Symbolic Dynamics: Entropy = Dimension = Complexity

Transcription

1 Symbolic Dynamics: Entropy = Dimension = omplexity Stephen G. Simpson Pennsylvania State University simpson@math.psu.edu Worshop on Infinity and Truth Institute for Mathematical Sciences National University of Singapore July 5 9, 0

2 Symbolic dynamics. Let G be (N d,+) or (Z d,+) where d. Let A be a finite set of symbols. We endow A with the discrete topology and A G with the product topology. The shift action of G on A G is given by (S g x)(h) = x(g +h) for g,h G and x A G. A subshift is a nonempty set X A G which is topologically closed and shift-invariant, i.e., x X implies S g x X for all g G. Symbolic dynamics is the study of subshifts. If X A G and Y B G are subshifts, a shift morphism from X to Y is a continuous mapping Φ : X Y such that Φ(S g x) = S g Φ(x) for all x X and g G. By compactness, any shift morphism Φ is given by a bloc code, i.e., a finite mapping φ : A F B where F is a finite subset of G and Φ(x)(g) = φ(s g x F) for all x X and g G.

3 Some new (!?!) results on subshifts: Let d be a positive integer, let A be a finite set of symbols, and let X be a nonempty subset of A G where G is N d or Z d. The Hausdorff dimension, dim(x), and the effective Hausdorff dimension, effdim(x), are defined as usual with respect to the standard metric ρ(x,y) = F n where n is as large as possible such that x F n = y F n. Here F n is {,...,n} d if G = N d, or { n,...,n} d if G = Z d. We first state some old results.. effdim(x) = sup x X effdim(x).. effdim(x) = liminf n K(x F n ). F n 3. effdim(x) = dim(x) provided X is effectively closed, i.e., Π 0. Here K denotes Kolmogorov complexity. 3

4 Theorem (Simpson 00). Assume that X is a subshift, i.e., X is closed and shift-invariant. Then effdim(x) = dim(x) = ent(x). Moreover dim(x) lim sup n and K(x F n ) F n for all x X, dim(x) = lim n K(x F n ) F n for many x X. Remar. Here ent(x) denotes entropy, log ent(x) = lim {x F n x X}. n F n This is nown to be a conjugacy invariant. Note. In the above theorem, there is no finiteness or computability hypothesis on the subshift X. Moreover, X can be a G-subshift where G is N d or Z d for any positive integer d. 4

5 Remar. The proof of Theorem involves ergodic theory (Shannon/McMillan/Breiman, the Variational Principle, etc.) plus a combinatorial argument which is similar to the proof of the Vitali overing Lemma. Remar. Theorem seems so fundamental that it could have been noticed long ago. Nevertheless, I have not been able to find it in the literature. So far as I can tell, everything in the theorem is new, except the following result of Furstenberg 967: dim(x) = ent(x) provided G = N. The proof of this special case is much easier. Remar. Theorem is an outcome of my discussions at Penn State during February April 00 with many people including John lemens, Mie Hochman, Dan Mauldin, Jan Reimann, and Sasha Shen. 5

6 Some new results on partial randomness. Bacground: Let h : {0,} [0, ) be computable. Say that x {0,} N is strongly h-random if c n(ka(x {,...,n}) > h(x {,...,n}) c). Here KA = a priori Kolmogorov complexity, i.e., KA(σ) = log m(σ) where m is a universal left r.e. semimeasure. Note that KA is similar but not identical to KP = prefix-free Kolmogorov complexity. Strong h-randomness has been studied by alude/staiger/terwijn, APAL, 006, and Reimann/Stephan, Proceedings of the 9th Asian Logic onference, 006. When h(σ) = s σ this is closely related to effective Hausdorff dimension as pioneered in Reimann s Ph.D. thesis,

7 New results: Theorem (Simpson 0). Assume that x is strongly h-random and x T y where y is Martin-Löf random relative to z. Then x is strongly h-random relative to z. Remar. The special case h(σ) = σ, i.e., when x is Martin-Löf random, is due to Joseph S. Miller and Liang Yu, TAMS, 008. Theorem 3 (Simpson 0). Let I be a countable index set. Assume that ( i I)(x i is strongly h i -random). Then, we can find a PA-oracle z such that ( i I)(x i is strongly h i -random rel. to z). Remar. The special case when h(σ) = σ and I = is due to Rodney G. Downey and Denis Hirschfeldt and Joseph S. Miller and André Nies, JML, 005. Remar. I do not now whether Theorems and 3 hold with KA replaced by KP. 7

