Effective symbolic dynamics, random points, statistical behavior, complexity and entropy

EFFECTIVE SYMBOLIC D YNAMICS, RANDOM POINTS, ST A TISTICAL BEHA VIOR, COMPLEXITY AND ENTROPY STEF ANO GALA TOLO, MA TH IEU HO YR UP, AND CRIST ´ OBAL ROJAS Abstract. W e consider the dynamical b eha vior of Martin- L¨ of random points in dynamical systems o v er m etric s paces with a computable dynamics and a computable i n v ari an t measure. W e use computable partitions to define a sort of effectiv e symbolic mo del for the dynamics. Through this construction we prov e that s uc h points ha v e typical statistical behavior (the b ehavior which is t ypical i n the Bi rkhoff ergodic theorem) and are r ecurren t. W e introduce and compare some notions of complexit y for orbits in dynamical systems and prov e: (i) that the complexit y of the or bits of random p oints equals the K ol mogoro v- Sina ¨ ı en trop y of the system, (ii) that the supremum of the complexity of orbits equals the topological entrop y . Contents 1. Int ro duction 2 2. Preliminarie s 4 2.1. Partial recurs ive functions 4 2.2. Algorithms on finite ob jects 5 2.3. Computabilit y ov er the reals 5 2.4. Computable Metric Spaces 5 2.5. Computable Pro bability Spa ces (CPS) 7 2.6. Algorithmic ra ndomness 9 2.7. Kolmogor ov complexity 9 3. Effective symbolic dynamics and statistics o f random p oints 10 3.1. Sym bolic dynamics of random p o ints 10 3.2. Some statistical pro p erties of random p oints 13 4. Measure-theo r etic entropies 15 4.1. Ent ropy with Shannon information 15 4.2. Ent ropy with Kolmo g orov information 16 4.3. Equiv alenc e b etw een lo cal entropies 17 4.4. Orbit complexity v s entrop y 18 4.5. Orbit complexity 19 5. Equiv a lence of the tw o notio ns of orbit co mplex it y for random p oints 20 6. T op o logical entropies 22 6.1. Ent ropy as a capa c ity 22 6.2. Ent ropy as a dimension 23 6.3. Orbit complexity v s entrop y 23 References 26 P AR TL Y SUPPOR TED BY ANR GRAN T 05 2452 260 OX 1 2 STEF ANO GALA T OLO, MA THIEU HOYR UP, AND CRIST ´ OBAL ROJAS 1. Introduction The randomness of a particular outcome is alwa ys rela tive to some statistical test. The notion of alg orithmic rando mnes s, defined b y Ma rtin-L¨ of in 1966, is an attempt to have an “a bsolute” notion of r a ndomness. This abs oluteness is actually relative to all “effective” statistica l tests, and lies on the hypothesis that this clas s of tests is sufficiently wide. Martin-L¨ of ’s original definition was given for infinite s ymbolic sequences. With this notion each single ra ndo m seq ue nc e b ehav es a s a ge ne r ic s equence of the prob- ability space fo r ea ch effective statistical test. In this wa y many probabilistic theo- rems having almo st everywhere s ta tement s ca n b e translated to statements which hold for e ach rando m s e quence. As an exa mple w e cite the fact that in each infinite string of 0’s and 1’s which is r andom for the uniform mea s ure, all the dig its app ea r with the s ame limit frequency . This is a particular case, rela ted to the stro ng law of larg e num bers (or B irkhoff er go dic theor em). A g eneral statement of this kind was given by V’yugin (Birkhoff erg o dic theorem for individual ra ndom s equences, see [ V’y97 ] and lemma 3 .2.1 be low). Recently the notion of Martin-L¨ of randomness was g eneralized to computable metric spa ces endowed with a measur e ([ G´ ac05 , HR07 ]). Computable metric s pa ces are separ able metric spaces where the distance can b e in so me sense effectively com- puted (see section 2.4 ). In those spaces, it is also p ossible to define “computable” functions, whic h a re functions who se b ehavior is in some sense g iven by an a lg o- rithm, and “ computable” meas ures (there is a n alg orithm to ca lc ulate the mea s ure of nice sets). The space of infinite symbolic se q uences, the real line or euclidea n spaces, a re examples of metric spa ces which b ecome computable in a very natural wa y . A pa rticularly interesting class of general sta tionary sto chastic pr o cesses is c o n- stituted by thos e generated by a measure- preserving map on a metric space, these are the ob jects studied by er go dic theory . In this pap er we co ns ider s ystems of the t ype ( X , T , µ ), where X is a computable metric space, µ a computable probability measure and T a co mputable endomorphis m. The ab ov e co ns idered symbolic shifts on s paces of infinite sequences which preser ve a computable mea sure ar e systems of this kind. In the classical er go dic theor y , a p ow erful techn ique (symbolic dynamics) a llows to asso cia te to a gene r al system as ab ov e ( X , T , µ ) a shift on a space of infinite strings having similar statistica l prop erties. In section 3 we use the alg orithmic features of computable metric space s to define computable measur able partitions and construct effective symbolic mo dels for the dynamics . In this mo dels r andom p oints ar e asso ciate d t o r ando m infinite st rings . This to ol allows to generalize theorems which are prov ed in the symbolic setting to the mor e g eneral setting of maps and metr ic space s. F or e xample the above cited V’yugin theorem becomes a Birk hoff theorem for r andom p oints. On this line, we a lso pr ov e a Poincar´ e recurrence theorem for r andom p oints. Those sta temen ts (see thm. 3.2.1 and pro p. 3.2.1 ) can b e summariz e d a s Theorem. L et ( X, µ ) b e a c omputable pr ob ability sp ac e. If x is µ -r andom, then it is r e curr ent with r esp e ct t o every me asu re pr eservi ng endomorphi sm T on ( X , µ ) . THE DYNAMICS O F ALGORITHMICALL Y RANDOM P OINTS 3 Mor e over, e ach µ -r andom p oint x is typic al for every er go dic endomorphism T , i.e. lim n →∞ 1 n n − 1 X j =0 f ( T j x ) = Z f dµ (1) for every c ont inuous b ounde d r e al-value d f . In the remaining par t of the pap er these too ls a re also used to prov e rela tio ns betw een v ar ious definitions of o r bit complexity and entrop y of the systems. In [ Bru83 ], Brudno defined a notion of algo rithmic complexity K ( x, T ) for the orbits o f a dynamical system on a co mpact spa c e. It is a measure of the infor mation rate which is necessary to describ e the b ehavior of the orbit of x . In this point- wise definition the infor ma tion is mea sured b y the Kolmo gorov information conten t. Later, White ([ Whi93 ]) also intro duced a slig ht ly differen t version K ( x, T ). Brudno then prov ed the following r esults, later improved by White: Theorem (Brudno , White) . L et X b e a c omp act top olo gic al sp ac e and T : X → X a c ont inuous map. (1) F or any er go dic pr ob ability me asur e µ the e quality K ( x, T ) = K ( x, T ) = h µ ( T ) holds for µ - almost al l x ∈ X , (2) F or al l x ∈ X , K ( x, T ) ≤ h ( T ) . where h µ ( T ) is the Kolmogo rov-Sina ¨ ı entrop y of ( X, T ) with r esp ect to µ and h ( T ) is the topo logical en tropy of ( X, T ). This result seems miraculous as no computability as sumption is requir ed o n the spa ce or on the transfor mation T . Actually , this miracle lies in the compactness of the s pa ce, which makes it finite when observ ations are made with finite precis ion (open covers of the space can be reduced to fin ite ope n cov ers). Indeed, when the space is not compa ct, it is po ssible to construct systems for which the algo rithmic complexit y o f orbits is correla ted in no wa y to their dynamica l co mplexity . In [ Gal00 ], Brudno’s definition was generalized to non- c ompact computable metric spaces. This definition co incides with Brudno’s one in the compact ca se a nd will b e given in section 4.5 . The above definitions of orbit c o