Patterns and Cycles in Dynamical Systems. Kate Moore Dartmouth College June 30, 2017

Transcription

1 Patterns and Cycles in Dynamical Systems Kate Moore Dartmouth College June 30,

2 Function Iteration Example: Let f (x) = 4x(1 x). Then (x, f (x), f 2 (x), f 3 (x)) = (.30,,, ) 2

3 Function Iteration Example: Let f (x) = 4x(1 x). Then (x, f (x), f 2 (x), f 3 (x)) = (.30,.84,, ) 2

4 Function Iteration Example: Let f (x) = 4x(1 x). Then (x, f (x), f 2 (x), f 3 (x)) = (.30,.84,.53, ) 2

5 Function Iteration Example: Let f (x) = 4x(1 x). Then (x, f (x), f 2 (x), f 3 (x)) = (.30,.84,.53,.99) 2

6 Function Iteration Example: Let f (x) = 4x(1 x). Then (x, f (x), f 2 (x), f 3 (x)) = (.30,.84,.53,.99) and so Pat(.3, f, 4) = st(.30,.84,.53,.99) =

7 Patterns and Dynamical Systems Theorem (Bandt-Keller-Pompe): Every piecewise-monotone map f : [0, 1] [0, 1] has forbidden patterns, i.e. patterns that never arise as iterates. # Allowed patterns complexity (i.e. topological entropy) 3

8 Patterns and Dynamical Systems Theorem (Bandt-Keller-Pompe): Every piecewise-monotone map f : [0, 1] [0, 1] has forbidden patterns, i.e. patterns that never arise as iterates. # Allowed patterns complexity (i.e. topological entropy) Example: Let f (x) = 4x(1 x). 321 / Allow(f ) 4321, 1432, 54213,... / Allow(f ) }{{} contain consecutive 321 3

9 Sarkovskii s Theorem An n periodic point of a map is a point such that f n (x) = x and f i (x) x for all 1 i < n. Theorem (Sarkovskii): If a continuous map f of the unit interval has an m-periodic point and l m in the Sarkovskii ordering n 5 2 n 3 2 n then f must also have an l-periodic point. Question: Is there a similar order for the permutation structure of periodic points? 4

10 Cycle Type Let x be a periodic point of order n and Pat(x, f, n) = π. The cycle type of x is ˆπ C n where π = π 1 π 2... π n ˆπ = (π 1, π 2,..., π n ). 5

11 Cycle Type Let x be a periodic point of order n and Pat(x, f, n) = π. The cycle type of x is ˆπ C n where π = π 1 π 2... π n ˆπ = (π 1, π 2,..., π n ). Example: Consider G 2 (x) = { 2x}. 5

12 Cycle Type Let x be a periodic point of order n and Pat(x, f, n) = π. The cycle type of x is ˆπ C n where π = π 1 π 2... π n ˆπ = (π 1, π 2,..., π n ). Example: Consider G 2 (x) = { 2x}. A 3-periodic orbit of G 2 is: ( 8 (x, G 2 (x), G2 2 (x)) = 9, 2 9, 5 ) 9 Giving Pat( 8 9, G 2, 3) = 312 and ˆπ = (3, 1, 2) = 231 5

13 The Shape of Cycles The representative of a 6-periodic orbit of F 3 (x) = {3x} is x =

14 The Shape of Cycles The representative of a 6-periodic orbit of F 3 (x) = {3x} is x = 13 ( 13 Pat(x, F 3, 6) = st 14, 11 14, 5 14, 1 14, 3 14, 9 ) = The cycle type of the orbit is ˆπ = (6, 5, 3, 1, 2, 4) =

15 The Shape of Cycles The representative of a 6-periodic orbit of F 3 (x) = {3x} is x = 13 ( 13 Pat(x, F 3, 6) = st 14, 11 14, 5 14, 1 14, 3 14, 9 ) = The cycle type of the orbit is ˆπ = (6, 5, 3, 1, 2, 4) = ˆπ =

16 Forcing Order For a family of interval maps F, a cycle ˆπ forces a cycle ˆτ if, for any f F, whenever ˆπ AlCyc(f ) then ˆτ AlCyc(f ) as well. 7

17 Forcing Order For a family of interval maps F, a cycle ˆπ forces a cycle ˆτ if, for any f F, whenever ˆπ AlCyc(f ) then ˆτ AlCyc(f ) as well. (1,3,4,2) (1,5,2,3,4) (1,2,4,3) (1,2,3,4) (1,4,3,2) (1,4,3,2,5) (1,4,2,3,5) (1,2,4,3,5) (1,5,4,2,3) (1,5,2,4,3) (1,2,3) (1,3,2) (1,3,4,2,5) (1,5,3,2,4) (1,3,2,4) (1,4,2,3) (1,2) The forcing relations for continuous maps, (Baldwin). (1) 7

18 Beta Shifts and Negative Beta Shifts For β > 1, consider the classes of functions, F β (x) = {βx} and G β (x) = 1 {βx} = { βx}. The graphs of F β (x) = {βx} for (a) β = , (b) β = 2, and (c) β = 2.5. The graphs of G β (x) = 1 {βx} = { βx} for (a) β = , (b) β = 2, and (c) β =

19 Itineraries Example: F 3 (x) = {3x} Name the monotonic intervals: 0, 1, 2 (x, F 3 (x), F 2 3 (x),...) = (.13,.39,.17,.51,.53,.59,.77,...)

