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in his nowadays classical studies on the kinetic theory of gases , the danish physicist m. knudsen experimentally observed that , no matter how an inert gas was injected into a pipeline , the direction in which a molecule rebounds from the pipeline s solid wall is asymptotically independent of its initial trajectory . that is , the fraction of particles leaving the surface in a given direction is proportional to @xmath0 , where @xmath1 is the angle that such a particle s trajectory defines , measured with respect to the normal to the surface ( @xcite ) . this behaviour is referred to as the * _ knudsen s ( cosine ) law _ * ever since . in the experiment , the gas is injected at a very low pressure so that interactions between particles are negligible . the physical justification of knudsen s law is the following . first of all , one assumes that particles bounce at the pipeline s wall _ elastically_. this means that the energy of a particle is preserved in a collision which , in turn , implies the _ refection law _ : the angle that the incident direction forms with the normal to a surface , @xmath2 , equals the angle @xmath3 of the reflected direction , @xmath4 . it has been proved that this law is valid in a first approximation if we do not take into account thermal effects . since we also assume that particles do not interact with one another , we must conclude that the microscopic irregularities on the pipeline s surface are responsible for the destruction of any particular pattern in the original gas distribution . indeed , even if we assume that the bounces are perfectly elastic , microscopic holes in the boundary of the pipeline and imperfections in relief are dimensionally comparable to the molecules of the gas and have therefore a disruptive ( i.e. , unpredictable ) effect on particle collisions : after many bounces , we are in the so - called _ knudsen s regime _ in which the reflected direction is independent of the incident one . however , this argument does not explain by itself why the reflected angles are distributed according to the _ cosine law_. the theory of _ billiards _ helps clarify this point ( @xcite , @xcite ) . the irregularities of the pipeline surface can be reasonably modelled as cavities or microscopic cells with a dispersive geometry such that , once a particle has entered one of them , it comes out of it with a rather arbitrary direction , even if all the collisions inside that cavity are elastic . if the pipeline is made of a uniform material , we can model one such cell as a billiard table and the pipeline wall as an infinite row of such billiard tables that a particle moving freely enters and exits . with this description , knudsen s cosine law can be then seen as a consequence of the fact that , for sufficiently dispersive billiard geometries , the liouville measure is the unique measure preserved by the billiard flow ( see @xcite for more details ) . unfortunately , this need not be the case for polygonal ( i.e. , non - dispersive ) tables such as fig . [ fig pipeline ] . so polygonal geometries require a slightly different approach .    since the characteristic size of this cell is infinitely small compared with the diameter of the pipeline , the exact position along the open side of the cell ( dotted line in fig . [ fig pipeline ] ) at which the particle enters a cell coming from the previous one is considered to be randomly distributed with uniform probability . following @xcite , we will refer to these billiards as _ random billiards_. more concretely , the dynamics of a particle bouncing inside the pipeline are given by the _ first return map _ to the open side of a the billiard cell , which defines a markov process . thus , the dynamics are characterised by a transition operator @xmath5 such that , for any reflected direction @xmath6 $ ] , @xmath7 is the probability that a molecule takes a direction in @xmath8 $ ] after the next rebound . these concepts will be reviewed in sections [ section preliminaries ] and [ section random billiards ] . in this formalism , knudsen s law can be mathematically written as follows . let @xmath9 be the initial angle distribution with which the gas is injected into the pipeline . that is , if @xmath8 $ ] , @xmath10 represents the proportion of particles that will hit the pipeline s wall for the first time with an incident angle @xmath11 . denote by @xmath12 the distribution after @xmath13 collisions . in this context , knudsen s cosine law can be expressed in two different ways . the * _ strong knudsen s law _ * claims that , for almost any initial distribution @xmath9 , after many collisions @xmath12 converges to the cosine distribution , @xmath14 . \label{eq strong law}\ ] ] on the other hand , the * _ weak knudsen s law _ * states that if , after any collision , we count the number of particles reflected with a given angle @xmath1 and then divide by the total number of particles , this quantity is proportional to @xmath15 . explicitly,@xmath16 obviously ( [ eq strong law ] ) implies ( [ eq weak law ] ) , so the strong knudsen s law implies the weak one . deterministic dynamical systems can be sometimes understood as idealizations of _ real _ systems , the latter usually subjected to negligible perturbations in a first approximation . the use of a dynamical approach to study a gas is one such example . in this paper , we will study the properties of the random billiard obtained as a result of modelling the rough surface of the pipeline by a zig - zag geometry as shown in fig . [ fig pipeline ] . our billiard cell is an isosceles triangle with open side @xmath17 defined by an angle @xmath18 . this model was introduced in @xcite but has not been studied in depth yet . for example , in @xcite nothing is said about the ergodic properties of this billiard table when @xmath19 ( it is shown that it is not ergodic under the liouville measure if @xmath20 though ) . thus it is not at all clear if the dynamics resulting from a random billiard as in fig . [ fig billiard table ] will in fact follow knudsen s law ( weak or strong ) . in this paper , we will prove that it is actually the case . more concretely , the main contributions of our paper are the following :    1 . little is known about the ergodicity of the flow of deterministic billiards given by irrational polygon tables ( @xcite ) . remarkably , we show that , for a concrete example , randomising the billiard in a sensible way ( i.e. , choosing the point at which the particle enters the billiard randomly but uniformly distributed on one side of the table ) implies its ergodicity and exactness with respect to the liouville measure . 