8 Degrees of unsolvability (Muchni). Let X be any set of reals. We view X as a mass problem, viz., the problem of finding some x X. In order to interpret finding, we use Turing s concept of computability. Accordingly, we say that X is algorithmically solvable if X contains some computable real, or in other words, X. Similarly, we say that X is algorithmically reducible to Y if each y Y can be used as a Turing oracle to compute some x X. The degree of unsolvability of X, deg(x), is the equivalence class of X under mutual algorithmic reducibility. Reference: Albert A. Muchni, On strong and wea reducibilities of algorithmic problems, Sibirsii Matematichesii Zhurnal, 4, 963, 38 34, in Russian. 8

9 I have been applying recursion-theoretic concepts such as Muchni degrees and Kolmogorov complexity to obtain new results in symbolic dynamics. Muchni degrees of subshifts. A subshift X is of finite type if it is given by a finite set of excluded finite configurations: X = {x A G ( g G)(S g x F / E)} where E and F are finite. Recall that E w is the lattice of Muchni degrees of nonempty Π 0 classes, in antor space (or in Euclidean space). Recall also that E w includes many specific, natural degrees which are associated with foundationally interesting topics. A picture of E w is on slides, 4,...,. Theorem 4 (Simpson 007). The Muchni degrees in E w are precisely the Muchni degrees of Z -subshifts of finite type. 9

10 Proof. One direction is trivial, because subshifts of finite type may be viewed as Π 0 classes. My proof of the other direction uses tiling techniques which go bac to Berger 966, Robinson 97, Myers 974. Another proof, due to Durand/Romashcheno/Shen 008, uses self-replicating tile sets. orollary (Simpson 007). We can construct an infinite family of Z -subshifts of finite type which are strongly independent with respect to shift morphisms, etc. Proof. This follows from the existence of an infinite independent set of degrees in E w. The existence of such degrees is proved by means of a priority argument. Thus we have an application of recursion theory (tiling methods plus priority argument) to prove a result in symbolic dynamics which does not mention computability concepts. 0

11 A possibly interesting research program: Given a subshift X, explore the relationship between the dynamical properties of X and the degree of unsolvability of X, i.e., its Muchni degree, deg(x). For example, the entropy of X is a well-nown dynamical property which serves as an upper bound on the complexity of orbits. In particular ent(x) > 0 implies ( x X)(x is not computable). By contrast, the degree of unsolvability of X serves as a lower bound on the complexity of orbits. For instance, deg(x) > 0 ( x X)(x is not computable). Theorem (Hochman). If X is of finite type and minimal (i.e., every orbit is dense), then deg(x) = 0. More generally, the theorem holds for all subshifts, not necessarily of finite type. Π 0

12 inf(r,) = deg (PA) w r = d = d s g d inf(a,) α+ α A picture of E w. Each blac dot except inf(a, ) represents a specific, natural degree in E w. We shall explain some of these degrees. 0

13 Two subshifts are said to be conjugate if they are topologically isomorphic, i.e., there is a shift isomorphism between them. The basic problem of symbolic dynamics is: classify subshifts up to conjugacy invariance. Muchni degrees can help, because the Muchni degree of a subshift is a conjugacy invariant. In particular, each degree in E w including 0,, r, d, d, d, s, g, inf(r,), inf(b α,), and even inf(a,) may be viewed as a conjugacy invariant for subshifts of finite type. It is interesting to compare the Muchni degree of a subshift X with other conjugacy invariants, e.g., the entropy of X. Generally speaing, the Muchni degree of X represents a lower bound on the complexity of the orbits, while the entropy of X is an upper bound on the complexity of these same orbits. 3

14 inf(r,) = deg (PA) w r = d = d s g d inf(a,) α+ α A picture of E w. Here a = any r.e. degree, r = randomness, b = LR-reducibility, = complexity, d = diagonal nonrecursiveness. 0 4

15 We now explain some degrees in E w. The top degree in E w is = deg(pa) where PA is the problem of finding a complete consistent theory which includes Peano arithmetic (or ZF, etc.). We also have inf(a,) E w where a is any recursively enumerable Turing degree. Moreover, a < b implies inf(a,) < inf(b,) We have r E w where r = deg(mlr), MLR = {x N x is Martin-Löf random}). We also have inf(r,) E w where r = deg({x N x is -random}), i.e., random relative to the halting problem. Also d E w where d = deg({f f is diagonally nonrecursive}), i.e., n(f(n) ϕ n (n)). 5