mplexities follow a topo logical approach. W e show that the measur e-theoretic setting als o provides a natural notion of orbit com- plexity K µ ( x, T ) defined by computable partitions. This kind of o rbit c o mplexity will b e defined almost everywhere and in particular a t ea ch µ -ra ndom p oint. F o r this notion the first result in Brudno and White’s theorem co mes eas ily . W e go further in showing: Theorem ( 4.4 .2 ) . Le t T b e an er go dic endomorphism of the c omputable pr ob abili ty sp ac e ( X , µ ) , K µ ( x, T ) = h µ ( T ) for al l µ -r andom p oint x . W e then prov e tha t the tw o notio ns of orbit complexity coincide on Martin-L¨ of random po ints: Theorem ( 5.0.1 ) . L et T b e an er go dic endomorphism of the c omputable pr ob abili ty sp ac e ( X , µ ) , wher e X is c omp act, K µ ( x, T ) = K ( x, T ) for al l µ -r andom p oint x . In the top olo gical context, we then consider K ( x, T ) and stre ngthen the second part of Brudno’s theor em, showing: 4 STEF ANO GALA T OLO, MA THIEU HOYR UP, AND CRIST ´ OBAL ROJAS Theorem ( 6.3.1 ) . L et T b e a c omputable map on a c omp act c omputable metric sp ac e X , sup x ∈ X K ( x, T ) = sup x ∈ X K ( x, T ) = h ( T ) Remark that this was already implied by B rudno’s theore m, us ing the v a riational principle: h ( T ) = sup { h µ ( T ) : µ is T -in v ar ia nt } . Nevertheless, our pro o f uses purely top o lo gical and algorithmic arg uments and no meas ure. In particular, it do es not use the v ar iational pr inciple, a nd c a n b e thought as an alterna tive pro of of it. Many o f these statements requir e that the dy namics and the inv ariant measure are computable. The first ass umption can b e ea sily check ed on concr ete systems if the dyna mics is given by a map which is effectively defined. The second is more delica te: it is well known that given a map on a metr ic space, there ca n b e a co ntin uous (even infinite dimensional) space of pr obability measures which are inv ariant for the map, and many of them will b e no n computable. An impo rtant par t o f the theory of dynamical systems is devoted to selecting measures which are particular ly meaning ful. F r om this p oint of view, an imp ortant clas s o f these mea s ures is the class of SRB in v aria nt measur es, which ar e measure s b eing in some se nse the “physically meaningful ones”(for a survey on this topic see [ Y ou0 2 ]). It can b e proved (see [ GHR07b ] and [ GHR07a ] a nd their re fer ences e.g.) that in several classe s of dy na mical systems wher e SRB measure s are proved to exist, these measures are als o computable fr om o ur formal po int of view, hence providing several classes of nontrivial c o ncrete examples wher e our results can b e applied. 2. Preliminaries 2.1. P ar tial recursiv e functions. The notion of algo rithm working on integers has b een formalized independently b y Marko v, Ch urch, T uring among o thers. Each constructed mo del defines a set of par tial (not defined everywhere) integer functions which c an b e c omputed by some effe ctive mec hanical or algorithmic (w.r.t. the mo del) pr o cedure. Later, it has been prov ed that all this mo dels define the sa me class of functions, namely: the set of p artial r e cursive fun ct ions . This fact s upp o r ts a working h yp o thesis known as Church’s Thesis, which states that e very (in tuitiv ely formalizable) algor ithm is a partial recurs ive function. It gives the co nnection betw een the informal notion of algor ithm a nd the forma l definition of recursive function. Let us say then that a r e cursive function is a function (on in tegers) that can be co mputed in some effe ctiv e or algorithmi c wa y . F or forma l definitions s ee for example, [ Rog87 ]. With this intuitiv e description it is mor e or less clear that there exists an effectiv e pro cedure to en umerate the cla ss of all partial recursive functions, asso ciating to each of them its G¨ odel nu mb er , which is the num ber of the progr am computing it. Hence there ex ists a universal r ecursive function ϕ u : N → N sa tisfying for all e, n ∈ N , ϕ u ( h e, n i ) = ϕ e ( n ) where e is the g¨ odel nu m ber of ϕ e and h , i : N 2 → N is some re cursive bijection. In c la ssical recursion theory , a set of natural num bers is ca lled r e cursively enu mer able ( r.e for short) if it is the range o f so me partial recursive function. That is if ther e exists an algorithm listing the set. W e denote b y E e the r.e set a sso ciated to ϕ e . Namely E e = rang ( ϕ e ) = { ϕ u ( h e, n i ) : n ∈ N } . THE DYNAMICS O F ALGORITHMICALL Y RANDOM P OINTS 5 2.2. Algorithms on fini te ob jects. Strictly sp eaking, algo rithms only work on int egers. Ho w ever, when the ob jects of some class hav e be e n ident ified with in tegers, it makes sense to sp eak ab out alg o rithms acting on these ob jects. Definition 2. 2.1. A Nu mb er e d Set O is a c o untable s e t together with a surjection ν O : N → O called the nu mb ering . W e wr ite o n for ν ( n ) and ca ll n the name o f o n . Of cours e, the p otential of algor ithms dep ends on the choice of the num bering, since it determines to what extent a n ob ject can b e algor ithmica lly r ecov ered from its name. If the ob jects of some co llection c a n be characterized by a finite num b er of integers, then the collection is a num bered s et since its ob jects can b e indexed using standard recursive bijections from N ∗ to N a nd from N k to N , which we will bo th denote h . i . Examples. (1) Q , with so me standard num b er ing ν Q is a num bered set. (2) The se t of partial recursive functions R = { ϕ e : e ∈ N } is a num bered set, G¨ odel num b er s b eing the names. (3) The c o llection { E e = r ang ( ϕ e ) : e ∈ N } o f all r.e subsets of N is a n um ber ed set. Definition 2.2.2. L et O b e a num bered set. T o any rec ursive function ϕ : N → N we asso cia te an algori thm A ϕ : N → O defined by A ϕ ( i ) = o ϕ ( i ) . Given a total alg orithm A : N → O (i.e A = A ϕ for some to ta l ϕ ), we say that the sequence of finite ob jects ( A ( n )) n ∈ N is enumer ate d by A , or that A is an algorithmic enu mer ation of this sequence. 2.3. Computability ov e r the reals. Definition 2 .3.1. Let x be a rea l num ber. W e s ay that: • x is l ow er semi-c omputable if the set E := { i ∈ N : q i < x } is r .e, • x is u pp er semi-c omputable if the set E := { i ∈ N : q i > x } is r .e, • x is c omputable if it is lower and uppe r semi-co mputable. Equiv a lently , a real num ber is computable if a nd only if there exis ts an a lgorith- mic enum eration of a sequence of r ational num bers converging exp onentially fast to x . That is: Prop ositi on 2.3.1. A r e al n umb er x is c o mputable if and only if ther e exists an algorithm A : N → Q such that |A ( i ) − x | < 2 − i , for al l i . Definition 2. 3 .2. Le t ( x n ) n be a s equence of computable reals. W e say that the s e quence is uniformly computable or that x n is computable uniforml y in n if there ex ists an algor ithm A : N → Q such that for a ll n and i it holds |A ( h n, i i ) − x n | < 2 − i . Uniform sequences of low er (upper ) semi-computable reals are defined in the same wa y . 2.4. Computable Metric Spaces. Definition 2.4.1. A c omputable metri c sp ac e (CMS) is a tr iple X = ( X , d, S ), where • ( X , d ) is a separ able complete metric space . • S = ( s i ) i ∈ N is a num b er ed dense subset of X (called ide al p oi nts ). • The rea l n um ber s ( d ( s i , s j )) h i,j i∈ N are all computable, unifor mly in h i, j i . 