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

lim g d g f x

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

b ourho o d U M satisfying f U U and invariant attracting set f U we have

g d g f x lim

SBR

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

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

of subintervals fI g and the length of overlap of inverse images of the subintervals pro duced the

matrix approximation

f I I

j 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

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

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 for all n and n as n Dene partitions of M with max diam

ir n

n n

m f

i j

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 n

E p

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

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

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

r n

n n n

m mf

j j

P x

m m

n n

where is the unique partition set containing x M The transition function P has as its

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

with P x to b e thought of as the probability that the random image of x lies in the set M

The following denition is taken from

Denition The Markov chains governed by a family of transition functions P are called smal l

if for every continuous function g M R random perturbations of f M

lim sup g y P x dy g f x

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

Proof Dene

sup fjg x g y j kx y k g

xy M

sup fkf x f y k kx y k g and

xy M

sup fkf x f y k kx y k g

xy M

Z Z r n

n n

X y

g y dmy g f x P x dy g y g f x

m m

M M

r n Z

n n

g y dmy g f x

m m

Z r n

n n

jg y g f xj dmy

m m

r n

n n

n as n n

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

j j

n 2

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

Prop osition Let f g b e a sequence of invariant densities obtained from the sequence of tran

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

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

M M

The rst and third terms go to zero by weak convergence and the middle term go es to zero as fP g

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

d sup h f

M M

where h f is the measuretheoretic entropy of f and M M is the space of all f invariant Borel

u u

det D f xj probability measures For the weight function x log where E is the unstable

d is the SBR measure subspace at x the measure which maximises the quantity h f

We will use symb olic dynamics to compute an approximation to this maximal measure known as the

equilibrium state for the weight function For brevity of notation we drop the n dep endence of

the variables and consider a xed Markov partition P f g Dene

if Int f Int

i j

otherwise

A denes a subset fx A for all i Zg f r g of allowable

A n x x n

i i+1

sequences Given a biinnite sequence x x x there is a unique p oint in x M

with the prop erty that f x for all i Z We denote the mapping from a sequence x to

x A

its corresp onding p oint x M by We have the following commutative diagram M is

continuous and surjective

A A

y y

M M

If f is transitive resp mixing then is transitive resp mixing see Prop osition If we

remove the b oundary p oints of the partition along with all of their foward and inverse images

is bijective on the remaining residual set see Theorem That is is bijective on M n

S S

j Z i

For M R Holder there is a unique equilibrium state Theorem and the equilibrium

state for is the pro jection of the equilibrium state for R under that is

Thus we may nd the equilibrium state for the smo oth dynamical system M f with Holder weight

function by pro jecting down the equilibrium state from a suitably chosen subshift of nite typ e

the one derived from a Markov partition and weight function the one dened by In fact

what we are going to do is use the relative areas of intersection of the partition sets with their inverse

images to provide us with an estimate of the weight function M R which will approximate

and We then lift this to a R calculate the corresp onding equilibrium state

n n A

map this down to the equilibrium state for In the next section we see how to approximate

the weight function using only the Markov partition and its inverse images

Approximation of the weight function

For starters we deal with the much simpler case of expanding maps to give an idea of the direction

well take for the Anosov case

Lemma The Expanding Case

n n

x x

0 1

log

uniformly as n

Proof In the expanding case x log j det D f xj as all directions are expanding We want

to estimate the Jacobian of f over the set f Let mE denote the Riemannian volume of

j i

a Borel subset E of our Riemannian manifold M Clearly

j det D f xj sup inf j det D f xj

n n

1 mf

i j j i

j i

n n n n n n

for all i j n since P f g is Int or f Int as either f

i j i j

a Markov partition From we have

n n

i j

inf sup

n n

j det D f xj m j det D f xj

j i j

j i

and since f is C as the diameter of the partition sets go es to zero

n n

i j

n n

m j det D f xj j

j i

uniformly If we put

n n

x x

1 0

x log

u u

then M R is an approximation of with uniformly as n since

n n n

f is C and is uniformly continuous on its domain of denition Note that x is a piecewise

constant approximation of j det D f xj constant on each of the sets in P f P 2

Remark Theorem holds if

n n

i j

where is any probability measure equivalent to m m and m

In the pro ofs section we nd a similar estimate for twodimensional Anosov systems

