Henk Bruin (University of Vienna) Carlo Carminati (University of Pisa) Stefano Marmi (Scuola Normale di Pisa) Alessandro Profeti (University of Pisa)

Transcription

1 Self-similarity in the entropy graph for a family of piecewise linear maps. Henk Bruin (University of Vienna) Carlo Carminati (University of Pisa) Stefano Marmi (Scuola Normale di Pisa) Alessandro Profeti (University of Pisa) Salzburg, February 2018

2 The family Q γ For fixed slope s > 1 and γ [0, 1], take: { x + 1, x γ Q γ (x) = 1 + s(1 x), x > γ 1 γ 1 There is matching if there are κ ± 1 such that Q κ+ γ (γ + ) = Q κ γ (γ ), (Q κ+ γ ) (γ + ) = (Q κ γ ) (γ ). The number is the matching index. = κ + κ

3 The family Q γ Facts about matching for Q γ (proved in our paper, not in this talk): Matching holds for an open dense set of full Lebesgue measure.

4 The family Q γ Facts about matching for Q γ (proved in our paper, not in this talk): Matching holds for an open dense set of full Lebesgue measure. Matching occurs for every s-adic rational: γ = ps q for p, q N. (In this case Q n γ (γ + ) = Q n γ (γ ) = 1 is fixed for all n large enough.)

5 The family Q γ Facts about matching for Q γ (proved in our paper, not in this talk): Matching holds for an open dense set of full Lebesgue measure. Matching occurs for every s-adic rational: γ = ps q for p, q N. (In this case Q n γ (γ + ) = Q n γ (γ ) = 1 is fixed for all n large enough.) The non-matching set E has Hausdorff dimension dim H (E) = 1, but dim H (E \ [0, δ)) < 1 for every δ > 0.

6 The family Q γ Metric and topological entropy for s=2 0.5 topological metric topological metric Figure: Topological & metric entropies of Q γ for s = 2 as functions of γ.

7 Metric and topological entropy for s=2 topological metric topological metric The family Q γ Theorem: Topological and metric entropy decreasing if < 0; h µ (Q γ ) and h top (Q γ ) are constant if = 0; increasing if > 0, as function of γ within matching intervals. (Recall = κ + κ.)

8 Plan of this talk Aim of this talk: Explain why matching happens;

9 Plan of this talk Aim of this talk: Explain why matching happens; Describe the matching intervals (components of [0, 1] \ E) precisely pseudo-centers period doubling cascades computing the matching index

10 Plan of this talk Aim of this talk: Explain why matching happens; Describe the matching intervals (components of [0, 1] \ E) precisely pseudo-centers period doubling cascades computing the matching index Tuning windows and explanation of the self-similarity of the entropy graphs.

11 Why there is matching Let g(x) := s(1 x) mod 1 and R : (0, 1) (0, 1) be the first return of Q γ to [0, 1). Lemma: R(x) = { g(x) if x (0, γ) g 2 (x) if x (γ, 1) γ = 1 8 s = 2

12 Why there is matching γ = 1 8 Lemma: For fixed γ [0, 1], the following conditions are equivalent: (i) g k (γ) < γ for some k N; (ii) matching holds for γ. In other words, the bifurcation set is E = {γ [0, 1] : g k (γ) γ k N}. R

13 Pseudo-centers Motivation: Find exact formulas for matching intervals J and their matching indices for slope s = 2 (also works for 2 s N).

14 Pseudo-centers Motivation: Find exact formulas for matching intervals J and their matching indices for slope s = 2 (also works for 2 s N). Let Q dyd be the set of dyadic rationals in (0, 1]. Definition The pseudo-center of an interval J (0, 1) is the (unique) dyadic rational ξ Q dyd with minimal denominator.

15 Pseudo-centers Motivation: Find exact formulas for matching intervals J and their matching indices for slope s = 2 (also works for 2 s N). Let Q dyd be the set of dyadic rationals in (0, 1]. Definition The pseudo-center of an interval J (0, 1) is the (unique) dyadic rational ξ Q dyd with minimal denominator. Definition For binary string u, let ǔ be the bitwise negation of u. For ξ Q dyd \ {1} and let w be the shortest even binary expansion of ξ and v be the shortest odd binary expansion of 1 ξ.