6 STEF ANO GALA T OLO, MA THIEU HOYR UP, AND CRIST ´ OBAL ROJAS W e now recall so me imp or tant examples of computable metric spaces: Examples. (1) the Cantor space (Σ N , d, S ) with Σ a finite alpha b e t and d the usual distance . S is the set of ultimately 0- stationary sequences. (2) ( R n , d R n , Q n ) with the euclidean metric and the standard num bering o f Q n . (3) if ( X 1 , d 1 , S 1 ) and ( X 2 , d 2 , S 2 ) ar e t wo computable metric s paces, the dis- tance d (( x 1 , x 1 ) , ( y 1 , y 2 )) = max( d 1 ( x 1 , y 1 ) , d 2 ( x 2 , y 2 )) on the pro duct space ( X 1 × X 2 , d, S 1 × S 1 ) makes it a computable metric space. F or fur ther examples we refer to [ W ei93 ]. Let ( X , d, S ) be a computable metric space. The computable str ucture of X assures that the who le space can b e “ r eached” using a lgorithmic means. Sin ce ideal points (the finite ob jects of S ) are dense, they can approximate any x at any finite precision. Then, x itself can b e identified to a s equence of ideal p oints conv erging to x in an effectively controlled wa y . Let us say that a sequence of ideal po ints ( s i n ) n is fast if d ( s i n , s i n +1 ) < 2 − n for all n . As the space is co mplete, a fast sequence has always a limit x , a nd d ( s i n , x ) < 2 − ( n − 1) for all n . Definition 2.4.2 (Computable p oints) . A p o int x ∈ X is sa id to b e c omputable if ther e exists a n algo rithm A : N → S which en umerates a fast seq uence whos e limit is x . As for rea l nu m ber s we can give the notion of unifor m sequence Definition 2.4 .3. Let ( x n ) n be a sequence o f co mputable p oints. W e s ay that the sequence is uniformly computable or that x n is co mputable uni for ml y in n if there exists an algorithm A : N → S such that for all n , the sequence ( A ( h n, i i )) i is fast and converges to x n . The num ber ed s et o f idea l p oints ( s i ) i induces the num bere d set of ide al b al ls B := { B ( s i , q j ) : s i ∈ S, q j ∈ Q > 0 } . W e denote by B h i,j i the ideal ba ll B ( s i , q j ). Computability can then b e extended from the num bered s et B to the space of op en subsets of X : such an op en subset U ⊆ X can b e identified to a colle c tion of ideal balls whose union is U . Definition 2.4 .4 (R.e op en s ets) . W e s ay that the set U ⊂ X is r . e op en if there is some r.e set E ⊂ N such that U = ∪ i ∈ E B i . R emark 2.4.1 . Let U b e a r.e o pe n se t. It is easy to see that there is a n algorithm to semi-de cide weather s o me ideal point belo ng s to U . That is, the a lgorithm will halt on input i iff s i ∈ U . This notion can b e extended to any p oint x in the following sense: The alg orithm seque ntially asks (from a n e x ternal user) fo r finite approximations o f x at required precisions . If x ∈ U the algorithm will even tually stop and answer “yes”, if x / ∈ U then the algo rithm will run and ask forever. F or formal definitions we refer to [ HR07 ]. Definition 2.4.5 . Let ( U n ) n be a sequence of r .e o p e n se ts . W e say that the sequence is uniformly r.e o r that U n is r.e op en uniformly in n if there exists an r.e set E ⊂ N such that for all n it holds U n = ∪ i ∈ E n B i , w her e E n = { i : h n, i i ∈ E } . Examples. (1) If the sequence ( U n ) n is uniformly r.e then the union ∪ n U n is a r.e op en set. THE DYNAMICS O F ALGORITHMICALL Y RANDOM P OINTS 7 (2) The universal rec ur sive function ϕ u makes the collectio n of a ll r .e op en sets (denoted U ) a sequence uniformly r.e. Indeed, de fine E := {h e, ϕ u ( h e, n i ) i : e, n ∈ N } . Then U = { U e : e ∈ N } where U e = ∪ i ∈ E e B i . (3) The n um ber ed s et U is clos ed under finite unions and finite intersections. F urther mo re, these ope r ations are effe ctive in the following sens e: there exists recursive functions ϕ ∪ and ϕ ∩ such tha t for all i, j ∈ N , U i ∪ U j = U ϕ ∪ ( h i,j i ) and the same holds for ϕ ∩ . Eq uiv alently: U i ∪ U j is r.e ope n uniformly in h i, j i . See [ HR07 ]. Definition 2.4.6 (Co ns tructive G δ -sets) . W e say that the set D ⊂ X is a c on- structive G δ -set if it is the intersection o f a sequence o f unifor mly r.e op en sets. Let ( X , S X , d X ) and ( Y , S Y , d Y ) b e computable metric spaces with U X and U Y the corr esp onding num bered sets of r .e o p en sets. Definition 2.4.7 (Computable F unctions) . A function T : X → Y is said to be c omputable if T − 1 ( B n ) is r .e op en uniformly in n . R emark 2.4.2 . W e remar k that this definitio n implies that the preimage o f a uniform sequence of r.e . op en sets is a unifor m sequence of r.e. o pe n sets. This co uld b e an alternative definition o f co mputable function. It follows that computable functions ar e con tin uous. Since we will w ork with functions which ar e not necessa rily contin uo us everywhere, we shall consider func- tions which a re co mputable on so me subset o f X . More precisely , a function T is said to b e c omp utable on D ( D ⊂ X ) if there is a uniform seq uence ( U X n ) n of r.e op en subsets of X such it holds T − 1 ( B n ) ∩ D = U X n ∩ D for the uniform s equence of ideal balls B n . D is called the domain of c omputability of T . remarks: • Since ide a l balls generate the top ology , a function is computable iff T − 1 ( B Y n ) is r.e o p en uniformly in n (or the intersection of D with a uniformly r .e op en set). • If T is computable then the images o f ideal p oints ca n b e uniformly com- puted, that is: T ( s X i ) is a computable p o int, uniformly in i . • More gener ally , if T is co mputable then there exists an a lgorithm which computes the ima ge T ( x ) o f a ny x in the following sense: the user en ters some ra tional ǫ to the algor ithm which, after ask ing finitely many times the user for finite approximations of x , halts outputting a finite approximation of T ( x ) up to ǫ . • The distance function d : X × X → R is a computable function. 2.5. Computable Probabilit y Spaces (CPS). When X is a computable metric space, the s pace of probability measur e s over X , denoted by M ( X ), c an b e endow ed with a str uc tur e of computable metric space (this will b e defined b elow, for mo re details, see [ G´ ac05 , HR07 ]). Then a co mputable measure can b e defined as a computable p oint of M ( X ). Some prerequisites from mea sure theo ry: W e say that µ n conv erge weakly to µ and write µ n → µ if µ n → µ iff µ n f → µf for a ll rea l contin uous b o unded f (2) 8 STEF ANO GALA T OLO, MA THIEU HOYR UP, AND CRIST ´ OBAL ROJAS where µf stands for R f dµ . Let us recall the Portmant eau theo rem. W e say that a Borel s e t A is µ -c ontinuous if µ ( ∂ A ) = 0, where ∂ A = A ∩ X \ A is the b oundary of A . Theorem 2.5.1 (Portman teau theorem) . L et µ n , µ b e Bor el pr ob ab ility me asur es on a sep ar able metric sp ac e ( X , d ) . The fol lowing ar e e quivalent: (1) µ n c onver ges we akly to µ , (2) lim sup n µ n ( F ) ≤ µ ( F ) for al l close d set s F , (3) lim inf n µ n ( G ) ≥ µ ( G ) for al l op en sets G , (4) lim n µ n ( A ) = µ ( A ) for al l µ -c ontinuity sets A . This theorem ea sily implies the following: when ( X, d ) is a s eparable metric space, weak conv ergence ca n b e prov ed using the following criter io n: Prop ositi on 2.5.1. Le t A b e a c oun table b asis of the top olo gy which is close d un der the formation of fin it e u nions. If µ n ( A ) → µ ( A ) for every A ∈ A , then µ n c onver ge we akly to µ . Let us intro duce o n M ( X ) the structure of a co mputable metric space. Let us endow M ( X ) with the weak to po logy , which is the top ology of weak convergence. As X is s e parable and complete, so is M ( X ). Let D ⊂ M ( X ) b e the set of tho se probability mea sures that are concentrated in finitely many p oints of S and assig n rational v a lues to them. It can