2 .   unlike other attempts to prove ( weak ) knudsen s law , where one usually assumes very irregular geometries at a microscopic level responsible for the dispersive effects , we deal with an extremely simple pattern . this has additional advantages as a simpler model capturing the main features of a system can be simulated and developed more easily . 3 .   in the literature , ergodicity of random billiards relies on the ergodicity of the first return map ( see @xcite ) . we use here a different approach and take advantage of a skew - type representation of random maps recently introduced in @xcite . the techniques used to prove the exactness of our model can be certainly adapted to study other ( random ) billiard tables and random maps in general . finally , we prove that the strong knudsen s law holds for our model . as we mentioned before , even if one assumes dispersive billiard geometries , one will only prove that the first return map is ergodic , i.e. , the weak law . the paper is structured as follows : in section [ section model ] we introduce the random billiard behind our model . we recall some basic definitions and properties of billiards in section [ section preliminaries ] and the concept of random billiard in section [ section random billiards ] . in section [ section random maps ] , we review the skew - type representation of random maps introduced in @xcite and show the relationship between this representation and the dynamical evolution of absolutely continuous measures . we prove that this skew - type representation is exact in section [ section exactness of s ] , which uses some auxiliary results presented separately in section [ section ergodicity of ss ] . finally , in section [ section knudsen s law ] , we illustrate our results with numerical simulations . suppose we have a gas of non - interacting particles moving freely inside a pipeline . our model consists of a two dimensional infinite pipeline whose rough walls are modelled as a sequence of cells built as a juxtaposition of a _ fundamental cell _ ( fig . [ fig billiard table ] ) . in this section , we are going to fix the geometry of such cell and introduce the main hypothesis behind the dynamics of the particles . the content of this section is extracted from @xcite . both the geometry and dynamics of our model are summarised in figure [ fig pipeline ] . we assume that the walls of the pipeline are not smooth but describe a serrated regular pattern built from a fixed isosceles triangle with an open side . we characterise that triangle by one of its angles @xmath21 . the dynamics of a particle moving freely inside the pipeline are as follows : the particle enters one of the cells with some angle @xmath2 and bounces on its sides elastically , that is , according to the reflection law . the particle eventually comes out of the cell with a given direction @xmath22 , then crosses the pipeline , and reaches another cell on the opposite wall , and the process recurs . since the diameter of the pipeline is several orders of magnitude bigger than the characteristic length @xmath17 of the cell , it is plausible to think that , every time the particle enters a new cell through the open side , it does so at a point uniformly distributed on @xmath17 .        in the literature , one encounters examples of particle dynamics on bounded domains where the random perturbation is introduced in the observations of @xmath22 every time the particles bounces off the wall ( see for example @xcite ) , which implies that collisions are no longer elastic . this perturbation is explained as a consequence of the roughness of the wall . observe that , in our model , randomness is introduced in a completely different way . we assume elastic collisions but , instead of considering dispersive geometries responsible for the random bounces , the noise is introduced in a sensible way without modifying the ( rather elementary ) geometry of the problem .    the trajectory that a particle describes after it comes out of a billiard cell until it enters another one on the opposite wall is completely irrelevant for dynamical purposes . in practice , we can better understand our dynamical system as the closed billiard table obtained by fixing together two triangular cells so that they share the same open side as in figure [ fig billiard table ] . when the particle crosses that open side , it retains its direction but it is assigned a different position on @xmath17 according to a uniform law .     at a uniformly distributed random point.,width=377,height=181 ]    the * _ incoming angle _ * @xmath2 ( resp . * _ outgoing angle _ * ) is then the angle in @xmath23 $ ] with which the particle enters ( resp . leaves ) the cell measured with respect to the normal to the open side @xmath17 . one often shifts @xmath23 $ ] by @xmath24 and measure @xmath1 in @xmath25 $ ] , i.e. with respect to horizontal . in @xcite , for any incoming angle @xmath26 $ ] , feres gives the different possible directions with which a particle may come out of the fundamental cell ( fig . [ fig billiard table ] ) . they are four , given by maps @xmath27\rightarrow\mathbb{r}$ ] defined as@xmath28 and any @xmath29 applies with certain probability @xmath30 . the probabilities associated to the maps ( [ eq tau maps ] ) are as follows:@xmath31{ll}1 & \theta\in\lbrack0,\alpha)\\ u_{\alpha}\left (   \theta\right )   & \theta\in\lbrack\alpha,\pi-3\alpha)\\ 2\cos(2\alpha)u_{2\alpha}(\theta ) & \theta\in\lbrack\pi-3\alpha,\pi-2\alpha)\\ 0 & \theta\in\lbrack\pi-2\alpha,\pi ] \end{array } \right . \smallskip\nonumber\\ p_{2}\left (   \theta\right )    &   = \left\ { \begin{array } [ c]{ll}0 & \theta\in\lbrack0,\pi-3\alpha)\\ u_{\alpha}\left (   \theta\right )   -2\cos(2\alpha)u_{2\alpha}(\theta ) & \theta\in\lbrack\pi-3\alpha,\pi-2\alpha)\\ u_{\alpha}\left (   \theta\right )   & \theta\in\lbrack\pi-2\alpha,\pi-\alpha)\\ 0 & \theta\in\lbrack\pi-\alpha,\pi ] \end{array } \right . \smallskip\nonumber\\ p_{3}\left (   \theta\right )    &   = \left\ { \begin{array } [ c]{ll}0 & \theta\in\lbrack0,2\alpha)\\ 2\cos(2\alpha)u_{2\alpha}(-\theta ) & \theta\in\lbrack2\alpha,3\alpha)\\ u_{\alpha}\left (   -\theta\right )   & \theta\in\lbrack3\alpha,\pi-\alpha)\\ 1 & \theta\in\lbrack\pi-\alpha,\pi ] \end{array } \right . \smallskip\nonumber\\ p_{4}\left (   \theta\right )    &   = \left\ { \begin{array } [ c]{ll}0 & \theta\in\lbrack0,\alpha)\\ u_{\alpha}\left (   -\theta\right )   & \theta\in\lbrack\alpha,2\alpha)\\ u_{\alpha}\left (   -\theta\right )   -2\cos(2\alpha)u_{2\alpha}(-\theta ) & \theta\in\lbrack2\alpha,3\alpha)\\ 0 & \theta\in\lbrack3\alpha,\pi ] , \end{array } \right . \tag{m2}\label{eq probabilities model}\ ] ] where@xmath32 and @xmath18 is assumed to be smaller than @xmath33 . as we will see later , this family of maps and probabilities is all we need to prove knudsen s law . this section aims at recalling the main concepts of the theory of ( deterministic ) billiards and one of its key results , namely , that the _ billiard map _ preserves the liouville measure . the content of this section was extracted from @xcite , which the reader is referred to for an exhaustive exposition on billiards .    