RENYI ENTROPIES OF APERIODIC DYNAMICAL SYSTEMS

FLORIS TAKENS AND EVGENY VERBITSKIY

Abstract In this pap er we continue the study of Renyi entropies of measure

preserving transformations started in Wehave established there that for er

go dic transformations with p ositiveentropy the Renyi entropies of order q q R

are equal to either plus innityq or to the measuretheoretic Kolmogorov

Sinai entropyq  The answer for nonergo dic transformations is dierent the

Renyi entropies of order q are equal to the essential inmum of the measure

theoretic entropies of measures forming the decomp osition into ergo dic comp onents

Thus it is p ossible that the Renyi entropies of order q are strictly smaller than

the measuretheoretic entropy which is the average value of entropies of ergo dic

comp onents

This result is a bit surprising the Renyi entropies are metric invariants which

are sensitive to ergo dicity

The pro of of the describ ed result is based on the construction of partitions with

indep endent iterates However these partitions are obtained in dierentways de

p ending on q for q we use a version of the wellknown Sinai theorem on

Bernoulli factors for the nonergo dic transformations for qwe use the notion

of collections of indep endent sets in RokhlinHalmos towers and their prop erties

Introduction

Alfred Renyi intro duced the generalization of the Shannon information entropy

in the b eginning of sixties His approach w as based on an axiomatic denition of

information and consisted of including the standard entropy function

n

X

H p p p log p

n i i

i

into a oneparameter family of generalized entropy functions

n

X

q

log H p p p q

q n

i

q

i

For a xed probability distribution p p the standard entropy is recovered

n

from the generalized entropies as follows

H p p limH p p

n q n

q

Since then the Renyi entropies have b een successfully used in information theory

and statistics and more recently in thermo dynamics and quantum mechanics In

dynamical systems Hentschel and Pro caccia suggested a oneparameter family of

generalized dimensions based on Renyis approach These dimensions proved to be

Date Decemb er

extremely useful in problems of multifractal analysis and characterization of chaotic

see eg

Some attempts were made to intro duce the generalized entropies of dy

namical system using Renyis approach The idea was to pro duce a suciently

rich family of invariants of a which will take into account the

nonuniform b ehavior of invariant measures However the prop osed way of gen

eralizing the KolmogorovSinai entropy using H instead of H turned out to be

q

nonpro ductive In we have established the following fact

Theorem For an ergodic dynamical system X BT with positive measure

theoretic entropy hT the Renyi entropies are given by the fol lowing formula

q

hT q

hT q

Also in we suggested another family of generalized entropies which recovers

the results rep orted in the physics literature

The pro of of Theorem relies heavily on Sinais theorem on Bernoulli factors

for which the assumptions of ergo dicity and p ositiveness of the measuretheoretic

entropy are crucial

In this pap er we prove a result similar to Theorem but without the ab ove

assumptions We consider ap erio dic measurepreserving automorphisms ie trans

formations T of some Leb esgue space X Bsuch that

n

x T xx for some n

Surprisinglythe result for such systems is dierent from the ergo dic case

Theorem Supp ose T is an aperiodic measurepreserving automorphism of the

R

Lebesgue space X B Let dmt be the decomposition of into ergodic

t

components and let

n o

h T messinf fhT g sup c mft hT cg

t t

Then the Renyi entropies are as fol lows

q

R

hT q

hT hT dmt q

t

h T q

This result is a bit surprising b ecause of the following an entropybased invariant

can detect ergo dicity However we are not aware of any interesting example where

this observation could be useful The rst candidates which come to mind are the

nonergo dic Markov shifts ie the shifts for which the transition probability matrix

vided hT h T of course P is not irreducible It is p ossible in this case pro

to show the Renyi entropies of order q are strictly smaller than the measure

theoretic entropy and thus the system is not ergo dic However this pro of is much

more involved than the standard one and follows the same idea

The pap er is organized as follows in the next section we give a formal denition

of the Renyi entropies and establish the basic prop erties in section w e recall facts

ab out the decomp osition into ergo dic comp onents We discuss a nonergo dic version

of Sinais theorem on Bernoulli factors and use it for the computation of the Renyi

entropies of order q in section In section wedevelop a notion of indep endent

partitions in RokhlinHalmos towers and subsequently prove the statementforq

Finally in the last section we p ose some op en questions ab out the p ossible connection

between the Renyi entropiesand the recently intro duced entropy convergence rates

