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Finite Approximation of Sinai-Bowen-Ruelle Measures For Finite Approximation of SinaiBowenRuelle Measures for Anosov Systems in Two Dimensions Gary Froyland Department of Mathematics The University of Western Australia Nedlands WA AUSTRALIA Abstract We describ e a computational metho d of approximating the physical or SinaiBowenRuelle measure of an Anosov system in two dimensions The approximation may either b e viewed as a xed p oint of an approximate PerronFrob enius op erator or as an invariant measure of a randomly p erturb ed system Keywords invariant measure PerronFrob enius op erator small random p erturbation Mathematics Subject Classication Primary F Secondary F D A Intro duction The existence and computation of imp ortant invariant measures of deterministic dynamical systems are still ma jor concerns in ergo dic theory In this note we do not address the problem of existence but provide a small step in the computation of imp ortant measures when they are known to exist In one dimension absolutely continuous invariant measures are considered to b e imp ortant from a computational p oint of view b ecause it is absolutely continuous measures that show up on computer simulations for most starting p oints If is an ergo dic absolutely continuous invariant measure for then by the Birkho Theorem we have any Borel measurable f Z n X i lim g d g f x n n i for almost all x and any given continuous g R Since Leb we may as well say that holds for Leb esgue almost all x is clearly the unique measure with this prop erty In this sense it seems natural to call absolutely continuous measures physical measures and ignoring roundo errors it is not so surprising that such measures commonly show up in computer simulations There is an analogue of absolutely continuous measures for wellb ehaved higher dimensional systems such as Anosov and Axiom A maps This measure is known as a SinaiBowenRuelle SBR measure denoted and has the prop erty that it is exhibited by Leb esgue almost all initial p oints SBR in the Anosov case and Leb esgue almost all p oints in a fundamental neighb ourho o d for Axiom A maps More formally we have the following theorem Theorem For f M a C Axiom A dieomorphism with fundamental neigh T i b ourho o d U M satisfying f U U and invariant attracting set f U we have i Z n X i g d g f x lim SBR n n M i for Leb esgue almost all x U and any given continuous g M R In the case of Anosov dieomor phisms U M Throughout the pap er f satises the conditions of this theorem SBR measures have clear physical signicance as they are exhibited by a large in the Leb esgue measure sense set of p oints Approx imations of the SBR measure are generally dened by exp erimentalists to b e the LHS of with g x evaluated up to some large n for a randomly chosen x Many statistical indicators such as x Lyapunov exp onents are estimated using this timeaverage A discussion of the shortcomings of time averaging and an application of our invariant measure approximation to the evaluation of Lyapunov exp onents may b e found in In TienYien Li resolved a conjecture of Ulam by showing that a unique absolutely continuous invariant measure of a onedimensional system could b e estimated using a nite approx imation of the PerronFrob enius op erator The unit interval was partitioned into a nite numb er m of subintervals fI g and the length of overlap of inverse images of the subintervals pro duced the i i matrix approximation f I I j i P ij I i to the innitedimensional PerronFrob enius op erator is onedimensional Leb esgue measure The invariant density of the Markov chain governed by P dened a piecewise constant approximation of the absolutely continuous invariant measure As the maximum length of the subintervals went to zero by rening the partition strong limit p oints of the invariant densities gave the unique absolutely continuous invariant measure Our goal was to extend this construction to dimensions to provide us with a nite approximation of the SBR measure of a given system when it exists The natural S r m with M and Int Int thing to do is to partition the space M into r sets f g i i j i i i for i j and use the obvious extension of namely m f j i P ij m i where m is normalised Riemannian volume on M By rening our partition so that the maximum diameter of the partition sets go es to zero we extract an invariant measure as a weak limit p oint of the invariant densities of the sequence of Markov chains governed by P Our result is that if at each stage of renement the partition is Markov then the limiting measure is the SBR measure More formally