a * _ billiard table _ * @xmath34 is the closure of a bounded open domain @xmath35 whose boundary @xmath36 is a finite union of smooth compact curves ( @xmath37 , @xmath38 ) . @xmath39 are called the * _ walls _ * of the billiard table and are assumed to intersect each other only at their endpoints or * _ corners_*. for any @xmath40 , @xmath41 is defined by a @xmath37 map @xmath42\subset \mathbb{r}\rightarrow\mathbb{r}^{2}$ ] such that the second derivative @xmath43 either vanishes or is identically zero . this condition prevents the wall @xmath41 from having inflection points or line segments . a wall such that @xmath44 is called * _ focusing _ * or * _ dispersing _ * if @xmath45 points inwards or outwards @xmath34 , respectively . otherwise @xmath46 and the wall is called * _ flat_*. in this framework , we consider the dynamics of a point particle moving freely within @xmath34 . let @xmath47 denote the position of a free particle moving within a billiard table . we suppose that the particle bounces at @xmath48 * _ elastically_*. that is , if it collides with the wall @xmath49 at a point @xmath50 that is not a corner , the incident angle @xmath2 that the velocity @xmath51 forms with the normal vector @xmath13 to @xmath49 at @xmath50 equals the angle of reflection @xmath3 between @xmath13 and @xmath51 after the collision , i.e. @xmath4 . this implies that @xmath51 has constant norm , which we will assume equal to @xmath52 for the sake of simplicity . under these assumptions , the * _ billiard flow _ * @xmath53 is the flow that this particle defines on the * _ phase space _ * @xmath54 . since the particle follows a straight trajectory between two consecutive collisions , the dynamical information of the system is contained in the geometry of the boundary @xmath48 and how the particle bounces off it . therefore , instead of studying the billiard flow , first one introduces the cross - section @xmath55 $ ] as the set of all postcollisional velocity vectors , where @xmath56 $ ] measures the angle between @xmath51 and the normal vector @xmath13 pointing to the interior of @xmath34 , and then considers the first return map @xmath57 that the billiard flow induces on @xmath58 . that is , given a point of @xmath58 , @xmath57 gives the next position at where the particle collides and its velocity after that collision . the map @xmath57 is often called the * _ billiard map_*.    it is a well known result in the theory of billiards that the billiard map preserves the * _ liouville measure _ * @xmath59 on @xmath58 , where @xmath60 is the lebesgue measure on @xmath48 and @xmath61 is the measure on @xmath23 $ ] given by @xmath62 , @xmath63\right )   $ ] ( ( * ? ? ? * ; * ? ? ? * lemma 2.35 ) ) . however , except for a few general results , little is known about the ergodicity of billiard maps with respect to the liouville measure for explicit examples of boundary geometries . for example , dispersive billiards are ergodic ( ( * ? ? ? * ; * ? ? ? * theorem 6.20 ) ) while regular polygons are not . finally , as we pointed out in section [ section model ] , it is customary in the literature to translate @xmath23 $ ] and measure @xmath1 in @xmath64 $ ] , in which case @xmath61 is given by @xmath65 , @xmath66\right )   $ ] . we will follow this convention throughout the paper . in this section we are going to get back to the concept of _ random billiard _ and its different representations . let @xmath34 be a billiard table with boundary @xmath67 . remove one of the walls of @xmath48 , for example @xmath68 , so that we obtain a billiard table with an _ open side_. without loss of generality , we can assume that the open side is just a line segment @xmath17 as in figure [ fig billiard table ] . this identification can be carried out by the map @xmath69\subset\mathbb{r}\rightarrow\mathbb{r}^{2}$ ] defining @xmath68 . on the other hand , suppose a particle enters the billiard from the open side at @xmath70 with an angle @xmath71 $ ] with respect to @xmath68 . denote by @xmath72 $ ] the angle associated to the velocity of that particle when it returns to the open side for the first time . the * _ first return map _ * ( to the open side ) is then the map @xmath73\rightarrow \overline{pq}\times\lbrack0,\pi]$ ] induced from the billiard map @xmath57 such that @xmath74 , i.e. , the particle leaves the billiard at @xmath75 with an angle @xmath76 . the liouville measure @xmath59 on @xmath77 $ ] induces a measure on @xmath78 $ ] by restriction in a natural way . we will continue referring to this measure as the liouville measure and denoting it by @xmath59 , where @xmath60 now stands for the lebesgue measure on @xmath17 . it can be proved that the first return map @xmath79 also preserves the liouville measure ( ( * ? ? ? * ; * ? ? ? * ; * ? ? ? * exercise i.5.1 ) ) . furthermore ,    [ lemma ergodicity first return]if the billiard map @xmath80\rightarrow\partial d\times\lbrack0,\pi]$ ] is ergodic with respect to @xmath59 , so is the first return map @xmath81\rightarrow\overline{pq}\times\lbrack0,\pi]$ ] . let @xmath82)$ ] be an invariant set , i.e. , @xmath83 . this set can be naturally regarded as a measurable set of @xmath77 $ ] since @xmath84 is identified with the wall @xmath68 . since @xmath57 is ergodic , @xmath85 has full @xmath86}$]-measure , which means that@xmath87}(({\textstyle\bigcup\nolimits_{n\geq1 } } f^{-n}\left (   a\right )   ) \bigcap\left (   \overline{pq}\times\lbrack 0,\pi]\right )   ) = 1.