Computing the approximate equilibrium state

For n large enough is a go o d approximation of Recall that the equilibrium state for is

n n

equal to the pro jection of the equilibrium state for under So lets nd the equilibrium state for

its a lo cally constant function dep ending only on the two symb ols to the right of centre and we

know how to do that

Theorem Let b e mixing so that A is irreducible and ap erio dic Supp ose that we

have a weight function R which dep ends only the on the two symb ols to the right of centre

that is x x x Dene

if A

e if A and x i x j

Let u v b e the left and right eigenvectors resp ectively corresp onding to the maximal nonnegative

eigenvalue so that uG u and Gv v Further dene

G v

ij j

The Markov measure generated by P is the equilibrium state for If pP p then fx

P P x a x a g a a a p

a a a a m m r a

r 2 r 1 0 1 0

Lemma The sto chastic matrix that generates the Markov measure that is the equilibrium state

for is given by

n n

i j

Proof By Theorem

n n

v G

j ij n

where

n n

i j n

e G

n n n

n n n n n n n

m and Q mf Now note that G L Q L where L m

i i j i

ij ij ij

Since Q is sto chastic and irreducible we know that the unique up to scalar multiples right

eigenvector with nonnegative eigenvalue is the vector with eigenvalue and so v

n n n n

m m m and Thus

r n

n n n

m mf

j i j n

n n

m m

j i

n n

i j

by and 2

This means that the equilibrium state is given by the Markov measure

n n n

a a p P P

n a a a a a

0 0 1 m2 m1

n n

for we merely where p is the invariant density of P To obtain the equilibrium state

pro ject down via that is

Remark The same argument applies to Anosov maps This is done in the pro ofs section

Remark In practice for E BM we will most likely take

i n

E p

for every i n has a uniform density on each of the partition sets and

i i

n n n

and distributing it uniformly What we are doing is taking the weight given to each partition set by

according to the natural volume measure over the entire partition set to obtain the measure

This extra error go es away as the size of the sets decreases since the distribution of mass within a

partition set do es not aect weak limits

Equation is a generalisation to higher dimensions of the approximation of the PerronFrob enius

op erator commonly used to estimate invariant densities of onedimensional systems with unique ab

solutely continuous invariant measures see Li for example What we have shown well were

nearly there is that this generalisation also provides us with an approximation of the absolutely con

tinous invariant measure of an ddimensional expanding map and the physical invariant measure

of an Anosov map later in two dimensions when the partition is Markov To show this we used

the relative volumes of inverse images of partition sets to estimate the Jacobian of the map in the

expanding directions The resulting piecewise constant approximation generated a sto chastic matrix

which in turn dened our estimate of the invariant measure This sto chastic matrix was nothing other

than the higher dimensional generalisation of Lis approximation So we can nd an approximate

equilibrium state of the Anosov dieomorphism simply by computing P and its left eigenvector

p To nish we see that we do indeed have an approximation in the sense that if is close to

in the uniform top ology C close then is close to in the weak top ology

(u)

Lemma weakly as uniformly

SBR n

Proof Let b e a sequence of approximations of obtained from shift systems so that k

n n

k The following argument is from the pro of of Lemma Cho ose some convergent

subsequence with f invariant We know that is Lipschitz with resp ect to the C

top ology for provided is nite see Theorem v or Theorem iv

lim by the Lipschitz prop erty

f f

h f lim

n n

u u

h f k k lim

n n

lim h f by weak convergence

h f

where the nal inequality follows from the upp er semicontinuity of the entropy map h f for

expansive f see Lemma or Theorem Now by we see that h f

(u)

In other words is the unique equilibrium state for and so we must have

(u)

SBR

So in our setup as we rene the partition the weight function converges to and we know

from p erturbation arguments that the approximations converge to an invariant measure What

Lemma tells us is that is in fact the equilibrium state for The measure that we obtain

as a weak limit of our approximations is the unique invariant measure that is exhibited by Leb esgue

almost all starting p oints in phase space All that is used to estimate this measure are the areas of

intersections of Markov partitions with their inverse images

Using Lemmata and Remark Theorem is proven in the expanding case

Remark The preceding theory may b e extended to twodimensional Axiom A attractors by

suitably dening a Markovian partition of M

Discussion

In this note we have shown that Lis formula may b e used to approximate the SBR measure in

twodimensional Anosov systems and expanding maps in ndimensions provided the partitions are