Renyi entropies of measure preserving transformations

The denition of the Renyi entropy of order q of a measurepreserving trans

formation go es along the lines of the standard denition of the measuretheoretic

KolmogorovSinai entropy and consists of steps the denition of the Renyi en

tropy of a nite partition Renyi entropy of an automorphism with resp ect to a par

tition and nally after taking the supremum over all nite partitions the Renyi

entropy of an automorphism which is a metric invariant

n

For any q R the entropy of order q of the partition f g is the number

i

i

n

X

q

log for q

i

q

i

H q

n

X

log for q

i i

i

q

with the standard convention for all q R and log

It is easy to check the following monotonicity prop erty

H q H q for any and q q

The Renyi entropy of order q with resp ect to a partition is dened as

n

hT q liminf H q

n

n

T

n

k n n

T with where T T is the partition into sets

i

k

k

i

k

Remark For q it is known see for example that the limit in exists

The pro of of this fact is based on a socalled subadditivity property of the Shannon

entropy H

H H H

for all partitions As it was shown by Renyi in the later is not the case

for any q This creates some additional problems in the treatment of the Renyi

entropies Nevertheless if and are indep endent partitions then

H q H q H q

for all q R W e will often exploit this fact

Finallywe dene the Renyi entropy of an automorphism T of order q as the number

hT q suphT q

where the supremum is taken over all nite partitions of X

Prop osition The Renyi entropies have the fol lowing properties

hT q for al l q

hT q hT q for q q

hT hT where hT is the measuretheoretic or Kolmogorov

Sinai entropy

n

hT q nhT q for any q R and every n

Prop erties follow easily from the denition of hT q and has been estab

lished in

Decomposition into ergodic components

For a measurable partition fC g Let X B be a Leb esgue space

t t

where can b e nite countable or uncountable we identify and the quotient or

factor X the space whose p oints are the elements of The set is a Leb esgue

space as well the set E is measurable if the set C is a measurable subset

tE t

of X and we obtain a measure m on by letting mE C A system of

tE t

measures f g t is called a canonical system of conditional measures belonging

t

to the partition fC g if

t t

is dened on some algebra B of subsets of C such that C B is a

t t t t t t

Leb esgue space

for any A B the set A C b elongs to B for malmost all t the function

t t

A C is a measurable function of t and

t t

Z

A A C dmt

t t

Supp ose T X X is a measurepreserving automorphism Then X B

can be decomp osed into ergo dic comp onents of T By this we mean the following

there exists a T invariant measurable partition fC g and a canonical system of

t

conditional measures f g such that for almost all t

t

C B Tj is ergo dic

t t t C

t

fC g is the decomp osition into ergo dic comp onents of X BT Supp ose

t

then

Z

hT hT dm

t

Consider the essential inmum and the essential supremum of measuretheoretic en

tropies of the measures from the decomp osition into ergo dic comp onents

t

h T messinffhT j t X g sup c mft hT cg

t t

h T messsup fhT j t X g inf c mft hT cg

t t

The quantity h T sometimes called the entropy rate has b een previously stud

ied in the literature in relation with the existence of nite generators gener

ating partitions for nonergo dic systems Awellknown theorem of Krieger states

that an ergo dic dynamical system with a nite measuretheoretic entropy hT ad

mits a nite generator with card exp hT It turns out that for non

ergo dic ap erio dic dynamical systems a similar result is true provided h T

a nite generator exists whose cardinalitydoes not exceed exp h T

Denote by fP P P g the set of all ordered partitions of X into m

m m

sets For any measure on X Bdene the partition pseudometric on as

m

follows

m

X

P Q P M Q P Q

k k m

k

If P Q then P and Q agree except on a set of measure and of course in

this case we say that P Q The space is a complete metric space

m

For an at most countable ordered partition P of X B the distribution vector

of P is given by

dP P P

Supp ose P and P are partitions into m sets of X B Y F resp ectively

then the distribution distance is

m

X

P P j j jdP dP j

k k

k

Supp ose we have a set f g of measures on X B For every t consider

t t