\ ] ] if @xmath88\right )   $ ] , then @xmath89 for some @xmath90 and , from the very definition of the first return map that , @xmath91 for some @xmath92 . therefore,@xmath93\right )   = \bigcup \nolimits_{m\geq1}t^{-m}\left (   a\right )   = a,\ ] ] where the last equality follows from the invariance of @xmath94 . consequently , @xmath95}(a)=1 $ ] and @xmath79 is ergodic . let @xmath96 be the probability space built upon @xmath17 with the normalised lebesgue measure @xmath60 on the borel @xmath97-algebra @xmath98 . the * _ random billiard _ * associated to @xmath34 is then the time - discrete random dynamical system @xmath99\rightarrow \lbrack0,\pi]~|~x\in\overline{pq}\right\ }   $ ] , where @xmath17 is regarded as a probability space . roughly speaking , the random billiard thus built models the outgoing state a free particle that entered the billiard through the open side at a point uniformly distributed along @xmath17 . when regarded as a probability space , @xmath17 will be denoted by @xmath100 . the random billiard @xmath99\rightarrow\lbrack0,\pi ] ~|~x\in\omega\right\ }   $ ] defines a transition probability kernel @xmath101\times\mathcal{b}\left (   [ 0,\pi]\right )   \rightarrow\lbrack0,1]$ ] given by@xmath102\text { and } a\in\mathcal{b}\left (   [ 0,\pi]\right )   . \label{eq probability kernel}\ ] ] we say that @xmath9 is * _ invariant _ * with respect to @xmath103\rightarrow\lbrack0,\pi]~|~x\in\omega\right\ }   $ ] if @xmath104}k(\theta , a)d\nu(\theta)\text { for any } a\in \mathcal{b}\left (   [ 0,\pi]\right)\ ] ] and that the random billiard is * _ ergodic _ * with respect to @xmath9 if @xmath79 is ergodic with respect to @xmath105 . for example , @xmath106 , @xmath107\right )   $ ] , is an invariant measure because @xmath59 is invariant with respect to the first return map @xmath79 ( ( * ? ? ? * ; * ? ? ? * ; * ? ? ? * proposition 2.1 ) ) . moreover , since dispersive billiards are ergodic with respect to the liouville measure , the corresponding random billiard is also ergodic with respect to @xmath61 ( lemma [ lemma ergodicity first return ] ) . in this situation , birkhoff s ergodic theorem implies that @xmath108 where @xmath109\rightarrow\lbrack0,\pi]$ ] is the projection onto the second factor . that is , the average number of particles reflected with some angle @xmath11 converges to @xmath110 so ( weak ) knudsen s law holds for dispersive billiards .    in general , for non - dispersive billiards , little is know about the ergodicity of the first return map . this is indeed the case of the random billiard introduced in section [ section model ] ( fig . [ fig billiard table ] ) . recall that , as far as we know , determining whether a general irrational triangular billiard is ergodic is still an open problem ( ( * ? ? ? * ; * ? ? ? * ; * ? ? ? * section 7 ) ) and only a few particular examples were proved to be ergodic ( @xcite ) . therefore , at this point , it is not clear at all if knudsen s law holds for the pipeline model introduced in section [ section model ] . to prove it , the first return map must be replaced with a more convenient representation . given a random billiard , the probability kernel ( [ eq probability kernel ] ) contains all the dynamical information of the system and determines its properties . this implies that , as far as the dynamics is concerned , the underlying probability space @xmath100 ( which in section [ section random billiards ] was equal to @xmath96 ) plays a secondary role . in particular , other probability spaces @xmath111 giving rise to the same probability kernel are available . indeed , if @xmath112 $ ] is a transition probability kernel on a measurable space @xmath113 , kolmogorov s existence theorem on markov processes guarantees that there exists a probability @xmath114 defined on the borel @xmath97-algebra of the topological space @xmath115 , @xmath116 for all @xmath117 , such that the chain @xmath118 is markovian and has transition probability kernel @xmath119 ( ( * ? ? ? * ; * ? ? ? * theorem 2.11 ) ) , where @xmath120 is the projection onto the @xmath121th factor . in this statement , @xmath122 has to be a @xmath97-compact hausdorff space and @xmath123 the borel @xmath97-algebra . unfortunately , @xmath124 is not very manageable to work with explicitly . for instance , the representation @xmath73\rightarrow\overline { pq}\times\lbrack0,\pi]$ ] given by the first return map introduced in section [ section random billiards ] only involves finite dimensional spaces and hence seems more convenient . we are now going to consider another skew - type representation for random maps recently introduced in @xcite . as we will show in section [ section exactness of s ] , this representation will be crucial to prove the asymptotic properties of the random map defined by our random billiard fig . [ fig billiard table ] . let @xmath125 be a general measure space and let @xmath126 be the unit interval regarded as a probability space , where @xmath60 stands for the lebesgue measure . later on , we will apply the results of this section to @xmath127 $ ] . for any @xmath128 , let @xmath129 and @xmath130 $ ] be measurable mappings such that @xmath131 is a measurable partition of the unity , i.e. , @xmath132 for any @xmath133 ( compare to ( [ eq probabilities model ] ) and ( [ eq tau maps ] ) in section [ section model ] ) . we define the * _ random dynamical system _ * @xmath134 such that @xmath135 with probability @xmath136 . the transition probability kernel of @xmath137 is given by@xmath138 this probability kernel defines the evolution of an initial distribution ( probability measure ) @xmath9 on @xmath113 under the random map @xmath137 iteratively as@xmath139}k(\theta , a)d\nu^{(n)}(\theta),~~n\geq1,\label{eq 15}\ ] ] where @xmath140 . a measure @xmath9 is called * _ _ * invariant if @xmath141 . following @xcite , consider now @xmath142 and set @xmath143 such that @xmath144 is a finite measurable partition of @xmath100 . we define the * _ skew - type representation _ * of the random map @xmath137 as the map @xmath145 such that@xmath146 where@xmath147 the map @xmath148 is well defined because if @xmath149 then @xmath150 necessarily . moreover , @xmath151 thus defined is @xmath152-measurable ( ( * ? ? ? * lemma 3.1 ) ) and it is a skew - type representation of the random dynamical system @xmath137 .    