A construction of strictly ergodic subshifts having entropy dimension (joint with K. K. Park and J. Lee (Ajou Univ.))

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1 A construction of strictly ergodic subshifts having entropy dimension (joint with K. K. Park and J. Lee (Ajou Univ.)) Uijin Jung Ajou University, Suwon, South Korea Pingree Park Conference, July 5, 204 Entropy Dimension (U.Jung) Uijin Jung (Ajou U.)

2 Topological and metric entropy Let (X, T ) be a topological dynamical system. The topological entropy of (X, T, U) given an open cover U is n h(t, U) = lim n n log N( T U), where N(V) is the minimum number of open sets of V covering X. i=0

3 Topological and metric entropy Let (X, T ) be a topological dynamical system. The topological entropy of (X, T, U) given an open cover U is n h(t, U) = lim n n log N( T U), where N(V) is the minimum number of open sets of V covering X. i=0 Let µ be an T -invariant probability measure of (X, T ). The metric entropy of (X, T, µ) given a partition P is n h µ (T, P) = lim n n log H( T P) i=0 ( = lim n n log A n i=0 T P ) µ(a) log µ(a).

4 Positive entropy systems Let h µ (T ) = h and P be a generator, i.e., h µ (T ) = h µ (T, P). The growth rates of orbits under P are exponential. T has a Bernoulli factor. T has Shannon-McMillan-Brieman property: µ(p n 0 (x)) 2 nh. T has Ornstein-Weiss return time property: n log R n(x) h for µ-a.e. x X If a system has the completely positive entropy, then it is disjoint from entropy zero systems (in the sense of Furstenberg). geodesic flow, Axiom A diffeomorphisms, Markov shifts,...

5 Zero entropy systems irrational rotation (Sturmian shifts), interval exchanges, horocycle flow, substitutional shifts,... General higher dimensional group action of zero entropy has many subactions of positive entropies. Growth rates of orbits are subexponential, but many examples we know well have poloynomial growth rates. How to measure intermediate (subexponential but not polynomial) complexity? sequence entropy (Kushnirenko, Goodman) Slow type entropy (Katok, Thouvenot) Entropy dimension (Dou, Huang and Park)

6 Topological Entropy Dimension Let (X, T ) be a topological dynamical system. Let U be an open cover of X. Then D(T, U) = inf{β 0 : lim n n β log N( T U) = 0}. The (topological) upper entropy dimension is defined by D(T ) = sup{d(t, U) : U open covers }. i=0

7 Topological Entropy Dimension Let (X, T ) be a topological dynamical system. Let U be an open cover of X. Then D(T, U) = inf{β 0 : lim n n β log N( T U) = 0}. i=0 The (topological) upper entropy dimension is defined by D(T ) = sup{d(t, U) : U open covers }. Similarly, the (topological) lower entropy dimension is defined by D(T ) = sup{d(t, U) : U open covers }, where D(T, U) = inf{β 0 : lim n n β log N( T U) = 0}. If D(T ) = D(T ) = α, then we say (X, T ) has entropy dimension α. It is a conjugacy invariant. i=0

8 Topological and Metric Entropy Dimension It is natural to define metric entropy analogously, by using D µ (T, P) = inf{β : lim n n β H( T P) = 0}, however D µ (T ) is always, so we have to find another invariant. i=0

9 Topological and Metric Entropy Dimension It is natural to define metric entropy analogously, by using D µ (T, P) = inf{β : lim n n β H( T P) = 0}, however D µ (T ) is always, so we have to find another invariant. Let S = {s < s 2 < } be an increasing sequence of integers. Define D(S) = inf{β 0 : lim sup n (s n) = 0} and similarly D(S) and D(S). β Given (X, T, U) and an S = {s < s 2 < }, S is called an entropy generating sequence of U if i=0 lim n n log N( T si U) > 0. If U is a finite generating open cover of (X, T ), then there is an entropy generating sequence S with D(X, T, U) = D(S). i=0