Markov The resulting estimate may b e viewed as either the xed p oint of an approximate nite

dimensional PerronFrob enius op erator or as an invariant measure of a small random p erturbation

of the original map Work on the former interpretation was done by Li for onedimensional piecewise

C expanding maps A Markov partition was not necessary using b ounded variation arguments Lis

work showed any nite partition would do Boyarsky and Lou extended Lis result to Jablonski

transformations a sp ecial class of piecewise expanding transformations in ndimensions Jablonski

transformations satisfy a typ e of separability condition and Boyarsky and Lou used Lis b ounded

variations arguments to obtain the result For more general transformations in higher dimensions the

problem b ecomes more dicult as the b ounded variation arguments are not able to b e used Baladi

and Young have shown for some mo del expanding maps and typ es of random p erturbations with

continuous densities that the PerronFrob enius op erators of the random systems approximate the

PerronFrob enius op erator of the unp erturb ed system suciently well to give convergence of densities

to the smo oth invariant measure as the noise go es to zero

Interpreting the approximation as an invariant measure of a randomly p erturb ed system results

have b een obtained by Kifer and Young Kifer has set out a numb er of technical hyp otheses

on the random p erturbations that guarantee convergence of the invariant measures of the p erturb ed

systems to the SBR measure of the unp erturb ed system Young has lo oked at random comp ositions

of p erturb ed maps and shown that as the p erturbations go to zero the invariant measures of the

random pro cess approach the SBR measure of the original map These two approaches are of little

computational use since the problem of computing the invariant measure of these random pro cesses

is just as dicult as the original problem of computing the SBR measure of the deterministic system

This is also the case for the work of where the invariant densities of a continuously p erturb ed

PerronFrob enius op erator are imp ossible to compute except in the simplest of cases For numerical

purp oses one really needs a nite dimensional approximation of the PerronFrob enius op erator or a

discontinuous random p erturbation

Our metho d allows a simple calculation of the invariant measure of the p erturb ed system as a left

eigenvector of a sto chastic matrix While this in principle provides a rigorous metho d of approximating

the SBR measure by computer the fact that Markov partitions need to b e constructed is a signicant

restriction for the practical application of the technique A matter for future work will b e to consider

the validity of the result if the Markov condition is relaxed Numerical results using nonMarkov

partitions are encouraging

Pro ofs

The Anosov case in two dimensions

We restrict ourselves to two dimensions as Markov partitions of Anosov dieomorphisms in dimension

three or ab ove are dicult to deal with see pages Each set in our partition has two

sides which are segments of stable foliations and two sides that are segments of unstable foliations see

App endix For any p oint on one of the stable b oundaries we have an unstable foliation running

through the p oint and across to an opp osing p oint on the other stable b oundary We have a similar

situation for the unstable b oundaries We intro duce some notation leaving out the n dep endence of

the partitions for convenience

b e the average unstable length of the partition set calculated by averaging the unstable Let

b e lengths integrating along one of the stable b oundaries of More precisely let

ileft

s u

R denote the length of a parametrisation of the left stable b oundary of and let

ileft

the unstable foliation passing through x Then

ileft

u u

i t

Here length means the length of the curve representing a segment of the unstable foliation from one

stable b oundary to the other Similarly dene

umax

Dene to b e the maximal length of an unstable foliation traversing the partition set

umin

from one stable b oundary to the other and put as the minimal such length Similarly dene

smax smin

and

i i

Throughout this pro of we will b e nding upp er and lower b ounds of lengths and areas Without

loss of generality we calculate everything in lo cal co ordinates as the upp er and lower b ounds will still

b e such for the average values when everything is transferred back to the manifold

i u s max

as At each p oint x we may calculate the sine of the angle b etween E and E Put sin