b e shown tha t this is a dense subset ([ Bil6 8 ]). W e c onsider the P rokhor ov metric ρ on M ( X ) defined by: ρ ( µ, ν ) := inf { ǫ ∈ R + : µ ( A ) ≤ ν ( A ǫ ) + ǫ for every Borel set A } . where A ǫ = { x : d ( x, A ) < ǫ } . This metric induces the weak top ology on M ( X ). F urthermore , it can b e shown that the triple ( M ( X ) , D , ρ ) is a computable metr ic space (see [ G´ ac05 ], [ HR07 ]). Definition 2.5.1 . A mea sure µ is co mputable if there is an algor ithmic enumer- ation of a fast se quence of ideal measures ( µ n ) n ∈ N ⊂ D converging to µ in the Prokho rov metric and he nc e , in the weak topo logy . The following theorem gives a characterization for the co mputabilit y of mea sures in terms of the co mputabilit y of the meas ure of sets (for a pro of see [ HR07 ]): Theorem 2.5.2. A me asur e µ ∈ M ( X ) is c omputable if and only if the me a- sur e µ ( B i 1 ∪ . . . ∪ B i k ) of finite u nions of ide al op en b al ls is lower-semi-c omputable uniformly in h i 1 , . . . , i k i . Definition 2.5.2. A Computable Pr ob abil i ty Sp ac e (CPS) is a pair ( X , µ ) where X is a computable metric s pa ce and µ is a computable Bor el pro bability measure on X . Definition 2.5 .3. Let ( X , µ ) and ( Y , ν ) b e t w o computable probability spaces. A morphism fro m ( X , µ ) to ( Y , ν ) is a measure -preserv ing function F : X → Y which is computable o n a constr uctive G δ -set of µ -mea s ure o ne. W e r ecall that F is measure-pr eserving if ν ( A ) = µ ( F − 1 ( A )) for every Bor e l set A . Co mputable pr obability structur e s can b e easily tr ansferred: Prop ositi on 2.5.2 . L et ( X , µ ) b e a c omputable pr ob abili ty sp ac e, Y b e a c omput able metric sp ac e and F : X → Y a function which is c omp utable on a c onstructive G δ - set of µ -me asur e one. Then the induc e d me asur e µ F on Y define d by µ F ( A ) = µ ( F − 1 ( A )) is c omputable and F is a morphism of c omputable pr ob abil ity sp ac es. THE DYNAMICS O F ALGORITHMICALL Y RANDOM P OINTS 9 2.6. Algorithmic randomness. Now we consider a g eneralizatio n of Martin- L ¨ of tests to computable pr o bability spaces. Let ( X , µ ) b e a co mputable pr obability space. Definition 2 .6.1. A Marti n-L¨ of µ -T est is a sequenc e ( U n ) n ∈ N of uniformly r.e op en sets which satisfy µ ( U n ) < 2 − n for all n . An y subset of T n U n is called a n effe ctive µ -nul l set . Definition 2.6.2. A point x ∈ X is called µ -r andom if x is co ntained in no effective µ -null se t. The set of µ -rando m p oints is denoted R µ . Note that µ ( R µ ) = 1. The follo wing is the generalizatio n for metric spaces of a classica l result in Can tor space due to Martin-L ¨ of. It says that the set of non- random p oints is not only a n ull set but an effectiv e null set. F or a pro of see [ HR07 ]. Theorem 2.6.1 . The union of al l effe ct ive µ -nul l sets, denote d by N µ , is again an effe ctive µ - n ul l set. Thu s, there is a single Ma rtin-L¨ of test (often called u niversal ) w hich tests no n- randomness, and R µ = N c µ . W e w ill need the following results, also taken from [ HR07 ]. Lemma 2 .6.1. Every µ -r andom p oint is in every r.e op en set of ful l me asur e. Prop ositi on 2.6.1 (Morphisms of CPS prese rve r a ndomness) . L et F b e a mor- phism of c omputable pr ob ability sp ac es ( X , µ ) and ( Y , ν ) . Then every µ -ra ndom p oint x is in the domain of c omputabil ity of F and F ( x ) is ν -r andom. 2.7. Kolmogorov complexity . The idea is to define, for a finite ob ject, the min- imal amo un t of a lgorithmic infor ma tion fro m whic h the ob ject can b e recovered. That is, the length of the shor test descr iption (code ) of the ob ject. Since this short- est description is s uppo sed to contain all necessary informa tio n to reconstruc t in an algorithmic wa y the co ded finite o b ject, the K olmogor ov Complexity is also calle d Algo rithmic Information Content . F or a complete intro duction to Kolmogo r ov com- plexity we refer to a standar d text [ L V93 ]. Let Σ ∗ and Σ N be the sets of finite a nd infinite words (ov er the finite alphab et Σ) resp ectively . A word w ∈ Σ ∗ defines the cylinder [ w ] ⊂ Σ N of all p ossible contin uations of w . A set D = { w 1 , w 2 , ... } ⊂ Σ ∗ defines an ope n set [ D ] = ∪ i [ w i ] ⊂ Σ N . D is called prefix-free if no word of D is pr efix of ano ther one, that is if the cylinders [ w i ] are pairwise dis jo int . Let X be Σ ∗ or N or N ∗ . Definition 2 .7.1. An i nterpr eter is a par tial r e c ursive function ϕ : { 0 , 1 } ∗ → X which has a prefix- fr ee domain. Definition 2.7 . 2. Let I : { 0 , 1 } ∗ → X b e an interpreter. The c omplex ity (o r Information Content ) K I ( x ) of x ∈ X is defined to b e K I ( x ) =  | p | if p is a s ho rtest input such that I ( p ) = x ∞ if there is no p such that I ( p ) = x It turns out that there exis ts an a lgorithmic enumeration of a ll interpreters, which entails the existence of a universal in terpreter U which is asymptotically optimal in the sens e that the invarianc e t he or em holds: 10 STEF ANO GALA T OLO, MA THIEU HOYR UP, AND CRIST ´ OBAL ROJAS Theorem 2.7.1 (Inv aria nce theorem) . F or al l interpr eter I ther e exists c I ∈ N such t hat for al l x ∈ X we have K U ( w ) ≤ K I ( x ) + c I . W e fix a universal interpreter U and we let K ( x ) = K U ( x ). 2.7.1. Estimates. Let us r e call some simple estimates of complexity . Let f , g b e real-v alued functions. W e say that g additively dominates f and write f < + g if there is a c o nstant c such that f ≤ g + c . As co des are alwa ys binary words, we use base- 2 logar ithms, which we de no te by log . W e define J ( x ) = x + 2 log( x + 1 ) for x ≥ 0. F or n ∈ N , K ( n ) < + J (log n ). F or n 1 , . . . , n k ∈ N , K ( n 1 , . . . , n k ) < + K ( n 1 ) + . . . + K ( n k ). The fo llowing pro p erty is a version of a result attributed to Kolmo gorov, stated in terms of prefix complexity instea d of plain co mplexity . Prop ositi on 2.7. 1. L et E ⊆ N × X b e a r.e. s et such that E n = { x : ( n, x ) ∈ E } is finite for al l n . Then for ( s, n ) with s ∈ E n , K ( s ) < + J (log | E n | ) + K ( n ) Prop ositi on 2.7. 2. L et µ b e a c omputable me asur e on Σ N . F or al l w ∈ Σ ∗ , K ( w ) < + − log µ ([ w ]) + K ( | w | ) Theorem 2. 7.2 (Co ding theorem) . L et P : X → R + b e a lower s emi-c omputable function such t hat P x P ( x ) ≤ 1 . Then K ( x ) < + − log P ( x ) , i.e. ther e is a c onstant c su ch that K ( x ) ≤ − log P ( x ) + c for al l x ∈ X . Moreov er, P x 2 − K ( x ) ≤ 1 as it is the Leb esg ue measur e of the doma in of the universal interpreter U . There is a relation betw een Kolmo g orov co mplexity and randomness, initial seg ments o f random infinite s trings b eing maximally co mplex. Theorem 2.7.3 (Chaitin, Levin) . L et µ b e a c omputable me asure . Then ω ∈ Σ N is a µ -r andom se quenc e if and only if ∃ m ∀ n K ( ω 1: n ) ≥ − log µ [ ω 1: n ] − m . The minimal such m , defined by d µ ( ω ) := sup n {− log µ [ ω 1: n ] − K ( ω 1: n ) } and called the r andomness defici ency of ω w.r.t µ , is not only finite a lmo st every- where: it has finite mean, that is R d µ ( ω ) dµ ≤ 1 . F or a pro o f see [ L V93 ]. 3. Effective symbolic dynamics and st a tistics of random points Let ( X , µ ) b e a computable pro bability spa ce and let R µ be the set of r a ndom po ints. The aim o f this section is to s tudy the se t R µ from a dynamical p o int of view. That is, we will put a dyna mic T on ( X , µ ) (a n endomorphism o f computable probability space), a nd lo ok at the abilities of rando m p oints (which are a prio ri independent of T ) to descr ib e the statistical pro per ties of T . W e r ecall that a Bo rel set A is called T -invariant if T − 1 ( A ) = A (mod 0) a nd that the transforma tion T is said to b e er go dic if every T -in v ariant set has measure 0 or 1. 