (1) with the probability function $\varphi(E)$ in
exponential form (8) is called a narrow resonance approximation.

##  3 Multiplicative integral and its application to a commutative algebra $A$

Basic structures related to a multiplicative integral occur in different areas
of mathematics, mechanics, physics and – as it follows from the sections 1 and
2 above – in the nuclear reactions and nuclear reactor physics as well.
However one cannot say that the multiplicative integral theory is currently
”particularly popular” among mathematicians and physicists. Let us define
multiplicative integral according to [1]. Let $A$ be an arbitrary associative
topological algebra with an identity $E$, and $f(t)$ is a function of real
variable $t$ which takes on the values in the algebra $A$,
$a,b\in\mathbf{R},a\leqslant b,[a,b]=\\{t\in\mathbf{R},a\leqslant t\leqslant
b\\}$, $T$ is the partition of an interval $[a,b]$ with the points
$t_{0}=a,t_{1},t_{2},\ldots,t_{n-1},t_{n}=b$, $t_{i}\leqslant
t_{i+1},i=0,1,2,\ldots,n-1$; $l(T)=\max(t_{i+1}-t_{i})$, $\Delta
t_{i}=t_{i+1}-t_{i}$.

Let us consider the following product:

$\prod\left(f,T\right)=\left(E+f(t_{0})\Delta
t_{0}\right)\times\left(E+f(t_{1})\Delta
t_{1}\right)\times\ldots\times\left(E+f(t_{n})\Delta t_{n}\right).$ (9)

If for any variation T such that $l(T)\to 0$ the product
$\prod\left(f,T\right)$ converges to some limit, this limit is called a
multiplicative integral of the function $f(t)$ over the interval $[a,b]$ and
is denoted by

$\int\limits_{a}^{{}^{\cup}b}E+f(t)dt.$ (10)

By analogy, for $f(t)$ and $T$ one can derive the following:

${}^{\cap}\prod\left(f,T\right)=\left(E+f(t_{n})\Delta
t_{n}\right)\times\left(E+f(t_{n-1})\Delta
t_{n-1}\right)\times\ldots\times\left(E+f(t_{0})\Delta t_{0}\right).$ (11)

If for any variation T such that $l(T)\to 0$ the product
${}^{\cap}\prod\left(f,T\right)$ converges to some limit, this limit gives
another type of the multiplicative integral for the function $f(t)$ over the
interval $[a,b]$ and is denoted by

$\int\limits_{a}^{{}^{\cap}b}E+f(t)dt.$ (12)

The multiplicative integrals (10) and (11) are called direct and backward
respectively.

Next let us consider a particular case of multiplicative integral application.
Let the algebra $A$ be commutative. Then, according to [1, 2],

$\displaystyle\int\limits_{a}^{{}^{\cap}b}E+f(t)dt=\lim\limits_{l(T)\to
0}\left(E+f(t_{0})\Delta t_{0}\right)\times\left(E+f(t_{1})\Delta
t_{1}\right)\times\ldots\times\left(E+f(t_{n})\Delta t_{n}\right)=$
$\displaystyle=\lim\limits_{l(T)\to 0}\left[E+\sum f(t_{i})\Delta
t_{i}+\sum\limits_{i<j}f(t_{i})f(t_{j})\Delta t_{i}\Delta
t_{j}+\sum\limits_{i<j<k}f(t_{i})f(t_{j})f(t_{k})\Delta t_{i}\Delta
t_{j}\Delta t_{k}+\ldots\right]=$ $\displaystyle=\lim\limits_{l(T)\to
0}\left[\sigma_{0}+\sigma_{1}+\sigma_{2}+\ldots+\sigma_{n}\right],$ (13)

where $\sigma_{0},\sigma_{1},\sigma_{2},\ldots,\sigma_{n}$ are the elementary
symmetric polynomials in variables $f(t_{0})\Delta t_{0}$, $f(t_{1})\Delta
t_{1}$, $f(t_{2})\Delta t_{2}$, $\ldots$, $f(t_{n})\Delta t_{n}$. Let
$\rho_{0}$, $\rho_{1}$, $\rho_{2}$, $\ldots$, $\rho_{n}$ be symmetric Newton’s
polynomials in the same variables:

$\rho_{k}=\sum\limits_{i}\left[f(t_{i})\Delta
t_{i}\right]^{k},~{}~{}k=1,2,\ldots,n.$ (14)

It is easy to see that

$\lim\limits_{l(T)\to 0}\rho_{k}=\begin{cases}0,&k>0,k\neq 1\\\
\int\limits_{a}^{b}f(t)dt,&k=1,\end{cases}$ (15)

where $\int\limits_{a}^{b}f(t)dt$ is the Riemann integral.