on The following fact will b e used later there exists a countable the metric

m

t

set which is dense in for almost every t

m m

t

The existence of such follows from the fundamental prop erties of the Leb esgue

m

spaces By denition a Leb esgue space X B admits a countable basis fB g

This in particular means that for any measurable set A B there exists a set C from

a minimal algebra generated by such that

C A and A n C

Denote by A the countable algebra generated by and let

n o

P P P P A

m m i

Hence is an at most countable collection of ordered partitions into m sets where

m

elements of these partitions are taken from A From we conclude that is

m

dense in Moreover for almost every t is dense in as well This

m m m

t

is a consequence of the following fact see also for almost every t the

countable collection of sets C is abasisinthe Leb esgue space C B

t t t t t

Renyi entropies of order q

In this section we are going to provethathT q h T forevery q We

start by showing that hT q h T

e have two invariant measures and Estimate from ab ove Supp ose that w

for an automorphism T We do not assume these measures to b e ergo dic Without

loss of generalitywe can assume that

hT hT

Consider now another invariant measure with The

measuretheoretic entropy of is given by see

hT hT hT

Note that due to hT hT Let be some nite partition For any

C one has

C C C

q q q

and therefore C C for q Hence for q

X X

q

q q

H q C C log log log

q q q

C C

q

log H q

q

From the ab ove one easily concludes that

n n

hT q liminf H q lim inf H q hT q

n n

n n

On the other hand due to the monotonicity of the Renyi entropies with resp ect to q

for qwe have

hT q hT q hT hT

Combining the two last inequalities we nally obtain that for any q

hT q hT

Thus we see that the Renyi entropy of a linear com bination of two measures do es

not exceed a minimum of the measuretheoretic entropies of these two measures It

is evident that the ab ove argument go es through in the case of a nite or countable

P P

where and decomp osition

k k k k

k k

Moreover the ab ove argument can b e equally easily generalized to the case of gen

erally uncountable decomp osition of an invariant measure into ergo dic comp onents

f g This is done in the following lemma

t

Lemma For a measure preserving system X BT one has

hT q h T

for every q

Proof Consider an ergo dic decomp osition of X BT as in section By the

denition of h T for every the set E ft hT h T g has

t

a p ositive mmeasure Supp ose there exists such that for any one

has mE

If such do es not exist then

hT h T for m aa t

t

As a result we immediately conclude that hT h T and using the fact that

hT hT q for any qwe obtain our claim

Assume such exists and chose any Since mE we can

dene

Z Z

dmt dmt

t t

c

mE mE

E E

It is clear that and are invariant probability measures Moreover hT

h T Using the ab ove argument for two measures and we conclude that

for any q

hT q h T

Since can be chosen arbitrary small we obtain the claim

Bernoulli factors of nonergo dic systems Let us recall a denition of a

Bernoul li automorphism

Denition An automorphism T of a Lebesgue space X B is cal led Ber

noul li if it is measuretheoretical ly isomorphic to a Bernoul li shift

If T is a Bernoulli automorphism then there exists a partition P of X such that

P is generating

n

fT P g is a sequence of indep endent partitions

nZ

Such partition P is called an indep endent generator for T A well known theorem

by Sinai states that for every ergo dic automorphism T with entropy hT and

every p ositive number h such that h hT there exists a Bernoulli factor with

entropy h A nonergo dic version of the Sinai theorem srt app eared in

Theorem Suppose T is an automorphism of a Lebesgue space X B Let

T be a Bernoul li automorphism of Y F with a nite independent generator P

cardP k Let fC g bea decomposition of X BT into ergodic components

t t

and m be a corresponding measure on the factor XfC g Assume that h T

t

hT Then there exists a partition Q cardQk such that

i

i fT Qg is a sequence of independent partitions

i i dQ dP

In fact using the techniques of one can establish a nonergo dic version of

the Ornstein fundamental lemma as well The strategy of generalizing

ergo dic results to the nonergo dic case consists of the following Supp ose that

fC g is the decomp osion of into ergo dic comp onents and that for almost every t