3.1. Sym bo lic dynamics of random p oints. Let T be a n endomo rphism o f the (Borel) pro ba bility space ( X , µ ). In the classical constr uction, one considers a ccess to the system g iven by a finite meas urable partition, that is a finite collection o f pairwise disjoint Borel sets P = { p 1 , . . . , p k } such that µ ( ∪ i p i ) = 1. Then, to ( X, µ, T ) is ass o ciated a symb olic dynamic al system ( X P , σ ) (called the s y mbolic THE DYNAMICS O F ALGORITHMICALL Y RANDOM P OINTS 11 mo del of ( X , T , P )). The set X P is a subset of { 1 , 2 , . . . , k } N . T o a p oint x ∈ X corres p o nds an infinite sequence ω = ( ω i ) i ∈ N = φ P ( x ) defined by: φ P ( x ) = ω ⇔ ∀ j ∈ N , T j ( x ) ∈ p ω j The transformatio n σ : X P → X P is the shift defined by σ (( ω i ) i ∈ N ) = ( ω i +1 ) i ∈ N . As P is a measur able par tition, the map φ P is mea surable a nd then the measure µ induces the meas ur e µ P (on the asso ciated symbolic mo del) defined by µ P ( B ) = µ ( φ − 1 P ( B )) fo r all meas urable B ⊂ X P . The requirement of φ P being measurable makes the s ymbolic mo del appropr iate from the measur e-theoretic view p oint, but is not enough to have a symbolic mo del compatible with the co mputational approa ch: Definition 3. 1.1. Let T be an endomo rphism of the co mputable probability s pace ( X , µ ) a nd P = { p 1 . . . , p k } a finite meas urable par tition. The asso ciated s ymbolic mo del ( X P , µ P , σ ) is said to b e an effe ctive symb olic mo del if the ma p φ P : X → { 1 , . . . , k } N is a morphism of CPS (here the space { 1 , . . . , k } N is endow ed with the standard computable structure). The sets p i are ca lled the atoms o f P and we denote by P ( x ) the ato m co nt aining x (if there is one). Observe that φ P is co mputable o n its domain only if the ato ms are op en r.e sets (in the domain): Definition 3.1.2 (Computable par titions) . A mea surable partition P is said to b e a c omputable p ar tition if its ato ms a re r.e op en sets. Conv ersely: Theorem 3.1. 1 . L et T b e an endomorphism of the CPS ( X, µ ) and P = { p 1 . . . , p k } a finite c o mputable p artition. Then t he asso ciate d symb olic mo del is effe ctive. Pr o of. Let D b e the domain of computability o f T (it is a full-measure co nstructive G δ ). Define the set X P = D ∩ \ n ∈ N T − n ( p 1 ∪ . . . ∪ p k ) X P is a full-mea sure co nstructive G δ -set: indeed, as p 1 ∪ . . . ∪ p k is r .e. and T is computable on D there a re uniformly r.e. op en sets U n such that D ∩ T − n ( p 1 ∪ . . . ∪ p k ) = D ∩ U n , so X P = D ∩ T n U n . As T is mea sure-pres erving, a ll U n hav e measure one. Now, X P ∩ φ − 1 P [ i 0 , . . . , i n ] = X P ∩ p i 0 ∩ T − 1 p i 1 ∩ . . . ∩ T − n p i n . This prov es that φ P is computable over X P . P rop osition 2.5.2 allows to co nclude.  After the definition an imp ortant question is: are there computable partitions ? the answer dep ends on the existence of op en r.e sets with a zero-mea s ure b oundary . Definition 3.1.3. A set A is said to b e almost de cidable if there are t w o r.e op en sets U and V s uch that: U ⊂ A, V ⊆ A c , µ ( U ) + µ ( V ) = 1 remarks: • a set is almo st decidable if a nd only if its co mplement is almos t decida ble, • an almo st decidable set is always a contin uit y set, • a µ -co ntin uity ideal ball is always almo st decidable, 12 STEF ANO GALA T OLO, MA THIEU HOYR UP, AND CRIST ´ OBAL ROJAS • unless the spa c e is disconnected (i.e. has no n-trivial clop en subsets), no set can be de cidable , i.e. s e mi-decidable (r.e ) and with a semi-decidable complement (such a set must b e clop en 1 ). Instead, a s et ca n b e decidable with pr ob abili ty 1 : ther e is an algorithm which decides if a p o int b elong s to the set or not, for almost every point. That is why we ca ll it almost de cidable . Ignoring computability , the ex is tence of op en µ -contin uit y sets directly follows from the fact that the collection of o pe n s ets is uncount able and µ is finite. The problem in the computable setting is tha t there are only c ountable ma ny op en r.e sets. F ortunately , there still always exis ts a basis of almost decidable balls. This result, first obtained in [ HR07 ] with o ther techniques, will b e use d many times in the seq uel, in par ticular it direc tly implies the e x istence of computable par titions. F or c ompleteness we prese nt a different, self-co ntained pr o of. Theorem 3.1.2 . Ther e is a family of u niformly c omputable r e als ( r i n ) i,n ∈ N such that for al l i , { r i n : n ∈ N } is dense in R + and such that for every i, n , the b al l B ( s i , r i n ) is almost de cid able. Pr o of. Let s i be an ideal point. Put I h j,k i = [ q j , q k ] with q j , q k po sitive rationa l nu m ber s. W e show that fo r every n = h j, k i we c a n compute, uniformly in n , a real r i n ∈ I n for whic h µ ( ∂ B ( s i , r i n )) = 0. First observe that for a closed interv al I = [ a, b ] ( a, b ∈ Q ), the co mplement of B I = B ( s i , b ) /B ( s i , a ), is r.e op en. Then by corollary 2 .5.2 , its measure is lower semi-computable and then w e ca n semi- decide fo r a given rational q the relatio n µ ( B I ) < q . The a lgorithm computing r i n enum erates a sequence of nested closed int erv a ls ( J k ) k ∈ N whose leng th tends to 0, with J 0 = I n , a nd such tha t for all k , µ ( B J k ) < 2 − k +1 . The n { r i n } = ∩ k ≥ 1 J k . It works as follows: In stag e k + 1 (the interv al J k = [ a, b ] has already b een found), put m = b − a 3 and test in par a llel µ ( B [ a,a + m ] ) < 2 − k and µ ( B [ b − m,b ] ) < 2 − k . Since µ ( B J k ) < 2 − k +1 , one of the tests must stop, and then provides the “go od” in terv a l J k +1 for whic h the condition holds.  W e de no te by B h i,n i the almost decidable ba ll B ( s i , r i n ). The family { B h i,n i : i, n ∈ N } is a basis for the topo logy . It is even effectively equiv ale nt to the basis o f ideal balls : ev ery ideal ball can b e expressed as a r .e. union of almos t decida ble balls, and vice- versa. W e finis h presenting s ome results that will b e needed in the next subsectio n. Corollary 3.1.1. On every c omputable pr ob a bility sp ac e, ther e ex ists a family of uniformly c omputable p artitions which gener ates the Bor el σ - algebr a. Pr o of. T ake P h i,n i = { B ( s i , r i n ) , X \ B ( s i , r i n ) } where B is the clos ed ba ll: as the almost decidable ba lls for m a ba sis of the top olog y , the σ -alg ebra genera ted by the P k is the Bo rel σ -field.  Prop ositi on 3.1.1 . If A is almost de cidable then µ ( A ) is a c omputable r e al numb er. 1 In Can tor s pace f or example (which is totally disconnecte d), ev ery cylinder (ball) is a decidable set. Indeed, deciding if some infinite sequence belongs to s ome cylinder reduces to a finite pattern- matc hing. THE DYNAMICS O F ALGORITHMICALL Y RANDOM P OINTS 13 Pr o of. Since U and V a re r.e op en, by theorem 2.5.2 their mea s ures are low er-semi- computable. As µ ( U ) + µ ( V ) = 1, their measures a re also upp er-se mi-computable.  