the main result is b e a C Anosov dieomorphism or expanding C Theorem Main Result Let f M map of a smo oth compact dimensional ddimensional Riemannian manifold M and denote by n n n g n b e a sequence of Markov m normalised Riemannian volume Let P f r n n n for all n and n as n Dene partitions of M with max diam ir n i n n m f i j n P ij n m i n n n n n Let p b e the normalised left eigenvector of P of eigenvalue that is p P p and dene r n n X mE n n i E p i n m i i n The sequence f g has a unique weak limit p oint namely the SBR measure of f An outline of our approach follows To show that the limiting measure is f invariant we rst cast the deterministic system f M as a randomly p erturb ed system governed by the nite state Markov chain n P We show that as the diameters of the partition sets decrease so do the random p erturbations and it is an easy matter to show f invariance of the limiting measure We then intro duce equilibrium states sp ecial invariant measures of f with resp ect to weight functions It is known that for a sp ecial weight function namely log lo cal expansion in unstable directions the equilibrium state of f is the SBR measure We use the relative areas of intersection of the partition sets with their inverse images to provide us with an approximation of the sp ecial weight function It then turns out that the appropriate matrix equation to solve to obtain an approximate equilibrium state is none other than Invariance of limit measures We show in this section that weak limit p oints of our construction are f invariant The rest of the pap er will b e devoted to showing that this invariant measure is the SBR measure and the uniqueness n follows The sto chastic matrix P may b e thought of as a transition matrix of a nite state Markov chain From this transition matrix we may dene a transition function P M BM R by n r n n n n X m mf x j j P x n n n m m x j j n n where is the unique partition set containing x M The transition function P has as its n x n n n after moves into may b e thought of as the probability that a p oint in invariant density P j i ij one iteration of f Insp ection of shows that this is not an unreasonable interpretation The construction of P allows us to discuss a concrete random p erturbation of the deterministic map f n with P x to b e thought of as the probability that the random image of x lies in the set M n The following denition is taken from Denition The Markov chains governed by a family of transition functions P are called smal l n if for every continuous function g M R random perturbations of f M Z lim sup g y P x dy g f x n n M xM The LHS of the dierence in represents the eect of random noise which is added after applying the function f to the p oint x M The integral averages the value of g over the allowed noise neigh b ourho o d The requirement for the Markov pro cess to b e a small random p erturbation of f is roughly that the noise neighb ourho o d applied after a transition from x coalesces ab out the deterministic image f x Lemma Our family of transition functions P is a small random p erturbation of f n Proof Dene sup fjg x g y j kx y k g g xy M sup fkf x f y k kx y k g and f xy M sup fkf x f y k kx y k g 1 f xy M Z Z r n n n n X y mf x j j g y dmy g f x P x dy g y g f x n n n m m M M x j j r n Z n n X mf x j g y dmy g f x n n m m x j j j Z r n n n X mf x j jg y g f xj dmy n n n m m x j j j r n n n X mf x j n n 1 g f f n m x j n as n n 1 g f f n n n n intersect and the greatest distance a p oint in such an then and f If mf x x j j n n n which is less than diameter of plus the diameter of f can b e from f x is less than x j j f n 2 1 f f We repro duce here the simple pro of of that weak limit p oints of invariant densities of small random p erturbations are invariant measures of the unp erturb ed map n Prop osition Let f g b e a sequence of invariant densities obtained from the sequence of tran n sition functions fP g and supp ose that weakly If the family fP g is a small random n n p erturbation of f then is f invariant Proof Let g C M Z Z Z Z n g f x dx g f x d x g f x dx g x dx M M M M Z Z g f x P x dy g y d x n n M M Z Z g x d x g x dx n M M The rst and third terms go to zero by weak convergence and the middle term go es to zero as fP g n is a small random p erturbation 2 So now we know that the measure that we extract from rep eated renements of our partition is f invariant The rest of the pap er is devoted to demonstrating it is the SBR measure Equilibrium states We consider the pressure of f with resp ect to a weight function M R dened by Z d sup h f f M M
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