t t

there exists a partition P of C into m elements which satises some required prop erty

t t

We recall that there exists a countable family of partitions which is dense in

m

t

the set all partitions into m elements for almost all t see section Using this family

one can construct a universal partition P such that P C P for almost all t

m t t

Estimate from below Now we can prove a lower estimate hT q

h T for all q Before we pro ceed with this estimate we would liketomake a

few remarks Firstly we compute the Renyi entropy of order q for a Bernoulli shift

Z

Let f kg and be a left shift Let be a Bernoulli measure on

generated by a probabilityvector p p p ie

k

f a a g p p

i m m n n a a

m n

for all m n and a a f kg Denote by the partition into the

m n

following cylinders

f ng f g

n i k

It is not very dicult to see that for q

k

X

q

hq log p

i

q

i

In particular if p p k then hq log k In this case since

k

h log k we immediately conclude that hq logk for q

We would also need the following statement

Lemma Suppose T Y Y is an automorphism preserving a measure Then

for any prime p one has

p

ph T h T

p

Proof Assume rst that T is ergo dic If T Y Y is ergo dic then there is

p p

nothing to prove since in this case h T hT phT

p

If T is not ergo dic then p there exist disjoint sets A A such that

p

p

p

is ergo dic on A with resp ect to Y A mo d T A A and T

i i i mo d p

i

p

jA Therefore h T phT

p

Hence we conclude that if T is ergo dic then h T phT forany prime

p p

R

Assume now that T is not ergo dic and let dm b e the decomp osition of

t

into ergo dic comp onents Applying the argumentabove to eachT we conclude

t

p p

that h T phT and therefore h T ph T

t t

Now let us pro ceed with the pro of of the inequality hT q h T for all

q Assume the opp osite ie there exists q such that hT q h T

Take a suciently large prime p such that there exists an integer k satisfying

p

phT q log kph T h T

Consider a Bernoulli shift dened as ab ove with p p k Then by

k

p

Theorem there exists a Bernoulli factor Q for T with Q Q k

k

p

Thus hT qQ log k but this is in contradiction with since

p p

phT q hT q sup hT qR

R

Hence hT q h T for q and together with this gives the

equality hT q h T for all q

R enyi entropies of order q

In this section we will prove the remaining part of Theorem The techniques

which we are going to use will be dierent from the previous section The reason is

that we do not want to assume h T or even hT In the case when

h T we can with the help of the nonergo dic version of Sinais theorem on

Bernoulli factors obtained in the previous section pro ceed as in

Our main goal is to construct partitions with arbitrarily large Renyi entropy of

order q q for every C we have to nd a partition such that

k

hT q liminf H q C

k

k

Since the Renyi entropies are monotonic in q we can restrict ourselves to q

First of all let us make an observation whichwillallow us to simplify the estimate

of the Renyi entropy of a partition from b elow

Denition The Renyi entropy of order q q of a nite partition f g

i

restricted to a set F F is the number

X

q

log F H q jF

i

q

i

It is easy to see that any each q and for any set F F one has

H q H q jF

In the next subsection we will showhow this can b e used when F is a base of some