x x

min

the supremum of these values taken over all of and similarly dene sin

Finally dene

umax smax

max max

A sin

i i

umin smin

min min

A sin

i i

and denote by A the area of

i i

Clearly

umin umax

i i

smin smax

i i

min max

A A A

i i

The imp ortant thing is that as the diameter of the partition sets decreases to zero we have convergence

of the maximum and minimum values That is

umin umax

i i

smin smax

i i

min max

A A A

i i

as the partition is rened This is b ecause the b oundaries of our partition sets are C and the angle

function is Holder For a C dieomorphism of a C manifold with a closed hyp erb olic invariant set

u s

the transition functions x E and x E are Holder continuous see p

x x

Lemma With the previous notation

un n n

m f

x x x

0 1 0

log

uniformly as n

Proof We want to put some b ounds on the value of the Jacobian of f restricted to the unstable

subspace Dene M R by x jD f xjE j We denote by an average value of

the restriction of to This average is calculated by integrating x over using suitably

i i

normalised Riemannian volume We wish to estimate the value of this is the average value of

f j i

x restricted to f Upp er b ounds rst We have that

j i

umax

f j

f j i

umin

as the right hand side is greater than or equal to the average value of x restricted to any of the

unstable foliations passing through We claim that

umax

max max

umin umax

A A

1 1 1

f j i f j i f j i

i i

umin umin umin min min

A A

i i

j j j

The second inequality is obvious the rst is true if

max

f j i

umin umax

min f j i

umax smax

max

sin

1 1

f i

f j i f j i

umax umin

f j i umin smin

min

sin

i i i

smin smax

min max

sin sin

i f j i

f j i

and the nal inequality always holds Thus we have proved our claim For the lower b ound we have

that

umin

f j

umax

f j i

and we wish to show that

umin

min min

umin umax

A A

1 1 1

f j i f j i f j i

i i

umax umax umax

max max

A A

i i j j j

Again the rst inequality is obvious the second is true if

min

f j i

umax umin

max

f j i

umin smin

min

sin

1 1

f j i

f j i f j i

umax umin

umax smax

max

f j i

sin

i i i

smin smax

min max

sin sin

f j i

The nal inequality is always true so holds

By and and and we have

min max

umax umin

A A

1 1

f j i f j i

i i

umax

f j i

max

min umin

i j

and clearly also

max min

umax umin

A A

1 1 1

f j i f j i f j i

i i

umax

max u

min umin

A A

i j j

In fact as the diameters of the partition sets go to zero the left and right hand sides of the inequality

converge to the average centre value Thus as the diameters of the partition sets go to zero

f j i

By putting

A 1

f x x

1 0

x log

0 x

and

n n

S S

j u

f we see that uniformly f is C and is uniformly continuous on M n

i n

j Z i

the set on which is invertible as n Again is just a piecewise constant approximation of

constant on the sets P f P 2

Remarks There is a simple interpretation of the ab ove estimate for 1 We are essentially

f j i

u u

Since the sets in the Markov partition are almost paral by A 1 A approximating

f j i

lelograms if they are small enough and b ecause of the way inverse images of partition sets overlap

it is simple to see how this works draw a picture

Also note the similarity b etween the expressions for the estimates of for the expanding and

f j i

hyp erb olic cases Recall that for f expanding we had

f j i

Our estimate of 1 in the Anosov case reduces to the ab ove estimate for expanding maps as here

f j i

coincides with A all directions are expanding and so

Lemma The sto chastic matrix that generates the Markov measure for our equilibrium state of

n n

i j n

Proof Set

f j i n

G e

i j

n n n

and Q L Q L where L As in the expanding case note that G

ij ij i ij

n n n

Since Q is sto chastic and irreducible we know that the unique up to scalar multiples A A

f j i

right eigenvector with nonnegative eigenvalue is the vector with eigenvalue and so

un un

n un n

v and Thus

j f j i n

un un

i j i

f j i

by and 2

By Lemmata and Remark we have proven Theorem in the Anosov case

Acknowledgements

The author would like to thank Anthony Quas Mark Pollicott and LaiSang Young for helpful dis

cussions

Note Added in Pro of A similar convergence result has b een proven by Ding and Zhou for

higher dimensional expanding maps using piecewise linear and piecewise quadratic approximations on

general partitions The author is grateful to a referee for bringing this work to his attention

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