The following r egards the computabilit y of inducing a mea s ure in a subset and will be used in the pro of of pro p. 3.2.1 Prop ositi on 3 .1.2. L et µ b e a c omputable m e asur e and A an almost de cidable subset of X . Then the induc e d me asur e µ A ( . ) = µ ( . | A ) is c omputable. F urthermor e, R µ A = R µ ∩ A . Pr o of. let W = B n 1 ∪ . . . ∪ B n k be a finite union of ideal balls. µ A ( W ) = µ ( W ∩ A ) /µ ( A ) = µ ( W ∩ U ) /µ ( A ). W ∩ U is a r.e op en set, so its measur e is low er semi- computable. As µ ( A ) is computable, µ A ( W ) is low er semi-computable. Note that everything is unifor m in h n 1 , . . . , n k i . The r esult follows from theorem 2.5.2 . Let U and V a s in the definitio n of an almos t decida ble set. Fir s t note that R µ ∩ A = R µ ∩ U , as R µ ⊆ U ∪ V by lemma 2.6.1 . Again by lemma 2.6.1 , R µ A ⊆ U , and as µ A ≤ 1 µ ( A ) µ , every µ -effective null set is a lso a µ A -effective null set, so R µ A ⊆ R µ . Hence, we hav e R µ A ⊆ R µ ∩ U . Conv ersely , R c µ A being a µ A -effective null set, its in tersection with U is a µ - effective null set, by definition of µ A . So R c µ A ∩ U ⊆ R c µ , which is equiv a le nt to R µ ∩ U ⊆ R µ A .  3.2. Some statis tical prop erti e s of random p oi n ts. With the to ols developed so far, it is p os sible to trans late many results o f the form µ { x : P ( x ) } = 1 , with P s ome predicate, into an “individual” result of the form: “If x is µ -random, then P ( x )” . In this sectio n we give tw o e x amples: r ecurrence and statistical typicalit y . Definition 3.2. 1 . Let X b e a metric space. A po int x ∈ X is said to b e r e curr ent for a transfor mation T : X → X , if lim inf n d ( x, T n x ) = 0. Prop ositi on 3.2.1 (Rando m p oints are recurrent) . L et ( X, µ ) b e a c omput able pr ob abil ity sp ac e. If x is µ -r andom, then it is r e curr ent with r esp e ct to every me a- sur e pr eserving endomorphism T on ( X , µ ) . Pr o of. take x ∈ R µ and B an almost dec ida ble neighbo rho o d of x . Then µ ( B ) > 0 and there is a r .e op en set U such that: [ n ≥ 1 T − n B = U ∩ D where D is the domain of computability o f T . By the Poincar´ e rec urrence theorem, this se t has full measure for µ B ( . ) = µ ( . | B ). By pro p osition 3.1.2 , x ∈ R µ B , so by lemma 2.6.1 , x is in U .  W e now prove that r andom p oints s atisfy a strong e r pro pe rty to b e used in the sequel: sta tistical typicalit y . Let us then introduce this concept. Let X b e a metric spa c e a nd T be a co ntin uo us transformation on X . Let C b ( X ) be the space o f bounded real- v alued co ntin uo us functions on X . F or f ∈ C b ( X ) define: 14 STEF ANO GALA T OLO, MA THIEU HOYR UP, AND CRIST ´ OBAL ROJAS f ( x ) := lim n →∞ 1 n n − 1 X j =0 f ( T j x ) (3) at the p o int s x wher e this limit e x ists. W e r ecall that a p oint x is called generic for T if f ( x ) is defined for every f ∈ C b ( X ). Every g eneric p oint x gener ates a proba bility measure µ x which is inv a riant for T , dually defined by: Z X f dµ x = f ( x ) for a ll f ∈ C b ( X ) . (4) In o ther words, x is gene r ic if the measure ν n = 1 n P j 0 and n ∈ N , the algorithmic infor ma tion that is needed to list a sequence of ideal p oints which follows the orbit of x for n steps at a distance less than ǫ is: K n ( x, T , ǫ ) := min { K ( i 0 , . . . , i n − 1 ) : d ( s i j , T j x ) < ǫ for j = 0 , . . . , n − 1 } where K is the s elf-delimiting Ko lmogorov c omplexity . W e then define the ma ximal and minimal gr owth-rates of this q uantit y : K ( x, T , ǫ ) := lim sup n →∞ 1 n K n ( x, T , ǫ ) K ( x, T , ǫ ) := lim inf n →∞ 1 n K n ( x, T , ǫ ) . As ǫ tends to 0, these quantities increase (or at lea st do not decr ease), hence they hav e limits (which can b e infinite). Definition 4.5 .1. The upp er and lower orbit c omplex ities of x under T are defined by: K ( x, T ) := lim ǫ → 0 + K ( x, T , ǫ ) K ( x, T ) := lim ǫ → 0 + K ( x, T , ǫ ) . R emark 4.5.1 . If T is computable, and assuming that ǫ takes only r ational v al- ues, the n firs t iterates of x could b e ǫ -shadow ed b y the orbit of a s ingle ideal po int instead of a ps eudo-orbit of n idea l points. Actually it is easy to see that it gives the same quantities K ( x, T , ǫ ) and K ( x, T , ǫ ): let K ′ n ( x, T , ǫ ) = min { K ( i ) : d ( T j s i , T j x ) < ǫ for j < n } , one has : K ′ n ( x, T , 2 ǫ ) < + K n ( x, T , ǫ ) + K ( ǫ ) K n ( x, T , ǫ ) < + K ′ n ( x, T , ǫ/ 2) + K ( n, ǫ ) Indeed, fro m ǫ a nd i 0 , . . . , i n − 1 some ideal po int can b e alg orithmically found in the constructive op en set B ( s i 0 , ǫ ) ∩ . . . ∩ T − ( n − 1) B ( s i n − 1 , ǫ ), unifor mly in i 0 , . . . , i n − 1 . Its n firs t itera tes 2 ǫ -sha dow the orbit of x , which proves the firs t inequality . F or the se c ond inequalit y , s ome i 0 , . . . , i n − 1 can be algorithmica lly found from n , ǫ , and a p o int s i whose n first iterates ǫ / 2-shadow the orbit of x , taking any s i j ∈ B ( T j s i , ǫ/ 2). R emark 4.5.2 . Under the same a s sumptions, one could define K ( B n ( s i , ǫ )) to b e K ( i, n, ǫ ), and replace K ( i ) by K ( B n ( s i , ǫ )) in the definition of K ′ n ( x, T , ǫ ), without changing the quantities K ( x, T , ǫ ) a nd K ( x, T , ǫ ). Indeed, K ( i ) < + K ( B n ( s i , ǫ )) < + K ( i ) + K ( n ) + K ( ǫ ) 20 STEF ANO GALA T OLO, MA THIEU HOYR UP, AND CRIST ´ OBAL ROJAS 5. Equiv alence of the two notions of orbit complexity for random points W e now prov e: Theorem 5.0 .1. Le t T b e an er go dic endomorphi sm of the c omputable pr ob ability sp ac e ( X , µ ) , wher e X is c omp act. Then for every Martin-L¨ of r ando m p oint x , K ( x, T ) = K µ ( x, T ) . Pr o of of K ( x, T ) ≤ K µ ( x, T ) . Let ǫ > 0. Cho os e a computable pa rtition ξ of diam- eter < ǫ (this is why we r equire X to b e compact). T o every cell of ξ , asso cia te an ideal p oint which is inside (as ξ is co mputable, this ca n b e done in a computable wa y , but we actually do not need that). The tra ns lation of symbolic seque nce s in sequences o f ideal p oints through this finite dictiona ry is co nstructive, and trans- forms the symbolic o r bit of a p oint x into a sequence o f ideal po ints which is ǫ -close to the orbit o f x . So K ( x, T , ǫ ) ≤ K µ ( x, T | ξ ). The ine q uality follows letting ǫ tend to 0.  T o prov e the other inequality , we reca ll some technical stuff. The self-delimiting Kolmogo rov complexity o f na tural num bers k ≥ 1 satisfies K ( k ) < + f ( k ) where f ( x ) = log x + 1 + 2 log(log x + 1) for all x ∈ R , x ≥ 1. f is a concav e increasing function and x 7→ xf (1 /x ) is an increas ing function on ]0 , 1] which tends to 0 as x → 0. W e r ecall that for finite sequences o f natur a l num b er s ( k 1 , . . . , k n ), one has K ( k 1 , . . . , k n ) < + K ( k 1 ) + . . . + K ( k n ) as the shortest descriptio ns for k 1 , . . . , k n can be extracted from their concatena - tion (this is one reas o n to use the self-delimiting complexit y instead of the plain complexity). Lemma 5. 0.1. L et Σ b e a finite alpha b et and n ∈ N . L et u, v ∈ Σ n and 0 < α < 1 / 2 such that the density of the set of p ositions wher e u and v differ is less than α , that is: 1 n # { i ≤ n : u i 6 = v i } < α < 1 / 2 Then   1 n K ( u ) − 1 n K ( v )   < αf (1 /α ) + c n wher e c is a c onst ant indep endent of u, v and n . Pr o of. Let ( i 1 , . . . , i p ) b e the o rdered sequence o f indices wher e u and v differ. By hypothesis, p/n < α . Put k 1 = i 1 and k j = i j − i j − 1 for 2 ≤ j ≤ p . W e now show that u can b e recov ered fro m v and roughly αf (1 /α ) n bits more. Indeed u ca n b e computed from ( v , k 1 , . . . , k p ), constructing the string which co in- cides with v ev erywhere but at pos itions k 1 , k 1 + k 2 , . . . , k 1 + . . . + k p , so K ( u ) < + K ( v ) + K ( k 1 ) + . . . + K ( k p ) < + K ( v ) + f ( k 1 ) + . . . + f ( k p ). Now, as f is a concav e increa sing function, one ha s: 1 p X j ≤ p f ( k j ) ≤ f   1 p X j ≤ p k j   = f  i p p  ≤ f  n p  THE DYNAMICS O F ALGORITHMICALL Y RANDOM P OINTS 21 As a result, 1 n K ( u ) ≤ 1 n K ( v ) + p n f  n p  + c n where c is some constant indep endent of u, v , n, p . As p/n < α < 1 / 2 and x 7→ xf (1 /x ) is increasing for x ≤ 1 / 2, one has : 1 n K ( u ) ≤ 1 n K ( v ) + αf (1 /α ) + c n Switc hing u and v gives the result ( c may be changed).  