RokhlinHalmos tower and is some sp ecial partition

RokhlinHalmos towers and indep endent collections of sets We have

assumed that T is an ap erio dic automorphism It is well known that for such au

tomorphisms one can construct RokhlinHalmos towers of any height and measure

arbitrarily close to

m

Let M X then fMTM T M g is called a RokhlinHalmos tower if

i j

T M T M for i j m

m

k

We will use the same letter for T M The height of the tower is said to be

k

m and mM is its measure

Wenowgive a denition of an indep endent collection of sets relative to a Rokhlin

Halmos tower We will asso ciate to such collections certain partitions which will b e

analogous to Bernoulli partitions

m

Denition Let fMTM T M g be a RokhlinHalmos tower We

say that acol lection I fI I g of subsets of is independent relative to if

N

I I for i j

i j

N

I g denote by the partition of X into the sets fI I I X n

j I N N

j

then

m m

k k k k k k

T I T M T I T M T T M

N I

k k

is a col lection of independent partitions of M

For convenience we wil l always assume that

M

k

for j N and k m I T M

j

N

Collections of indep endent sets exist in every tower This follows from the following

two observations Firstly since T is assumed to be ap erio dic the invariant measure

has no atoms Secondly for any Leb esgue space X B where has no atoms

for every measurable set A and each A one can nd a set B A with

B

It follows immediately from the denition that if I is a collection of indep endent

is the corresp onding partition then sets in and

I

o n

k

k

k

M j j f Ng M I T I T I

k j j j



k I                   m-1          T M

 

 

 

           TM

       M

   I    I

1 2 

m

Figure Indep endent sets I I in MTM T M

k k

is a partition of M into N sets of equal measure M N for every k m

m

Using we easily obtain an estimate on the Renyi entropy of

I

q

M

m m

m

q jM q H H log N

I I

m

q N

q

m log N log M

q

mq q

If the measure of the base of the tower M is not to o small say M N

m

then H q m log N

I

In the next subsection we estimate the Renyi entropy of a partition which is close

to some partition where I is a collection of indep endent sets

I

Approximation lemma Let I I I b e a collection of indep endent

N

m

fI I g be the sets in the RokhlinHalmos M T M and let

I N

corresp onding partition Supp ose that another partition E E is such

N

that the sets

E E

j

j

m

are close to the corresp onding I s for j N Since the partition has

j

I

a large Renyi entropy sub ject to a relation b etween N and of course then the

m

partition has a large Renyi entropy as well This can be rigorously formulated

in the following way

Lemma Let I fI I g be a col lection of independent sets in

N

N

m

M T M with m and let fI I I X n I g be the

I N N j

j

corresponding partition Suppose E E E is another partition of X

N N

such that

N

X

E I

j

j

N

j

Then for every q one has

q q q

m

H q log N log log logm

m q m q m

Proof This lemma is a generalization of lemmas and from and its pro of

follows quite closely the pro ofs of the corresp onding results in Nevertheless due

to the necessary mo dications and for the sake of completeness we provide a pro of