10 Metric Entropy Dimension Given (X, T, U) and an S = {s < s 2 < }, S is called an entropy generating sequence of U if lim n n log N( T si U) > 0. If U is a finite generating open cover of (X, T ), then there is an entropy generating sequence S with D(X, T, U) = D(S). i=0 Given (X, T, µ) with a partition P, the metric entropy dimension is defined by D µ (T, P) = sup{d(s)} and D µ (X, T ) = sup D µ (T, P) and similarly for D µ (T ) and D µ (T ). Now D µ (T ), D µ (T ) and D µ (T ) are isomorphism invariants and share many properties analogous to usual metric entropy. P

11 New Result Theorem Let α (0, ). There is a strictly ergodic subshift X with the following properties:. both the topological and the metric entropy dimension of X are α, 2. X is weakly mixing and rigid, 3. X has a variant of Shannon-McMillan-Brieman property as follows: For each x X, lim sup n log µ(pn(x)) nβ has the critical value β = /2. There is a set ˆX X with full measure with the property that for any τ [0, α], there is an increasing sequence {n i} i N such that for each x ˆX, has the critical value β = τ. lim i (n log i) β µ(pn (x)) i 4. Property 3 also holds if we replace log µ(p n (x)) with log R n (x), where R n (x) is the first return time of x to the cylinder x [0,n).

12 Sketch of the construction for α = /2 Let A be an alphabet of cardinality >. Take any large l N and a set C B l (A Z ) with N = C 2 m N large enough. This C is our basic block system for the first step. and

13 Sketch of the construction for α = /2 Let A be an alphabet of cardinality >. Take any large l N and a set C B l (A Z ) with N = C 2 m N large enough. This C is our basic block system for the first step. and Assume, for the j-th step, a set C j of words of length l j has been constructed with N j = C j so that log N j 2 j lj. Give an ordering on C j and write C j = {u (j), u(j) 2,, u(j) N j }.

14 Sketch of the construction for α = /2 Let A be an alphabet of cardinality >. Take any large l N and a set C B l (A Z ) with N = C 2 m N large enough. This C is our basic block system for the first step. and Assume, for the j-th step, a set C j of words of length l j has been constructed with N j = C j so that log N j 2 j lj. Give an ordering on C j and write C j = {u (j), u(j) 2,, u(j) N j }. Make a block system C j+ such that typical element u C j+ is given by u is a word obtained from the concatenation u (j) u(j) 2 u (j) N j by possibly permuting words u (j) 2, u (j) 2 3,, u (j) 2 [ N j] 2. Hence an element in C j+ is of the form u (j) u(j) 2 u(j) 3 u(j) γ(4) u(j) 5 u (j) 8 u(j) γ(9) u(j) 0 u(j) N j, where γ is a permutation on the set {i 2 : < i N j }.

15 Sketch of the construction for α = /2 We have C j = {u (j),, u(j) N j } with N j = C j, and l j = u for all u C j. We call a block u (j) k C j permuted (or unstable) if k = i 2 for some < i N j and unpermuted (or stable) otherwise. By using N j+ = ( N j )! and l j+ = l j N j, inductively we have log(n j+ ) 2 j l j+, so we can construct C k for all k N.

16 Sketch of the construction for α = /2 We have C j = {u (j),, u(j) N j } with N j = C j, and l j = u for all u C j. We call a block u (j) k C j permuted (or unstable) if k = i 2 for some < i N j and unpermuted (or stable) otherwise. By using N j+ = ( N j )! and l j+ = l j N j, inductively we have log(n j+ ) 2 j l j+, so we can construct C k for all k N. The sequence {log N j } has the property that log N j lj 2 j 0 but log N j (l j ) /2 ɛ 2 j (l j+) ɛ for all ɛ > 0. Since u (j) is a prefix of u (j+) for each j, there is a unique limit point w {0, } N of the sequence {u (j) } j N. Let X + be the orbit closure of w and X the (usual) inverse limit of X +. (X, σ) is minimal and uniquely ergodic.

17 Sketch of the construction for α = /2 The limit lim j (l j) β log N j has the critical value β = /2, that is, log N j lj 2 j 0 but log N j (l j ) /2 ɛ 2 j (l j+) ɛ for all ɛ > 0, but it does not say that the topological dimension is /2.