as we will see , the map @xmath151 is extremely useful to study the properties of @xmath137 . for example , @xmath9 is an invariant measure of @xmath137 on @xmath153 if and only if @xmath105 is an invariant measure of @xmath151 on @xmath154 ( ( * ? ? ? * lemma 3.2 ) ) . moreover , since we know that the sine @xmath65 is invariant by the transition probability kernel ( [ eq 16 ] ) of our billiard table , we conclude that @xmath61 is invariant by @xmath137 built from ( [ eq probabilities model ] ) and ( [ eq tau maps ] ) , and therefore @xmath59 is invariant by the corresponding map @xmath151 . in an abuse of terminology , we will continue calling @xmath59 the * _ liouville measure when referring to _ * @xmath151 . in this context , we will say that the random map @xmath137 with initial distribution @xmath9 is * _ ergodic _ * ( resp . * _ mixing _ * , * _ exact _ * ) if @xmath151 is ergodic ( resp . mixing , exact ) with respect to @xmath105 . in section [ section exactness of s ] , we are going to prove that the @xmath151 associated to ( [ eq probabilities model ] ) and ( [ eq tau maps ] ) is exact . [ h ]    s.jpg    unlike the first return map @xmath79 ( section [ section random billiards ] ) , the map @xmath151 has no dynamical interpretation and is a purely auxiliary tool to represent and study @xmath137 . therefore , we need to establish the relationship between @xmath151 and the transition probability kernel ( [ eq 16 ] ) , which carries all the dynamical information . this is the content of theorem [ theorem 1 ] . before stating that relationship , we need to introduce some notation and an auxiliary lemma , whose proof is included separately in the appendix for the sake of a clearer exposition . [ definition slices]we define iteratively the sets @xmath155 , where @xmath156 for any @xmath157 , as@xmath158    [ lemma slices go to zero]let @xmath159 the fibre through @xmath133 .  if @xmath160 , then@xmath161    [ theorem 1]let @xmath9 be a measure on @xmath113 and @xmath140 . then,@xmath162   : = \int_{[0,1)\times x}\mathbf{1}_{a}\left (   \pi_{2}\circ s^{n}\right )   d\left (   \lambda\otimes\nu\right ) = \nu^{(n)}\left (   a\right )   .\ ] ]    first of all , it is not difficult to check that@xmath163 this can be proved iteratively rewriting @xmath119 in terms of ( auxiliary ) dirac deltas , @xmath164 where @xmath165 , @xmath166 .    on the other hand , since @xmath167 is the disjoint union of the sets @xmath168 , @xmath156,@xmath169 where , in the second line , we have used that @xmath170 on the set @xmath168 . if now we apply fubini s theorem,@xmath171 but by lemma [ lemma slices go to zero ] , for any fixed @xmath133,@xmath172 therefore,@xmath173    [ corollary koopman]let @xmath174 , @xmath175 , be the * koopman operator * associated to @xmath151 , i.e. , @xmath176 . let @xmath177 be the projection onto the second factor . if @xmath9 is a measure on @xmath153 , then@xmath178}u_{s}^{n}\left ( 1_{\pi_{2}^{-1}(a)}\right )   d\left (   \lambda\otimes\nu\right )   , ~a\in \mathcal{b},~n\in\mathbb{n}.\ ] ]    let @xmath151 be the skew - type representation of the random billiard introduced in section [ section model ] and let @xmath179 be a probability measure absolutely continuous with respect to the liouville measure @xmath61 with radon - nikodym derivative @xmath180,\mu\right )   $ ] . one can easily check that @xmath181 , where @xmath182 and @xmath183 $ ] . if @xmath151 is mixing ( or exact ) then , from corollary [ corollary koopman ] , @xmath184 ( ( * ? ? ? * proposition 4.4.1 ( b ) ) ) , which implies the strong knudsen s law . we will prove that @xmath151 is exact in section [ section exactness of s ] and give more details about the strong law in section [ section knudsen s law ] . the proof uses the fact that the pull - back @xmath185 of a characteristic function @xmath186 , @xmath187   \right )   $ ] , can not be invariant by @xmath188 ( section [ section ergodicity of ss ] ) . it is worth observing that the strong law , unlike the standard approach to random billiards available in the literature from the first return map ( see [ eq 18 ] ) , is a consequence of the asymptotic properties of the skew - type representation @xmath151 . let @xmath151 be the skew - type representation of the random map ( [ eq tau maps ] ) and ( [ eq probabilities model ] ) . in this section , we are going to show that @xmath190\rightarrow\lbrack0,1)\times \lbrack0,\pi]$ ] can not leave invariant any characteristic function @xmath191 , where @xmath192   \right )   $ ] has probability @xmath193 and @xmath177 denotes the projection onto the second factor . this result will be used in the next section to prove that @xmath151 is exact . first , we will show that our model exhibits a very useful symmetry . [ prop symmetry]let @xmath151 be the skew - type representation of the random map ( [ eq tau maps ] ) and ( [ eq probabilities model ] ) . let @xmath194\rightarrow\mathbb{r}$ ] be a function such that @xmath195 is @xmath189-invariant , i.e.,@xmath196 and let @xmath197\rightarrow\lbrack0,\pi]$ ] be defined by @xmath198 . then @xmath199 is also @xmath189-invariant . it is not difficult to realise from ( [ eq probabilities model ] ) that@xmath200 just using that the the function @xmath201 introduced in ( [ eq definition function u ] ) has period @xmath202 . furthermore , straightforward computations shows that@xmath203 given an index @xmath204 , we define its conjugate index @xmath205 in the following manner:@xmath206    on the other hand , ( [ eq 13 ] ) is equivalent to@xmath207 we have to check that this property holds for @xmath208 . so let @xmath209 $ ] such that @xmath210 . since @xmath211 and @xmath212 for any @xmath213 , we have that @xmath214 implies@xmath215{r}p_{i}\left (   x\right )   = p_{\widetilde{i}}\circ\phi\left (   x\right )   > 0\\ p_{j}\left (   \tau_{i}\left (   x\right )   \right )   = p_{\widetilde{j}}\circ \phi\circ\tau_{i}\left (   x\right )   = p_{\widetilde{j}}\circ\tau_{_{\widetilde { j}}}\circ\phi\left (   x\right )   > 0 \end{array } \right\ }   \longrightarrow(p_{\widetilde{i}}\cdot(p_{\widetilde{j}}\circ \tau_{_{\widetilde{j}}}))\circ\phi\left (   x\right )   > 0.\ ] ] therefore , using the invariance of @xmath216 expressed in ( [ eq 34])@xmath217 so @xmath208 also satisfies ( [ eq 34 ] ) and is @xmath189-invariant . [ prop g invariant s2 and phi]let @xmath218,\mu\right ) $ ] and suppose that @xmath195 is invariant by @xmath189 and @xmath219 . if @xmath18 in ( [ eq probabilities model ] ) and ( [ eq tau maps ] ) is irrational , then @xmath216 is constant . let @xmath220\right/   _ { 0\sim\pi}$ ] be the quotient space obtained by identifying @xmath221 . this space is homeomorphic to the unit circle @xmath222 . the function @xmath223\rightarrow\mathbb{r}$ ] induces a function @xmath224:\left . [ 0,\pi]\right/   _ { 0\sim\pi}-\{[0]\}\rightarrow \mathbb{r}$ ] on the quotient that is well defined except at @xmath225 $ ] , the equivalence class of @xmath226 , because maybe @xmath227 . we are going to prove that @xmath224 $ ] is invariant by an irrational rotation . since constants are the only integrable functions invariant by irrational rotations , @xmath224 $ ] must be constant and @xmath216 too . recall that @xmath228 reads as@xmath229 we are going split the interval @xmath230   $ ] into subintervals and , for each of them , we will explore the invariance properties of @xmath216 implied by ( [ eq 34 ] ) . ( i ) : :    if @xmath231 , then    @xmath232    so@xmath233 ( ii ) : :    if @xmath234 , then    @xmath235 . writing    @xmath236 with    @xmath237,@xmath238 now ,    since @xmath216 is    @xmath219-invariant,@xmath239    therefore,@xmath240   \left (   \left [   x\right ]   \right )   = \left [   g\right ]    \left (   \left [   x+6\alpha\right ]   \right )   \text { if } x\in\lbrack\pi    -4\alpha,\pi-3\alpha).