here

We shall use the following notation let f Ng and

m m

for r r r I T I T I r

r r r m

m



m m

s E T E T E for s s s

s s s m

m



m

Weknowthatr M M N Since is close to in weexpectthe

I

m m

sets M to have approximately the same measure as the sets of M Let

m m

us mak e it precise We say that s s is a bad or a fat element of

if

m m

s M N M

and is go o d or thin otherwise We collect the indexes of all bad elements into

the set

m

S fs s is bad g

m

We will now show that bad elements of cover less than a half of M in measure

ie

M s M

sS

We intro duce the following notation

M s rs r M

m

k

s r T M s r

k

s s r

m

r

It is easy to see that

M s r M s t for r t

i j

T M s r T M s r for i j

s r s t for r t

m

Let s then

X

N N

E M I s E M I s r

j j

j j

j j

m

r

m

X X

N k

E M I T M s r

j

j

j

m

r

k

Consider the sets participating in the last sum separately We claim that

k

T M s r if s r

k k

k N

T M s r M I E

j

j

j

if s r

k k

For this it is sucient to show that

E I if s r

r k k

s

N

k

k

E M I E I

j r

j s

j

k

k

if s r

k k

The pro of is straightforward let j N and k m then

E M I E I E n I E I I n E E I

j r j r j r

j s j s j s

k k k

k k k

A B

r Then for j s r we have Supp ose rst that s

k k k k

I n E E A B E n I I

j j j j

j j

since I for j N

j

For j s r we have

k k

A B E n I E I n E I

j j r

j s j

k

k

since E E I I

j s j r

k k

Now consider the case s r If j s and j r then

k k k k

A B E E I I

j s j r

k k

If j s and j r then A E E but n I

k k s j

j

k

B I n E E I I E

j s r r

j s

k k k

k

since E E

j s

k

Similarly for j s and j r we conclude that B but A E I

k k r

s

k

k

Hence we proved and therefore

Using and the fact that T is measurepreserving we can simplify

X

N

E M I s d s rM s r

j H

j

j

m

r

where d s r fk s r g is the Hamming distance between s and r We

H k k

rewrite in the following form

m

X X

N

M s r i E M I s

j

j j

i

r d sri H

m i i i

Given s the number of rs such that d s r i is C N where C is

H

m m

the binomial co ecient Let us intro duce the following notation

X

M

i i

N C x s M s r y

i i

m

m

N

r d sri

H

P

m

x s Note that s M

i

i

m

Since M s r r M and r M M N for every i one has

X X

M M

i i

x s M s r C N y

i i

m

m m

N N

rd sri rd sri

H H

Furthermore for every s there exists k f mg such that

s

k m k

s s

X X X

y x s y

i i i

i i i

P P

m k

s

From and we conclude that x s y for all l and as a

i i

il il

result

k m

s

X X

ix s iy

i i

i i

m

We will show now that if s S then k Indeed if s S then by

s

denition of S

m m

s M N M

and from we have

P

k

s

m

X

x s

s M

i

i i m m

i

C N N

m

m

N M M

i

However Lemma see App endix below states that

k

X

i i m m

C N N

m

m

N

i

for all k m Hence k m s

Now for all s S we have

m

X

N

ix s by E M I s

i j

j

j

i

k k

s s

X X

M

i i

N by iC iy

i

m

m

N

i i

k

s

X

M m

i i

C N by Lemma

m

m

N N

i

k

s

X

m

y

i

N

i

m

X

m

x s by

i

N

i

Hence

X

N N

E M I E M I s

j j

j j

j j

sS

m

X X X X

m m

x s M s r

i

N N

m

i

sS sS r

X

m

s M

N

sS

Therefore using our assumption one has

X

N

N

s M E M I

j

j

j

m

sS

N

M

N m

ie bad elements s cover not more than a half of M

m

Now we can estimate the Renyi entropy of the partition

X

m m q

H q H q jM log s M

q

m

s

X

q

A

log s M

q

m

s nS

X

s M

A

log

q

m m

q

N M

m

s nS

M

log

q

m m

q

N M

m q

log N m log log M log

q q

q

q

m log N log log log m

q m q

This nishes the proofofLemma

Partitions with large Renyi entropy Consider q and take N N

q

N For the convenience of notation we put Take R N such that

q

R

N N N

We cho ose a sequence of RokhlinHalmos towers f g

k

m

k

M T M

k k k

Rk

of height m Rk and measure N For each k let

k k

I I k I k

k N

be a collection of indep endent sets in We dene a sequence of collections of

k

pairwise disjoint sets E E k E k as follows for j N let

k N

E

j

E k E k n I k for k

j j k j

For any j f N g the sequence of characteristic functions f g is

E k

j k

B Indeed we obviously have aCauchy sequence in L X

E k M E k

j j k

and hence for k k K we have

X

E k M E k as K

j j k

k K

From this we conclude that there exists E B such that

j

for k

E k E

j j

It follows from the construction that E E for i j Since we can neglect

j i

sets of measure zero wemay assume that E E and hence wehave a collection

i j

E E E of pairwise disjoint subsets of X

N

Furthermore for every j N and any Lk one has

L

X

k k k k k

E M I k E M E L E l M E l

j

j j j j j

l k

k

E k M I k

j

j

k

Moreover since E k I k E M E L as L and E l M

j j j j

j

E l we conclude that

j l

R

X

N

k

k

E M I k

j l k

j

R

N N N

l k

and hence

N

X

k

k

E M I k

j

j

N

j

N

Now let fE E E gwhereE X n E and applying Lemma

N N N j

j

we conclude that

q

m

k

log N H q log o

k

m q m

k k

log N o

where o as k

For any n N there exists k N such that

m Rk nRk m

k k

n m

k

Since H q H q we have

m

k

n m

k

H q H q log N o

n m m

k k

k

This proves that

n

h T q lim inf H log N q

n

n

Everywhere ab ove we have assumed that q However since

hT q hT

for all q and every partition we have obtained partitions with large Renyi

entropies of all orders q q as well Finally since N is an arbitrary integer we

proved the remaining part of Theorem

Final remarks

a Another version of Renyi entropies can b e dened using lim sup instead of lim inf