16 inf(r,) = deg (PA) w r = d = d s g d inf(a,) α+ α A picture of E w. Here a = any r.e. degree, r = randomness, b = LR-reducibility, = complexity, d = diagonal nonrecursiveness. 0 6

17 Let = {g N N g is recursive}. Let be any nice subclass of. For instance =, or = {g g is primitive recursive}. We have d E w where d = deg({f N N f is diagonally nonrecursive and -bounded}), i.e., ( g ) n(f(n) < g(n)). Also, d = deg({x N x is -complex}, i.e., ( g ) n(k(x {,...,g(n)}) n)}). Moreover, d < d whenever contains a function which dominates all functions in. For x N let effdim(x) = the effective Hausdorff dimension of x, i.e., effdim(x) = lim inf n K(x {,...,n}). n Given a right recursively enumerable real number s <, we have s E w where s = deg({x N effdim(x) > s}). Moreover, s < t implies s < t (Miller). 7

18 inf(r,) = deg (PA) w r = d = d s g d inf(a,) α+ α A picture of E w. Here a = any r.e. degree, r = randomness, b = LR-reducibility, = complexity, d = diagonal nonrecursiveness. 0 8

19 More generally, let g : N [, ) be an unbounded computable function such that g(n) g(n+) g(n)+ for all n. For example, g(n) could be n/ or n/3 or n or 3 n or logn or logn+loglogn or log log n or the inverse Acermann function. Define g = deg({x N x is g-random}), i.e., c n(kp(x {,...,n} g(n) c). Theorem 5 (Hudelson 00). g < h provided g(n)+logg(n) h(n) for all n. In other words, there exists a g-random real with no h-random real Turing reducible to it. This is a generalization of Miller s theorem on the difficulty of information extraction. References: Phil Hudelson, Mass problems and initial segment complexity, in preparation. Joseph S. Miller, Extracting information is hard, Advances in Mathematics, 6, 0,

20 inf(r,) = deg (PA) w r = d = d s g d inf(a,) α+ α A picture of E w. Here a = any r.e. degree, r = randomness, b = LR-reducibility, = complexity, d = diagonal nonrecursiveness. 0 0

21 Letting z be a Turing oracle, define MLR z = {x N x is random relative to z} and K z (τ) = the prefix-free Kolmogorov complexity of τ relative to z. Define y LR z MLR z MLR y and y LK z c τ (K z (τ) K y (τ)+c). Theorem (Miller/Kjos-Hanssen/Solomon). We have y LR z y LK z. For each recursive ordinal number α, let 0 (α) = the αth iterated Turing jump of 0. Thus 0 () = the halting problem, and 0 (α+) = the halting problem relative to 0 (α), etc. This is the hyperarithmetical hierarchy. We embed it naturally into E w as follows. Theorem 6 (Simpson 009). 0 (α) LR z every Σ 0 α+ set includes a Σ0,z set of the same measure. Moreover, letting b α = deg({z 0 (α) LR z}) we have inf(b α,) E w and inf(b α,) < inf(b α+,).

22 inf(r,) = deg (PA) w r = d = d s g d inf(a,) α+ α A picture of E w. Here a = any r.e. degree, r = randomness, b = LR-reducibility, = complexity, d = diagonal nonrecursiveness. 0

23 History: Kolmogorov 93 developed his calculus of problems as a nonrigorous yet compelling explanation of Brouwer s intuitionism. Later Medvedev 955 and Muchni 963 proposed Medvedev degrees and Muchni degrees as rigorous versions of Kolmogorov s idea. Some references: Stephen G. Simpson, Mass problems and randomness, Bulletin of Symbolic Logic,, 005, 7. Stephen G. Simpson, An extension of the recursively enumerable Turing degrees, Journal of the London Mathematical Society, 75, 007, Stephen G. Simpson, Mass problems and intuitionism, Notre Dame Journal of Formal Logic, 49, 008, Stephen G. Simpson, Mass problems and measure-theoretic regularity, Bulletin of Symbolic Logic, 5, 009, Stephen G. Simpson, Medvedev degrees of -dimensional subshifts of finite type, to appear in Ergodic Theory and Dynamical Systems. Stephen G. Simpson, Entropy equals dimension equals complexity, 00, in preparation. THE END. THANK YOU! 3