16 Pseudo-centers Motivation: Find exact formulas for matching intervals J and their matching indices for slope s = 2 (also works for 2 s N). Let Q dyd be the set of dyadic rationals in (0, 1]. Definition The pseudo-center of an interval J (0, 1) is the (unique) dyadic rational ξ Q dyd with minimal denominator. Definition For binary string u, let ǔ be the bitwise negation of u. For ξ Q dyd \ {1} and let w be the shortest even binary expansion of ξ and v be the shortest odd binary expansion of 1 ξ. Define the interval I ξ := (ξ L, ξ R ) containing ξ where, ξ L :=.ˇvv, ξ R :=.w. Also define the degenerate interval I 1 := (2/3, + ).

17 Pseudo-centers In short: I ξ := (ξ L, ξ R ) with ξ L :=.ˇvv, ξ R :=.w.

18 Pseudo-centers In short: I ξ := (ξ L, ξ R ) with ξ L :=.ˇvv, ξ R :=.w. If ξ = 1/2 then w = 10, v = 1 and ξ L =.01, ξ R =.10. (01) w = u01 ξ L =.u001ǔ110; (11) w = u11 ξ L =.u101ǔ010; (010) w = u010 ξ L =.u00ǔ11; (110) w = u110 ξ L =.u10ǔ01.

19 Pseudo-centers In short: I ξ := (ξ L, ξ R ) with ξ L :=.ˇvv, ξ R :=.w. If ξ = 1/2 then w = 10, v = 1 and ξ L =.01, ξ R =.10. (01) w = u01 ξ L =.u001ǔ110; (11) w = u11 ξ L =.u101ǔ010; (010) w = u010 ξ L =.u00ǔ11; (110) w = u110 ξ L =.u10ǔ01. ξ ξ R ξ L =.10 3 =.10 3 = =.01 3 =.01 9 = = = = = = = = = = = = =

20 Pseudo-centers Theorem: All matching intervals have the form I ξ, where ξ Q dyd are precisely the pseudo-centers of the components of [0, 2 3 ] \ E. The matching index is (ξ) = 3 2 ( w 0 w 1 ), where w a is the number of symbols a in w (the shortest even binary expansion of ξ).

21 Pseudo-centers Theorem: All matching intervals have the form I ξ, where ξ Q dyd are precisely the pseudo-centers of the components of [0, 2 3 ] \ E. The matching index is (ξ) = 3 2 ( w 0 w 1 ), where w a is the number of symbols a in w (the shortest even binary expansion of ξ). Proposition: If γ 1 6, then w 0 = w 1. In particular, all matching intervals in ( 1 6, 2 3 ) have matching index = 0

22 Pseudo-centers Theorem: All matching intervals have the form I ξ, where ξ Q dyd are precisely the pseudo-centers of the components of [0, 2 3 ] \ E. The matching index is (ξ) = 3 2 ( w 0 w 1 ), where w a is the number of symbols a in w (the shortest even binary expansion of ξ). Proposition: If γ 1 6, then w 0 = w 1. In particular, all matching intervals in ( 1 6, 2 3 ) have matching index = 0 Remark: For general slope s 2, the formulas are (ξ) = s s 1 s (s 1 2a) w a, M = ( s s, a=0 s s + 1 )

23 Pseudo-centers (period doubling) Pseudo-center ξ =.w (even exp.) and 1 ξ =.v (odd exp.). The matching interval is I ξ = [ξ L, ξ R ] for ξ L =.ˇvv and ξ R =.w.