W e a re now able to prov e the other inequa lity . Pr o of of K µ ( x, T ) ≤ K ( x, T ) . Fix so me computable par tition ξ . W e s how that for any β > 0 there is s o me ǫ > 0 suc h that for every Martin-L¨ of ra ndom point x , K µ ( x, T | ξ ) ≤ K ( x, T , ǫ ) + β . As K ( x, T , ǫ ) inc r eases as ǫ → 0 + and β is a rbitrary , the inequality follows. First take α < 1 / 2 such that αf (1 / α ) < β , and remar k that lim ǫ → 0 + µ  ( ∂ ξ ) ǫ  = µ ( ∂ ξ ) = 0 Hence there is so me ǫ such that µ  ( ∂ ξ ) 2 ǫ  < α . F rom a sequence o f idea l p oints we will reco nstruct the symbolic orbit of a random p oint with a density of erro r s less than α . Lemma 5.0.1 will then a llow to conclude. W e define an algo rithm A ( ǫ, i 0 , . . . , i n − 1 ) with ǫ ∈ Q > 0 and i 0 , . . . , i n − 1 ∈ N which o utputs a word a 0 . . . a n − 1 on the a lphab et ξ . T o co mpute a j , A semi-decides in a dov etail picture: • s i j ∈ C for every C ∈ ξ , • s ∈ C for every s ∈ B ( s i j , ǫ ) and every C ∈ ξ . The first test which stops provides some C ∈ ξ : put a j = C . Let x b e a random p oint whose itera tes are covered by ξ , a nd s i 0 , . . . , s i n − 1 be ideal p oints which ǫ -shadow the first n iterates of x . W e claim that A will halt on ( ǫ, i 0 , . . . , i n − 1 ). Indeed, as T j x belo ngs to some C ∈ ξ , C ∩ B ( s i j , ǫ ) is a non-empty op en se t and then co nt ains at least o ne ideal p o int s , w hich will b e even tually dealt with. W e now compare the symbolic or bit o f x with the symbolic sequence computed by A . A discrepancy at rank j can app ea r only if T j x ∈ ( ∂ ξ ) 2 ǫ . Indeed, if T j x / ∈ ( ∂ ξ ) 2 ǫ then B ( T j x, 2 ǫ ) ⊆ C where C is the cell T j x b elong s to. As d ( s i j , T j x ) < ǫ , B ( s i j , ǫ ) ⊆ B ( x, 2 ǫ ) ⊆ C , so the algor ithm g ives the right cell. Now, as x is typical, lim sup n →∞ 1 n # { j < n : T j x ∈ ( ∂ ξ ) 2 ǫ } ≤ µ  ( ∂ ξ ) 2 ǫ  < α so there is some n 0 such that for all n ≥ n 0 , 1 n # { j < n : T j x ∈ ( ∂ ξ ) 2 ǫ } < α . This implies tha t for all n ≥ n 0 and ideal p oints s i 0 , . . . , s i n − 1 which ǫ -shadow the first n iter ates o f x and with minimal complex it y , the a lgorithm A ( ǫ, i 0 , . . . , i n − 1 ) pro duces a symbolic string u which differs from the symbolic orbit v of x of leng th n with a density of erro r s < α . As K ( u ) < + K ( ǫ ) + K n ( x, T , ǫ ) a nd αf (1 /α ) < β , 22 STEF ANO GALA T OLO, MA THIEU HOYR UP, AND CRIST ´ OBAL ROJAS applying lemma 5.0.1 gives: 1 n K ( ξ n ( x )) = 1 n K ( v ) ≤ 1 n K ( u ) + αf (1 /α ) + c n ≤ 1 n ( K n ( x, T , ǫ ) + K ( ǫ ) + c ′ ) + β + c n where c ′ is indepe ndent of n . T a king the lim sup as n → ∞ gives: K µ ( x, T | ξ ) ≤ K ( x, T , ǫ ) + β  Combining theor e ms 5.0.1 and 4.4.2 , w e obtain a version o f Brudno’s theor em (theorem 1 ) for Mar tin-L¨ of ra ndo m po int s. Corollary 5.0. 1. Le t T b e an er go dic endomorphism of t he c omputable pr ob abi lity sp ac e ( X , µ ) , wher e X is c omp act. Then for every Martin-L¨ of r ando m p oint x : K ( x, T ) = h µ ( T ) 6. Topological entropie s Bow en’s definition o f top olo gical entrop y is r eminiscent of the ca pa city (or b ox counting dimensio n) o f a to tally b ounded subset of a metric space. In order to find r elations with orbit complexity w e will also use a nother characteriz a tion of top ologica l e nt ropy , expr essing it as a kind of Hausdorff dimension. W e first present Bow en’s definition. In this sectio n, X is a metric space and T : X → X a contin uo us map. 6.1. En trop y as a capacit y. W e r ecall the definition: for n ≥ 0, let us define the distance d n ( x, y ) = max { d ( T i x, T i y ) : 0 ≤ i < n } and the Bow en ball B n ( x, ǫ ) = { y : d n ( x, y ) < ǫ } , whic h is op e n by contin uit y of T . Giv en a tota lly b ounded se t Y ⊆ X and num bers n ≥ 0 , ǫ > 0 , let N ( Y , n, ǫ ) be the minimal ca rdinality of a cov er of Y b y Bow en balls B n ( x, ǫ ). A set of p o int s E such that { B n ( x, ǫ ) : x ∈ E } is a cov er of Y is a lso called an ( n, ǫ )-spanning s et of Y . One then defines: h 1 ( T , Y , ǫ ) = lim sup n →∞ log N ( Y , n, ǫ ) n which is non-dec reasing a s ǫ → 0 , s o the following limit exists: h 1 ( T , Y ) = lim ǫ → 0 h 1 ( Y , T , ǫ ) . When X is compact, the top olo gic al entr opy o f T is h ( T ) = h 1 ( T , X ). It meas ur es the exp onential g rowth-rate of the num ber o f distinguishable o rbits o f the system. R emark 6.1 .1 . The topolo gical en tropy can b e defined using separ ated sets in- stead of op en cov ers: a subs et A of X is ( n, ǫ )-sepa rated if for a ny distinct p oints x, y ∈ A , d n ( x, y ) > ǫ . Let us define M ( Y , n, ǫ ) as the maximal ca rdinality o f an ( n, ǫ )-separa ted subset of Y . It is ea sy to see that M ( Y , n , 2 ǫ ) ≤ N ( Y , n , ǫ ) ≤ M ( Y , n, ǫ ), and hence h 1 ( T , Y ) can b e alter natively defined using M ( Y , n, ǫ ) in place of N ( Y , n, ǫ ). THE DYNAMICS O F ALGORITHMICALL Y RANDOM P OINTS 23 6.2. En trop y as a dimensi on. It is p ossible to define a top olo gical entropy whic h is an analog of Haus do rff dimensio n. His definition coincides with the cla ssical o ne in the compact case. Hausdorff dimension has stronger stability pro p erties tha n box dimension, which has imp or tant consequences, as we will see in what fo llows. W e r efer the reader to [ Pes98 ], [ HK02 ] for more details. Let X b e a metric space and T : X → X a c o ntin uo us map. The ǫ -size of E ⊆ X is 2 − s where s = sup { n ≥ 0 : diam( T i E ) ≤ ǫ for 0 ≤ i < n } . It measure s how long the orbits starting fro m E ar e ǫ -clos e. As ǫ dec r eases, the ǫ -size of E is non-decr easing. The 2 ǫ -s ize o f a Bowen ball B n ( x, ǫ ) is less than 2 − n . In a wa y tha t is r eminiscent from the definition of Hausdorff measure, let us define m s δ ( Y , ǫ ) = inf G ( X U ∈G ( ǫ -size( U )) s ) where the infimu m is taken ov er all countable cov ers G of Y by open sets of ǫ - size < δ . This quantit y is monotonically increasing as δ tends to 0, so the limit m s ( Y , ǫ ) := lim δ → 0 + m s δ ( Y , ǫ ) exists and is a supremum . There is a cr itical v alue s 0 such tha t m s ( Y , ǫ ) = ∞ for s < s 0 and m s ( Y , ǫ ) = 0 for s > s 0 . Let us define h 2 ( T , Y , ǫ ) as this critical v alue: h 2 ( T , Y , ǫ ) := inf { s : m s ( Y , ǫ ) = 0 } = sup { s : m s ( Y , ǫ ) = ∞} . As le ss and less cov ers a re allow ed when ǫ → 0 (the ǫ -size of sets do es not decreas e), the following limit exists h 2 ( T , Y ) := lim ǫ → 0 + h 2 ( T , Y , ǫ ) and is a supr e m um. In [ Pes98 ], it is pr ov ed that: Theorem 6.2.1. When Y is a T -invariant c omp act set , h 1 ( T , Y ) = h 2 ( T , Y ) . In particular , if the space X is co mpa ct, then h ( T ) = h 1 ( T , X ) = h 2 ( T , X ). 6.3. Orbit com plexit y vs en trop y . Now we prov e the main theorem of the s ec- tion: Theorem 6.3.1 (T o p ologica l ent ropy vs orbit c o mplexity) . L et X b e a c omp act c omputable met ric sp ac e, and T : X → X a c omputable map. Then h ( T ) = sup x ∈ X K ( x, T ) = sup x ∈ X K ( x, T ) . In order to prove this theorem, we define a n e ffective version of the top ologic a l ent ropy , which is strong ly rela ted to the complex ity of orbits. 