\ ] ] ( iii ) : :    if @xmath241 then    @xmath242 .    writing @xmath243 with    @xmath237,@xmath244    using    @xmath245,@xmath246 that    is,@xmath240   \left (   \left [   x\right ]   \right )   = \left [   g\right ]    \left (   \left [   x+6\alpha\right ]   \right )   \text { if } x\in\lbrack\pi    -3\alpha,\pi-2\alpha).\ ] ] ( iv ) : :    suppose @xmath247 ,    @xmath248 with @xmath237 .    since @xmath249 on    @xmath250 , by ( [ eq    34]),@xmath251    since @xmath216 is invariant by    @xmath219,@xmath252 summarising ,    @xmath253 which    implies@xmath254\left (   [ x]\right )   = [ g]\left (   [ x+2\alpha]\right )   \text { if } x\in\lbrack\pi-2\alpha,\pi-\alpha).\ ] ]    now @xmath237 and we can use the    @xmath189-invariance expressed in the first item so    that@xmath240   \left (   \left [   x\right ]   \right )   = \left [   g\right ]    \left (   \left [   x+2\alpha\right ]   \right )   = \left [   g\right ]   \left (   \left [    x+6\alpha\right ]   \right )   .\ ] ] ( v ) : :    finally , let @xmath255 ,    @xmath256 with @xmath237 .    since @xmath257 on    @xmath258@xmath259    by ( [ eq 34 ] ) . using the @xmath219-invariance of    @xmath216,@xmath260 that is ,    @xmath261 , which again    implies@xmath254\left (   [ x]\right )   = [ g]\left (   [ x+2\alpha]\right )   \text { if } x\in\lbrack\pi-\alpha,\pi).\ ] ]    now @xmath262 and    @xmath263 as by assumption    @xmath264 ; we are in the situation of ( i )    so@xmath240   \left (   \left [   x\right ]   \right )   = \left [   g\right ]    \left (   \left [   x+2\alpha\right ]   \right )   = \left [   g\right ]   \left (   \left [    x+6\alpha\right ]   \right )   .\ ] ]    in conclusion , if @xmath216 is invariant by @xmath189 and @xmath219 , then@xmath240   \left (   \left [   x\right ]   \right )   = \left\ { \begin{array } [ c]{l}\left [   g\right ]   \left (   \left [   x+4\alpha\right ]   \right )   \text { \ if } x\in(0,\pi-4\alpha)\\ \left [   g\right ]   \left (   \left [   x+6\alpha\right ]   \right )   \text { \ if } x\in\lbrack\pi-4\alpha,\pi ) . \end{array } \right .   \label{eq 31}\ ] ] let @xmath265 such that @xmath266 . observe that @xmath267 because @xmath264 . applying the invariance expressed in ( [ eq 31 ] ) iteratively , we have@xmath240   \left (   \left [   x\right ]   \right )   = \left [   g\right ] \left (   \left [   x+\left (   4k+6\right )   \alpha\right ]   \right )   \text { for any } x\in\left (   0,\pi\right ) . \label{eq 39}\ ] ] in other words , @xmath268   $ ] is invariant by the rotation of angle @xmath269 . if @xmath270 is irrational , @xmath216 must be constant . [ propositon characteristic function invariant]let @xmath271\right )   $ ] and suppose that @xmath272 is @xmath189-invariant . if @xmath18 in ( [ eq probabilities model ] ) and ( [ eq tau maps ] ) is irrational , then @xmath273 equals either @xmath52 or @xmath226 . suppose that @xmath193 . we want to prove that @xmath272 can not be @xmath189-invariant . since constants are trivially @xmath189-invariant , we can subtract @xmath273 from @xmath186 so that @xmath274 is still @xmath189-invariant and has expectation @xmath226 . let @xmath275 and define @xmath276 , which is clearly @xmath219-invariant because @xmath277 . since @xmath195 is @xmath151-invariant , so are @xmath278 ( proposition [ prop symmetry ] ) and @xmath279 . by proposition [ prop g invariant s2 and phi ] , @xmath280 is constant and equal to its expectation @xmath281   $ ] . but @xmath282 = \operatorname*{e}\nolimits_{\mu}\left [   g\right ]   = 0 $ ] because @xmath283 and @xmath284 are invariant by @xmath219 . therefore , @xmath285   = 0 $ ] and @xmath280 , which is constant , must be equal to @xmath226 . that is,@xmath286    since@xmath287{l}1-\mu\left (   c\right )   > 0\text { if } x\in c\\ -\mu\left (   c\right )   < 0\text { if } x\notin c , \end{array } \right.\ ] ] ( remember that we assumed @xmath193 ) , we conclude from ( [ eq 25 ] ) that @xmath288 and@xmath289{l}g\left (   x\right )   = 1/2\longleftrightarrow x\in c,\\ g(x)=-1/2\longleftrightarrow x\in c^{c}. \end{array}\ ] ] moreover , looking carefully at the proof of proposition [ prop g invariant s2 and phi ] , we have@xmath240   \left (   \left [   x\right ]   \right )   = \left\ { \begin{array } [ c]{l}\left [   g\right ]   \left (   \left [   x+4\alpha\right ]   \right )   \text { \ if } x\in(0,\pi-4\alpha)\\ -\left [   g\right ]   \left (   \left [   x+6\alpha\right ]   \right )   \text { \ if } x\in\lbrack\pi-4\alpha,\pi ) , \end{array } \right.\ ] ] where instead of @xmath245 we have now used @xmath290 . therefore , ( see ( [ eq 39]))@xmath240   \left (   \left [   x\right ]   \right )   = -\left [   g\right ] \left (   \left [   x+\left (   4k+6\right )   \alpha\right ]   \right )   \text { for any } x\in(0,\pi ) , \label{eq 40}\ ] ] where @xmath267 is such that @xmath291 . the minus sign in ( [ eq 40 ] ) tells us that the rotation @xmath292 of angle @xmath293 sends @xmath294 to @xmath295 and vice versa . let @xmath296 and @xmath297 the fourier expansions of @xmath186 and @xmath298 respectively . since @xmath299}$ ] , we have that@xmath300 on the other hand , we already argued that @xmath301 . imposing that the fourier coefficients of @xmath186 and @xmath298 are unique , we deduce that@xmath302 as @xmath303 is irrational , @xmath304 for any @xmath305 , which implies that @xmath306 for any @xmath305 so @xmath186 is constant a.s .. but this is clearly a contradiction because ( [ eq 25 ] ) implied @xmath307 . in this section , we are going to prove that @xmath151 is exact . this will imply knudsen s strong law for the random billiard introduced in section [ section model ] . let @xmath308 be a measurable transformation of a probability space @xmath309 . from the measurability of @xmath79 , we have the chain of @xmath97-algebras@xmath310 the map @xmath79 is called * _ exact _ * if the @xmath97-algebra @xmath311 only contains sets of measure either @xmath226 or @xmath52 . it is not difficult to prove that sets in @xmath312 are characterised by the property @xmath313 this characterisation , in turn , implies that a map @xmath79 is exact if and only if@xmath314 for any @xmath315 such that @xmath316 ( ( * ? ? ? * section 2.2 ) ) . equation ( [ eq 32 ] ) reads at the level of @xmath317 spaces as@xmath318 where @xmath319 and @xmath320 is the * _ koopman operator _ * defined on @xmath317 . if a map is exact , then@xmath321 which means that @xmath322   $ ] in @xmath317 as @xmath323 ( see ( * ? ? ? * section 2.5 ) ) . if @xmath9 is a probability absolutely continuous with respect to @xmath61 with radon - nikodym derivative @xmath324 , then@xmath325   \underset{n\rightarrow \infty}{\longrightarrow}\mu\left (   a\right )   \operatorname*{e}\nolimits_{\mu } \left [   f\right ]   = \mu(a),\ ] ] that is , the sequence of measures @xmath326 converges ( weakly ) to @xmath61 for any absolutely continuous measure @xmath179 ( ( * ? ? ? * proposition 4.4.1 ( b ) ) , ( * ? ? ? * section 2.6 ) ) . we want to show that the skew - type representation @xmath327\rightarrow\lbrack0,1)\times\lbrack0,\pi]$ ] of the model introduced in section [ section model ] is exact with respect to the liouville measure to conclude that , for any absolutely continuous measure @xmath179 on @xmath328\right )   $ ] and any @xmath329)$ ] , @xmath330}\mathbf{1}_{a}\left (   \pi_{2}\circ s^{n}\right )   d\nu=\int_{[0,\pi]}u_{s}^{n}\left (   \mathbf{1}_{\pi_{2}^{-1}(a)}\right )   d\lambda\otimes\nu\underset{n\rightarrow\infty}{\longrightarrow } \mu\left (   a\right )   , \label{eq 33}\ ] ] i.e. , that the strong knudsen s law holds . observe from ( [ eq 33 ] ) that we only need to consider the action of the koopman operator @xmath331 on square integrable functions that are the pull - back by @xmath332 of functions in @xmath333,\mathcal{b}\left (   [ 0,\pi]\right ) , \mu\right )   $ ] . we are going to denote @xmath334,\mathcal{b}\left (   [ 0,\pi]\right )   , \mu\right )   $ ] simply by @xmath335\right )   $ ] for the sake of a clearer notation . consequently , we do not need to check that @xmath151 is exact for the borel @xmath336-algebra of @xmath337 $ ] but it is enough to consider a smaller one , namely , the smallest @xmath97-algebra that makes both the functions in @xmath338\right )   \right )   $ ] and @xmath151 measurable . for any @xmath339 , let @xmath340\right )   \right\ }   \right )   , \ ] ] where @xmath168 are as in definition [ definition slices ] . for any @xmath341\right )   $ ] , the sets @xmath342 and @xmath343 will be called * generators * ( of @xmath344 ) . the sequence of @xmath97-algebras @xmath345 define a filtration,@xmath346 indeed , for any @xmath341\right )   $ ] and any @xmath347 , a generator @xmath348 can be written as@xmath349 which implies @xmath350 . we define @xmath351 as the limit of this filtration . @xmath352    @xmath353\rightarrow\lbrack0,1)\times\lbrack0,\pi]$ ] is @xmath351-measurable . for any @xmath354\right ) $ ] , @xmath355 is also @xmath351-measurable . the second statement of the proposition is obvious . to prove that @xmath151 is @xmath351-measurable it is enough to show that @xmath356 for the generators of @xmath351 . so let @xmath357 such that @xmath358 where @xmath341\right )   $ ] and @xmath347 . then , by definition of the sets @xmath168@xmath359 which clearly belongs to @xmath351 .    the following two lemmas aim at getting a better insight of the structure of the @xmath97-algebra @xmath351 . as an immediate consequence of them , it will be enough to show that @xmath151 is exact by looking at its action on the generators of @xmath351 . [ prop finite union generators]any set @xmath360 can be expressed as a finite union of disjoint generators . first to all , observe that , for a fixed @xmath339 , the sets @xmath168 are disjoint and form a partition of @xmath361 $ ] . the complement of a generator @xmath362 is@xmath363   \backslash u\right )   , \ ] ] where @xmath364 denotes the disjoint union , so it can be expressed as a finite union of generators . on the other hand , if we take a countable union of generators @xmath365 , @xmath366,@xmath367 where @xmath368 is in @xmath369 . but@xmath370 where the indices @xmath371 can only take finite number of possibilities . therefore , a countable union of generators reduces to a finite union . the union @xmath372 of a filtration @xmath373 is an algebra of sets . the @xmath97-algebra generated by an algebra of sets @xmath374 is characterised by containing all the sets in @xmath374 and the limit of all monotone sequence of sets . that is , @xmath375 coincides with the monotone class generated by @xmath374 ( ( * ? ? ? * ; * ? ? ? * ; * ? ? ? * section 1.3 , theorem 1 ) ) . this observation is the key to proving the following lemma :    [ lemma a contains generator]if @xmath376 has positive probability , then @xmath94 contains a generator of positive probability . let @xmath377 . if @xmath378 , then @xmath360 for some @xmath339 . by proposition [ prop finite union generators ] , @xmath94 can be expressed as a finite union of generators and at least one of them must have positive probability . if @xmath379 , then there exists a monotone sequence of sets @xmath380 such that @xmath381 . we are going to deal with the cases @xmath382 increasing or decreasing separately .    * if @xmath382 is increasing , then @xmath383 . since @xmath384 , at least one @xmath385 must have positive measure , @xmath386 . but @xmath387 for some @xmath388 , so it contains a generator of positive probability . * let @xmath389 be a decreasing sequence such that @xmath390 . suppose that @xmath391 ( otherwise there is nothing to prove ) . we have@xmath392 so that @xmath94 can be expressed as the complement of a countable union of disjoint sets . rename @xmath393 and @xmath394 , and express any @xmath395 as a disjoint union of generators ( lemma [ prop finite union generators]),@xmath396 in this decomposition , we only consider sets of strictly positive probability . then,@xmath397 where @xmath398 . we claim that there exists a generator @xmath399 of strictly positive probability that does not intersect @xmath400 , i.e. , @xmath401 . + let @xmath402 be a sequence appearing in the decomposition ( [ eq 38 ] ) . if @xmath403 , then@xmath404   \right\backslash \pi_{2}^{-1}\left ( \bigcup\nolimits_{m\in i_{i_{1}^{n} ... i_{r}^{n}}}u_{m}\right )   \right ) \bigcap j_{i_{1}^{n} ... i_{r}^{n}}\ ] ] is a generator of strictly positive measure that does not intersect @xmath400 . if @xmath405 for any finite sequence @xmath402 in ( [ eq 38 ] ) , then @xmath406 a.s . and since we assumed that @xmath407 , there must exist some set @xmath408 with positive probability that does not appear in the decomposition of @xmath400 . that is , @xmath409\right )   \bigcap j_{i_{1} ... i_{k}}\subseteq a$ ] .    