in In principle due to the lack of subadditivityofH there might exist a nite

q

partition such that

n

H q hT q lim sup

q n

n

n

lim inf H q hT q

n

q n

for some q R q However using the results of Theorems and one can

easily show that

h T q sup hT q sup hT q hT q

nite nite

for all q R

Since hT q for q in the ergo dic and ap erio dic cases the claim is

obviously true for q

To complete the pro of we have to show that for q

sup hT q hT

nite

in the ergo dic case and

sup hT q h T

nite

in the ap erio dic case The rst inequality follows immediately from the mono

tonicity prop erties Prop osition and the fact that for q standard entropy

the limit in exists

The second inequality is proved exactly in the same manner as an inequality

hT q h T in section

b Formally sp eaking the pair of metric invariants hT hT q q can

detect ergo dicity if hT hT q then T cannot b e ergo dic However

we were not able to nd any relevant examples where this could b e useful

In our opinion an example of a measurepreserving system X BT where the

nonergo dicity can be decided from the p ositiveness of hT hT q would be

interesting

c The dierence b etween ergo dicity and nonergo dicityislessinteresting than the

dierence between ergo dicity and weak As it is well known weak mixing

of T is equivalent to the ergo dicity of any direct pro ducts of T with an ergo dic

T is ergo dic but not weakly mixing Then there exists automorphism S Supp ose

an ergo dic measurepreserving dynamical system Y FS suchthatX Y G

T S is not ergo dic Unfortunately the Renyi entropies are not able to detect

nonergo dicityof such systems for q one has

hT S q hT qhS q hT hS hT S

where the rst and the third equalities are standard facts for entropylike c haracter

istics and the second equality follows from Theorem

d Entropy convergence rates were intro duced in

Let X B be a Leb esgue space and T be a measurepreserving automorphism

Supp ose that X BThas zero entropy Hence for any nite partition one has

n

hT lim inf H

n

n

Let c and a n is a sequence of p ositive numb ers such that a

n n

Denote by the set of all nontrivial partitions X Binto two sets

The automorphism T is said to be

of typ e LI c for a if for every

n

n

lim inf H c

n

a

n

of typ e LS cfora if for every

n

n

lim sup H c

a

n

n

Similarly one denes typ es LI c LS c LI c etc Clearly the typ e of a

measurepreserving transformation is a measuretheoretic invariant

It was shown in that there are no ap erio dic transformations of typ e LI

k

for a where a on n Every totally ergo dic transformation ie T is

n n

ergo dic for every k is of typ e LS forg log n where g R

is p ositive monotone increasing and

Z

g x

dx

x

Also in FBlume constructed a class of weakly mixing systems which can be

distinguished by these invariants

It would interesting to know if the corresp onding notions for the Renyi entropies

b oth in the case of q and q can pro duce useful convergence rates which are

dierent from the case of standard entropy

Appendix Auxiliary results

Throughout this section we assume that N N N x denotes an integer part

k

of x and C will denote the binomial co ecient

m

m

k

C

m

k m k

For k m let

k k

X X

i i i i

a k N C N b k N iC N

m m

m m

i i

The following result is a straightforward generalization of Lemma from

m

Lemma Let m be an integer m and put k Then

for k k one has

a k N

m

m m

N

m

N

for k k m one has

N a k N

m

b k N m

m

Acknowledgments

We are grateful to F Blume and EGF Thomas for valuable disscusions The

second author acklowledges the supp ort of the Dutch Organization for Scientic Re

search NWO grant

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