18 Sketch of the construction for α = /2 The limit lim j (l j) β log N j has the critical value β = /2, that is, log N j lj 2 j 0 but log N j (l j ) /2 ɛ 2 j (l j+) ɛ for all ɛ > 0, but it does not say that the topological dimension is /2. For l N, take j with l j l < l j+ = l j N j. When l = k l j, log B l (X) log( l N klj j P k ) log k N j = klj 2 log N j. l j

19 Sketch of the construction for α = /2 The limit lim j (l j) β log N j has the critical value β = /2, that is, log N j lj 2 j 0 but log N j (l j ) /2 ɛ 2 j (l j+) ɛ for all ɛ > 0, but it does not say that the topological dimension is /2. For l N, take j with l j l < l j+ = l j N j. When l = k l j, log B l (X) log( l N klj j P k ) log k N j = klj 2 log N j. l j Now for each u B l (X), u is a subblock of a block Q Q k+ formed by concatenating k + blocks in C j, and the maximal number of permuted positions in the j-th level is k +. Hence log B l (X) log(l j+ N l klj j P k+ ) klj log l j+ + klj log N j k+ 3 2 l j log N j from which the system X has topological dimension /2.

20 Sketch of the construction for α = /2 For x X and j N, write x = B (j) B(j) 0 B(j) B(j) 2 with B (j) i C j. Denote by B (j) 0 the block occurring at positions containing the 0-th coordinate. (Note: B (j) 0 is a subblock of B (j+) 0 ) Recall that we say the block B (j) i (x) unpermuted if it belongs in the un-permuted parts of any C j+ -blocks. Otherwise permuted.

21 Sketch of the construction for α = /2 For x X and j N, write x = B (j) B(j) 0 B(j) B(j) 2 with B (j) i C j. Denote by B (j) 0 the block occurring at positions containing the 0-th coordinate. (Note: B (j) 0 is a subblock of B (j+) 0 ) Recall that we say the block B (j) i (x) unpermuted if it belongs in the un-permuted parts of any C j+ -blocks. Otherwise permuted. For µ-a.e. x X, B (j) 0 (x) are unpermuted for all large j. If u u k C j are concatenation of unpermuted blocks occurring in C j+, we have µ(u ) = = µ(u k ) = µ(u u k ). By considering the maximal number of permuted blocks in C j occurring in B (j) 0 B (j) k is k, we have µ(p n (x)) (l j ( N j From which we have, for each x X, lim sup n n β log µ(p n(x)) k+2)). has the critical value β = /2.

22 Sketch of the construction for α = /2 For µ-a.e. x X, B j (x) are stable for all large j. Let L j be the set of all x X such that B (j) 0 (x) is a permuted block in C j. Then L = k= j k L j has measure 0. Let ˆX L C be the set of all x L C such that for all large j, (a) B (j) 0 (x) lies in a consecutive unpermuted blocks, and (b) B (j) 0 (x) does not lie in the last /p portion of that consecutive unpermuted blocks. We also have µ( ˆX) =. Let τ [0, α] be given. Since lim j (l j) β log N j has the critical value β = /2 and lim j (l j 4 N j) β log N j = 0 for all β > 0, there is {n j } with () l j n j l j 4 N j and (2) lim j (n j) β log N j has the critical value β = τ. Then lim j (n j) β log µ(p nj (x)) has the critical value τ. Return time property can be shown similarly.

23 A system of entropy dimension α Theorem Let α (0, ). There is a strictly ergodic subshift X with the following properties:. both the topological and the metric entropy dimension of X are α, 2. X is weakly mixing and rigid, 3. X has a variant of Shannon-McMillan-Brieman property as follows: For each x X, lim sup n log µ(pn(x)) nβ has the critical value β = /2. There is a set ˆX X with full measure with the property that for any τ [0, α], there is an increasing sequence {n i} i N such that for each x ˆX, has the critical value β = τ. lim i (n log i) β µ(pn (x)) i 4. Property 3 also holds if we replace log µ(p n (x)) with log R n (x), where R n (x) is the first return time of x to the cylinder x [0,n).

24 Thank You!