24 Pseudo-centers (period doubling) Pseudo-center ξ =.w (even exp.) and 1 ξ =.v (odd exp.). The matching interval is I ξ = [ξ L, ξ R ] for ξ L =.ˇvv and ξ R =.w. But ξ L is also the right end-point of I ξ1 for ξ 1 =.ˇvv. We call this period doubling. It repeats countably often, converging to ξ. ξ (ξ 2 ) L = (ξ 3 ) R (ξ 1 ) L = (ξ 2 ) R ξ L = (ξ 1 ) R ξ ξ R I ξ2 I ξ1 I ξ

25 Pseudo-centers (period doubling) Pseudo-center ξ =.w (even exp.) and 1 ξ =.v (odd exp.). The matching interval is I ξ = [ξ L, ξ R ] for ξ L =.ˇvv and ξ R =.w. But ξ L is also the right end-point of I ξ1 for ξ 1 =.ˇvv. We call this period doubling. It repeats countably often, converging to ξ. ξ (ξ 2 ) L = (ξ 3 ) R (ξ 1 ) L = (ξ 2 ) R ξ L = (ξ 1 ) R ξ ξ R I ξ2 I ξ1 I ξ Lemma: The pseudo-center of the next period doubling can be obtained from the previous using the substitution: χ : w ˇvv ˇw v ˇv v vw ˇv ˇv ˇw. Thus the limit ξ has s-adic expansion ξ =.ˇv ˇwv ˇvvw ˇv ˇwvw ˇvv ˇv ˇw...

26 Pseudo-centers (period doubling) Metric and topological entropy for s=2 0.5 topological metric topological metric Remark: This proposition explains constant entropy on all matching intervals in M = ( 1 6, 2 3 ). A no devil s staircase argument gives: h µ (Q γ ) = log( ) and h top (Q γ ) = 2 log 2, 2 3 for all γ [ 1 6, 2 3 ].

27 Pseudo-centers (tuning windows) Pseudo-center ξ =.w (even expansion) and 1 ξ =.v (odd exp). The matching interval is I ξ = [ξ L, ξ R ] = [.ˇvv,.w]. The tuning interval is T ξ = [ξ T, ξ R ] for ξ T =.ˇv ˇw. ξ T ξ ξ L I ξ ξ ξ R

28 Pseudo-centers (tuning windows) Pseudo-center ξ =.w (even expansion) and 1 ξ =.v (odd exp). The matching interval is I ξ = [ξ L, ξ R ] = [.ˇvv,.w]. The tuning interval is T ξ = [ξ T, ξ R ] for ξ T =.ˇv ˇw. ξ T ξ ξ L I ξ ξ ξ R Theorem: Let K(ξ T ) = {x : g k (x) ξ T k}. Then x K(ξ T ) T ξ if and only if x =.σ 1 σ 2 σ 3 σ 4... for σ 1 {w, ˇv}, σ j {w, v, ˇw, ˇv} describing a path in the diagram. w v ˇv ˇw

29 Pseudo-centers (tuning windows) Pseudo-center ξ =.w (even expansion) and 1 ξ =.v (odd exp). The matching interval is I ξ = [ξ L, ξ R ] = [.ˇvv,.w]. The tuning interval is T ξ = [ξ T, ξ R ] for ξ T =.ˇv ˇw. ξ T ξ ξ L I ξ ξ ξ R Theorem: Let K(ξ T ) = {x : g k (x) ξ T k}. Then x K(ξ T ) T ξ if and only if x =.σ 1 σ 2 σ 3 σ 4... for σ 1 {w, ˇv}, σ j {w, v, ˇw, ˇv} describing a path in the diagram. If (ξ) = 0, then all matching in T ξ is neutral. w v ˇv ˇw

30 Metric and topological entropy for s=2 topological topological metric metric Shape of the entropy function Question: Known: γ h(µ γ ) is Hölder. Is γ h top (Q γ ) Hölder?

31 Metric and topological entropy for s=2 topological topological metric metric Shape of the entropy function Question: Known: γ h(µ γ ) is Hölder. Is γ h top (Q γ ) Hölder? Conjecture: The neutral tuning windows are exactly the plateaus of (topological and metric) entropy.

32 Metric and topological entropy for s=2 topological topological metric metric Shape of the entropy function Question: Known: γ h(µ γ ) is Hölder. Is γ h top (Q γ ) Hölder? Conjecture: The neutral tuning windows are exactly the plateaus of (topological and metric) entropy. Corollary: The shape of the entire entropy function (i.e., pattern of increase/decrease) is repeated in every tuning window T ξ with (ξ) > 0, and reversed in every tuning window T ξ with (ξ) < 0.