6.3.1. Effe ctive entr opy as an effe ct ive dimension. Befo r e defining an effective ver- sion, we give a simple characterizatio n whic h w ill accommo da te to effectivisation. Definition 6 .3.1. A nul l s -c over of Y ⊆ X is a set E ⊆ N 3 such that: (1) P ( i,n,p ) ∈ E 2 − sn < ∞ , (2) for ea ch k , p ∈ N , the set { B n ( s i , 2 − p ) : ( i, n, p ) ∈ E , n ≥ k } is a cov er of Y . The idea is simple: every null s -cov er induces op en cov ers of a r bitrary small size and ar bitrary sma ll w eight. Remark that a ny null s - cov er of Y is also a null s ′ -cov er for all s ′ > s . 24 STEF ANO GALA T OLO, MA THIEU HOYR UP, AND CRIST ´ OBAL ROJAS Lemma 6 .3.1. h 2 ( T , Y ) = inf { s : Y has a nu l l s -c over } . Pr o of. Suppo s e s > h 2 ( T , Y ). W e fix p, k ∈ N a nd put ǫ = 2 − p and δ = 2 − k . As m s δ ( Y , ǫ ) = 0, there is a co ver ( U j,k,p ) j of Y by op en se ts of ǫ -siz e δ j,k,p < δ with P j δ s j,k,p < 2 − ( k + p ) . Let s i be any ideal point in U j,k,p . If δ j,k,p > 0, then δ j,k,p = 2 − n for so me n . If δ j,k,p = 0, take any n ≥ ( j + k + p ) /s . In b oth cases, U j,k,p is included in the B ow en ball B n ( s i , ǫ ). W e define E k,p as the set of ( i, n, p ) obtained this way , a nd E = S k,p E k,p . By construction, for ea ch k , p , { B n ( s i , 2 − p ) : ( i, n, p ) ∈ E , n ≥ k } is a cov er o f Y . Moreov er, P ( i,n,p ) ∈ E k,p 2 − sn ≤ P j δ s j,k,p + P j 2 − ( j + k + p ) ≤ 2 − ( k + p )+2 , so P ( i,n,p ) ∈ E 2 − sn < ∞ . Conv ersely , if Y ha s a null s - cov er E , take ǫ, δ > 0 and p, k such that ǫ > 2 − p +1 and δ > 2 − k . F or all k ′ ≥ k , the family { B n ( s i , 2 − p ) : ( i, n, p ) ∈ E , n ≥ k ′ } is a cov er of Y by op en sets o f ǫ - size smalle r than 2 − n ≤ δ . Mo reov er, P ( i,n,p ) ∈ E ,n ≥ k ′ 2 − sn tends to 0 as k ′ grows, so m s δ ( Y , ǫ ) = 0 . It follows that s ≥ h 2 ( T , Y ).  By an effe ctive n ull s -cov er, we mean a null s -c ov er E which is a r.e. subset o f N 3 . Definition 6 .3.2. The effe cti ve top olo gic al entr opy of T on Y is defined by h eff 2 ( T , Y ) = inf { s : Y has an effective null s -cover } As less null s -covers are a llow ed in the e ffective version, h 2 ( T , Y ) ≤ h eff 2 ( T , Y ). Of course, if Y ⊆ Y ′ then h eff 2 ( T , Y ) ≤ h eff 2 ( T , Y ′ ). W e now pr ov e: Theorem 6. 3.2 (Effectiv e top ologica l entrop y vs low er orbit complexity) . L et X b e an effe ctive metric sp ac e and T : X → X a c ont inuous map. F or al l Y ⊆ X , h eff 2 ( T , Y ) = s up x ∈ Y K ( x, T ) which implies in pa rticular that h eff 2 ( T , { x } ) = K ( x, T ): the restriction of the system to a sing le orbit may have p ositive effectiv e top o lo gical entrop y . This kind of result has already b een obta ined for the Hausdorff dimension of subsets of the Ca nt or space, proving that the effective dimension of a set A is the supremum o f the low er growth-rate of Kolmo gorov c o mplexity of sequences in A (whic h corresp onds to theorem 6.3.2 for sub-shifts). This r emark able prop erty is a counterpart of the countable stability prop erty of Haus dorff dimensio n (dim Y = sup i dim Y i when S i Y i = Y ) (s e e [ CH94 ], [ May01 ], [ Lut03 ], [ Rei0 4 ], [ Sta05 ]). Theorem 6.3.2 is a direct consequence o f the tw o following lemmas. Lemma 6 .3.2. L et α ≥ 0 and Y α = { x : K ( x, T ) ≤ α } . One has h eff 2 ( T , Y α ) ≤ α . Pr o of. Let β > α b e a ratio na l n um ber. W e define the r.e. set E = { ( i, n, p ) : K ( i, n, p ) < β n } . Let p ∈ N a nd ǫ = 2 − p . If x ∈ Y α then K ( x, T , ǫ ) ≤ α < β so for infinitely man y n , there is some s i such that x ∈ B n ( s i , ǫ ) and K ( i, n, p ) < β n . So for a ll k , { B n ( s i , 2 − p ) : ( i, n, p ) ∈ E , n ≥ k } covers Y α . Moreover, P ( i,n,p ) ∈ E 2 − β n ≤ P ( i,n,p ) ∈ E 2 − K ( i,n,p ) ≤ 1. E is then an effective null β -cov er of Y α , so h eff 2 ( T , Y α ) ≤ β . And this is true for every rationa l β > α .  Lemma 6 .3.3. L et Y ⊆ X . F or al l x ∈ Y , K ( x, T ) ≤ h eff 2 ( T , Y ) . THE DYNAMICS O F ALGORITHMICALL Y RANDOM P OINTS 25 Pr o of. Let s > h eff 2 ( T , Y ): Y has an effective null s -cov er E . As P ( i,n,p ) ∈ E 2 − sn < ∞ , b y the co ding theorem K ( i, n, p ) ≤ sn + c for some co nstant x , which do es not depe nd on i, n, p . If x ∈ Y , then for each p, k , x is in a ball B n ( s i , 2 − p ) for some n ≥ k with ( i, n, p ) ∈ E . Then K n ( x, T , 2 − p ) ≤ sn + c fo r infinitely many n , so K ( x, T , 2 − p ) ≤ s . As this is true for all p , K ( x, T ) ≤ s . As this is true fo r all s > h eff 2 ( T , Y ), we can conclude.  Pr o of of the or em 6.3.2 . By lemma 6.3.3 , α := s up x ∈ Y K ( x, T ) ≤ h eff 2 ( T , Y ). Now, as Y ⊆ Y α , h eff 2 ( T , Y ) ≤ h eff 2 ( T , Y α ) ≤ α by lemma 6.3.2 .  The definition o f an effective null α -cov er inv olves a summable computable se- quence. The univ ersality of the sequence 2 − K ( i ) among summable lo w er semi- computable sequence s is a t the core of the pro o f of the preceding theorem, which states that there is a universal effective n ull α -cover, for every α ≥ 0. In other words, ther e is a maximal set of effectiv e top o logical en tropy ≤ α , and this set is Y α = { x ∈ X : K ( x, T ) ≤ α } . The definition of the top olo gical entropy as a capac it y co uld b e also made effec- tive, res tr icting to effective cov ers. Clas sical c apacity do e s no t s ha re with Hausdor ff dimension the countable stability . F or the same reaso n, its effective version is no t related w ith the orbit complexit y as strongly as the effective top o logical ent ropy is. Nev ertheless, a w eaker r elation holds, which is sufficien t for our purpo se: the upper complexity of orbits is b ounded by the effective capacity . W e do not develop this and only sta te the needed prop erty (which implicitly uses the fact that the effective ca pacity coincides with the classica l capacity for a compact computable metric space): Lemma 6.3.4 . L et X b e a c omp act c omput able metric sp ac e. F or al l x ∈ X , K ( x, T ) ≤ h 1 ( T , X ) . Pr o of. W e fir st constr uct a r.e. set E ⊆ N 3 such that for each n , p , { s i : ( i, n, p ) ∈ E } is a ( n, 2 − p )-spanning set and a ( n, 2 − p − 2 )-separated s et. Let us fix n and p and enumerate E n,p = { i : ( i, n, p ) ∈ E } , in a unifor m way . The algo rithm s tarts with S = ∅ a nd i = 0. A t step i it analyzes s i and dec ide s to add it to S or no t, and go es to step i + 1. E n,p is the set o f p o ints which are e ventually added to S . Step i : for each ideal p o int s ∈ S , test in par a llel d n ( s i , s ) < 2 − p − 1 and d n ( s i , s ) > 2 − p − 2 : at least one of them must stop. If the first one s tops first, r eject s i and g o to Step i + 1. If the seco nd one stops first, g o on with the o ther p o int s s ∈ S : if a ll S has b een consider ed, then add s i to S and go to Step i + 1. By construction, the set o f selected ideal p oints forms a ( n, 2 − p − 2 )-separated set. If ther e is x ∈ X which is at distance at least 2 − p from every selected p o int, then let s i be an ideal po int s i with d n ( x, s i ) < 2 − p − 1 : s i is at distance a t least 2 − p − 1 from every selected p oint, so at step i it must have b een selec ted, as the fir s t test co uld not stop. This is a co nt radiction: the s e le cted po int s form a ( n, 2 − p )-spanning set. F ro m the prop erties of E n,p it follows that N ( X, n, 2 − p ) ≤ | E n,p | ≤ M ( X, n, 2 − p − 2 ), and then sup p  lim sup 1 n log | E n,p |  = h 1 ( T , X ) If β > h 1 ( T , X ) is a ratio nal num ber, then for eac h p , there is k ∈ N suc h tha t log | E n,p | < β n for all n ≥ k . 26 STEF ANO GALA T OLO, MA THIEU HOYR UP, AND CRIST ´ OBAL ROJAS Now, for s i ∈ E n,p , K ( i ) < + log | E n,p | + 2 log log | E n,p | + K ( n, p ) by propo si- tion 2.7.1 . 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