Expanding the polynomials $\sigma_{k}$ in terms of $\rho_{l}$
($k,l=1,2,\ldots,n$) and applying the equality (15), we calculate

$\lim\limits_{l(T)\to 0}\sigma_{k}.$ (16)

It is known that

$\rho_{m}-\rho_{m-1}\sigma_{1}+\rho_{m-2}\sigma_{2}+\ldots+\rho_{1}\sigma_{m-1}(-1)^{m-1}+(-1)^{m}\sigma_{m}=0,~{}~{}~{}m=1,2,\ldots,n,$
(17)

whence it follows that

$\displaystyle\lim\sigma_{1}$
$\displaystyle=\lim\rho_{1}=\int\limits_{a}^{b}f(t)dt,$
$\displaystyle\lim\sigma_{2}$
$\displaystyle=\frac{1}{2!}\left(\int\limits_{a}^{b}f(t)dt\right)^{2},$
$\displaystyle\ldots,$ $\displaystyle\lim\sigma_{n}$
$\displaystyle=\frac{1}{n!}\left(\int\limits_{a}^{b}f(t)dt\right)^{n}$ (18)

and as a result (taking into account that
$e^{x}=1+\frac{x}{1!}+\frac{x^{2}}{2!}+\frac{x^{3}}{3!}+\ldots+\frac{x^{n}}{n!}+\ldots$
and that $n\to\infty$ when $l(T)\to 0$) we obtain

$\int\limits_{a}^{{}^{\cup}b}E+f(t)dt=\exp\left(\int\limits_{a}^{b}f(t)dt\right).$
(19)

Due to Eq.(13) symmetry, for the backward multiplicative integral we,
obviously, obtain the following equality:

$\int\limits_{a}^{{}^{\cap}b}E+f(t)dt=\exp\left(\int\limits_{a}^{b}f(t)dt\right).$
(20)

##  4 Multiplicative integral and its application to the derivation of a
probability function for a moderating neutron to avoid the resonant absorption

According to the stated above, it is easy to establish the correspondence
between the theory of a probability function for a moderating neutron to avoid
the resonant absorption which is a real scalar function (and therefore a
particular case of commutative algebra $A$) and a multiplicative integral.
Indeed, according to the section 3, the identity $E$ of this commutative
algebra $A$ is the real unit 1, and
$f(E)=-\dfrac{\Sigma_{a}}{\overline{\xi}\left(\Sigma_{S}+\Sigma_{a}\right)E}$
is a scalar function of a real argument $E$ which takes on the values in the
algebra $A$, $E_{0},E_{f}\in\mathbf{R}$, $E_{0}<E_{f}$,
$[E_{0},E_{f}]=\\{E\in\mathbf{R},E_{0}\leqslant E\leqslant E_{f}\\}$, $T$ is
the partition of an interval $[E_{0},E_{f}]$ with the points
$t_{0}=E_{0},t_{1},t_{2},\ldots,t_{n-1},t_{n}=E_{f}$, $t_{i}\leqslant
t_{i+1},i=0,1,2,\ldots,n-1$; $l(T)=\max(t_{i+1}-t_{i})$, $\Delta
t_{i}=t_{i+1}-t_{i}$.

Then for the probability function to avoid the resonant absorption, according
to (4) with the partition $T$ of an interval $[E_{0},E_{f}]$ we obtain:

$\varphi(E)=\lim\limits_{n\to\infty}\prod\limits_{i=1}^{n}\left(1-\dfrac{\Sigma_{a}}{\overline{\xi}\left(\Sigma_{S}+\Sigma_{a}\right)E_{i}}\Delta
E_{i}\right)=\lim\limits_{l(T)\to\infty}\prod\limits_{i=1}^{n}\left(1+f(E_{i})\Delta
E_{i}\right)=\int\limits_{E_{0}}^{{}^{\cap}E_{f}}E+f(t)dt,$ (21)

where
$f(E)=-\dfrac{\Sigma_{a}}{\overline{\xi}\left(\Sigma_{S}+\Sigma_{a}\right)E}$.

Consequently, taking into account (20), we derive the final expression:

$\varphi(E)=\exp\left(-\int\limits_{E}^{E_{f}}\dfrac{\Sigma_{a}}{\overline{\xi}\left(\Sigma_{S}+\Sigma_{a}\right)E}dE\right).$
(22)

##  5 Conclusion

In the present paper we showed that the probability function for a moderating
neutron to avoid resonant absorption may be written down in a form of a
multiplicative integral. Using the results of the multiplicative integral
theory application for a commutative algebra the analytical expression for the
probability function for a moderating neutron to avoid the resonant absorption
was rigorously obtained. The expression for Fermi spectrum incorporating the
obtained probability function turned out to coincide with the expression for
Fermi spectrum known as the narrow resonance approximation. However, the
former one is an exact expression and therefore does not have any
restrictions.

We should also note here that since the narrow resonance approximation is used
for another types of nuclear reactions, the obtained results may be
generalized for a broader range of nuclear physics problems.

## References

  * [1] J.D. Dollard, Ch.N. Friedman. Product integration with applications to differential equations. London: Addison-Wesley Publ. Co., 1979
  * [2] O.V. Manturov The product integral, Journal of Mathematical Sciences, Vol.55, No.5, (1991), pp. 2042-2076; DOI: 10.1007/BF01095674
  * [3] G.G. Bartolomey, G.A. Bat’, V.D. Babaykov and M.S. Altukhov, Basic theory and methods of nuclear power installations calculation Moscow: Energoatomizdat, 1989
  * [4] S.V. Shirokov, Nuclear reactor physics (in Russian) Kiev: Naukova dumka, 1992