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 attractors 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 dynamical system 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