for any @xmath357 , @xmath410 in other words , @xmath353\rightarrow\lbrack0,1)\times \lbrack0,\pi]$ ] is an exact endomorphism of @xmath411,\mathcal{f},\lambda\otimes\mu\right )   $ ] . let @xmath357 be such that @xmath412 . by lemma [ lemma a contains generator ] , @xmath94 contains a generator @xmath413 of positive measure for some @xmath341\right )   $ ] , @xmath414 . it is an immediate consequence of the definitions that , after @xmath13 iterations , the set @xmath415 has all its fibers of length @xmath52 , that is,@xmath416    let @xmath417 and let @xmath159 be the fiber at @xmath209 $ ] . looking carefully at ( [ eq tau maps ] ) and ( [ eq probabilities model ] ) , one can see that@xmath418 where @xmath419 denotes the identity on @xmath25 $ ] , and@xmath420 these remarks imply that , if @xmath421 , @xmath422 since all the fibers of @xmath423 have length @xmath52 , @xmath424 and , iteratively,@xmath425 define @xmath426 . obviously @xmath427 , the fibers of @xmath294 have all length @xmath52 , and @xmath428 . we want to show that @xmath429 . observe that @xmath430 because @xmath431 . from @xmath427 we have@xmath432 but @xmath294 and @xmath433 have the same measure ( @xmath189 is @xmath59-preserving ) , so @xmath186 is a @xmath189-invariant function a.s .. since @xmath434 a.s . ( the fibers of @xmath294 have all length @xmath52 a.s . ) , @xmath435 has full measure by proposition [ propositon characteristic function invariant ] . therefore , @xmath436 and @xmath437 this proves that @xmath189 is exact . in general , we have@xmath438 so that , using again that @xmath151 is @xmath59-preserving,@xmath439 which proves that @xmath440 is increasing and converges to @xmath52 , i.e. , @xmath151 is also exact .    to sum up , the skew - type representation @xmath441\rightarrow\lbrack0,1)\times\lbrack0,\pi]$ ] associated to the random map ( [ eq tau maps ] ) and ( [ eq probabilities model ] ) is exact which implies , by ( [ eq 33 ] ) , that @xmath442 as @xmath323 for any initial distribution @xmath9 on @xmath25 $ ] absolutely continuous with respect to the law @xmath443 , @xmath444\right )   $ ] . since @xmath445 on @xmath446 , @xmath61 and the lebesgue measure @xmath60 are absolutely continuous with respect to each other on @xmath447 . in other words , _ strong knudsen s law _ @xmath442 _ holds for any initial distribution absolutely continuous with respect to _ @xmath448__. _ _ in this section , we are going to show that numerical simulations are in accordance with theoretical results . with this aim , we take an arbitrary initial distribution on @xmath230   $ ] such as in the following picture and we make it evolve according to our random system ( [ eq tau maps ] ) and ( [ eq probabilities model ] ) .    initial.jpg + initial normalised distribution of particles . [ fig initial distribution ]    since we can only simulate a finite number particles in a computer , in this experiment we take a total amount of 30000 balls and divide @xmath449   $ ] in subintervals of the same length ( 45 in our experiment ) . that is , we approximate the initial distribution by a step function . in each subinterval , we put the proportion of balls according to the probability density function above . the initial angle associated to any of those balls is the middle point of the interval where they fall . the simulation then goes as follows . at the @xmath121-_th _ step , we take the @xmath450-_th _ particle with angle @xmath451 and a random number @xmath452 uniformly distributed between @xmath226 and @xmath52 , one for each particle . if @xmath453 , then @xmath454 , where @xmath455 are as in ( [ eq tau maps ] ) , and so on . after a long number of iterations , the distribution stabilises as follows :    final.jpg + final normalised distribution of particles . in red , @xmath456 . [ fig final distribution ]    as we can see , the outline of the final distribution tends to the graph of @xmath457 . the small inaccuracy is explained by the fact that only a finite ( i.e. , a discrete ) number of initial angles is considered . the smaller the subintervals in which @xmath230   $ ] is divided are , the better the final distribution approximates @xmath457 . we have repeated this experiment over several initial distributions obtaining always similar results , which experimentally confirms the validity of knudsen s law for our model . [ proof of lemma [ lemma slices go to zero]]to start with , will prove that @xmath458 from the very definition of @xmath151 and @xmath155 , we have@xmath459 on the other hand , let @xmath460 . then @xmath461 for some @xmath462 . let @xmath463 . it is obvious that @xmath464 as @xmath465 because @xmath466 is not empty . moreover , @xmath467 . therefore @xmath468 and@xmath469 identifying both fibers @xmath466 and @xmath470 as subsets of @xmath471 , the map@xmath289{rcl}s_{x}:i_{x}\bigcap j_{i_{1}\cdots i_{n}}\subset\lbrack0,1 ) & \longrightarrow & i_{\tau_{i_{n}}(x)}\bigcap j_{i_{1}\cdots i_{n-1}}\\ y & \longmapsto & \varphi_{i_{n}}(y,\tau_{i_{n}}(x))=\frac{1}{p_{i_{n}}(x)}\left (   y-{\textstyle\sum\nolimits_{j=1}^{i_{n}-1 } } p_{j}(x)\right )   , \end{array } \label{eq 37}\ ] ] is then a diffeomorphism . after having identified @xmath472 as a subset of @xmath471 , we can compute its lebesgue measure @xmath473 . consequently,@xmath474 applying iteratively ( [ eq 19 ] ) , we obtain@xmath475              n. chernov and r. markarian . _ introduction to the ergodic theory of chaotic billiards . _ monografas del instituto de matemtica y ciencias afines * 19*. instituto de matemtica y ciencias afines ; pontificia universidad catlica del per , lima , 2001 .
we prove knudsen s law for a gas of particles bouncing freely in a two dimensional pipeline with serrated walls consisting of irrational triangles . 
 dynamics are randomly perturbed and the corresponding random map studied under a skew - type deterministic representation which is shown to be ergodic and exact .    * keywords : * knudsen s law , random billiards , random maps , irrational polygons , ergodic billiards .