20 Itineraries Example: F 3 (x) = {3x} Name the monotonic intervals: 0, 1, 2 (x, F 3 (x), F3 2 (x),...) = (.13,.39,.17,.51,.53,.59,.77,...) Why? Applying F 3 is now a shift of the word. Σ(w 1 w 2 w 3...) = w 2 w Pat(x, F 3, 4) = Pat( , Σ 3, 4) = st( , , , ) =

21 Beta and Negative Beta Expansions For F β (x) = {βx}, itineraries correspond to β-expansions: x = w 1 β + w 2 β 2 + w 3 β

22 Beta and Negative Beta Expansions For F β (x) = {βx}, itineraries correspond to β-expansions: x = w 1 β + w 2 β 2 + w 3 β For G β (x) = { βx}, itineraries correspond to ( β)-expansions: ( w1 + 1 x = ( β) + w ( β) 2 + w ) ( β) Alternating Order: In odd positions, 0 is low and β is high, in even positions, β is low and 0 is high < alt < alt < alt

23 Segmentations Ask: Is ˆπ a cycle of F N (x) = {Nx}? An N-segmentation of ˆπ is a sequence 0 = e 0 e 1 e N = n such that each segment ˆπ et+1ˆπ et+2... ˆπ et+1 is increasing. A 3-segmetation of ˆπ = (6, 1, 4, 3, 2, 5) = From this, define a word ω by π = ω = 11

24 Segmentations Ask: Is ˆπ a cycle of F N (x) = {Nx}? An N-segmentation of ˆπ is a sequence 0 = e 0 e 1 e N = n such that each segment ˆπ et+1ˆπ et+2... ˆπ et+1 is increasing. A 3-segmetation of ˆπ = (6, 1, 4, 3, 2, 5) = From this, define a word ω by π = ω =

25 Segmentations Ask: Is ˆπ a cycle of F N (x) = {Nx}? An N-segmentation of ˆπ is a sequence 0 = e 0 e 1 e N = n such that each segment ˆπ et+1ˆπ et+2... ˆπ et+1 is increasing. A 3-segmetation of ˆπ = (6, 1, 4, 3, 2, 5) = From this, define a word ω by π = ω =

26 Segmentations Ask: Is ˆπ a cycle of F N (x) = {Nx}? An N-segmentation of ˆπ is a sequence 0 = e 0 e 1 e N = n such that each segment ˆπ et+1ˆπ et+2... ˆπ et+1 is increasing. A 3-segmetation of ˆπ = (6, 1, 4, 3, 2, 5) = From this, define a word ω by π = ω = Theorem (Archer-Elizalde): Pat(ω, Σ N, n) = π min{n : ˆπ AlCyc(F N )} = 1 + des(ˆπ) 11

27 Negative Segmentations Ask: Is ˆπ a cycle of G N (x) = { Nx}? A N-segmentation of ˆπ is a sequence 0 = e 0 e 1 e k = n such that each segment ˆπ et+1ˆπ et+2... ˆπ et+1 is decreasing. A 3-segmetation of ˆπ = (6, 5, 2, 1, 4, 3) = π = ω =

28 Negative Segmentations Ask: Is ˆπ a cycle of G N (x) = { Nx}? A N-segmentation of ˆπ is a sequence 0 = e 0 e 1 e k = n such that each segment ˆπ et+1ˆπ et+2... ˆπ et+1 is decreasing. A 3-segmetation of ˆπ = (6, 5, 2, 1, 4, 3) = π = ω = Theorem (Archer-Elizalde): If ω is primitive, then Pat(ω, Σ N, n) = π min{n : ˆπ AlCyc(G N )} = 1 + asc(ˆπ) + ɛ(ˆπ) 12

29 Negative Segmentations Ask: Is ˆπ a cycle of G N (x) = { Nx}? A N-segmentation of ˆπ is a sequence 0 = e 0 e 1 e k = n such that each segment ˆπ et+1ˆπ et+2... ˆπ et+1 is decreasing. A 3-segmetation of ˆπ = (6, 5, 2, 1, 4, 3) = Collapsed cycles: π = ω = ω = Theorem (Archer-Elizalde): If ω is primitive, then Pat(ω, Σ N, n) = π min{n : ˆπ AlCyc(G N )} = 1 + asc(ˆπ) + ɛ(ˆπ) 12

30 β-shifts and β-shifts Theorem: Let B p (ˆπ) = inf{β : ˆπ AlCyc(F β )}. Then B p (ˆπ) is equal to the largest real root of n p ω (x) = x n 1 w j x n j, j=1 where ω = w 1 w 2... w n is the word defined by the unique (1 + des(ˆπ))-segmentation of ˆπ. 13

31 β-shifts and β-shifts Theorem: Let B p (ˆπ) = inf{β : ˆπ AlCyc(F β )}. Then B p (ˆπ) is equal to the largest real root of n p ω (x) = x n 1 w j x n j, j=1 where ω = w 1 w 2... w n is the word defined by the unique (1 + des(ˆπ))-segmentation of ˆπ. Theorem: Let B p (ˆπ) = inf{β : ˆπ AlCyc(G β )}. Then B p (ˆπ) is equal to the largest real root of n p ω (x) = ( x) n 1 + (w j + 1)( x) n j, where ω = w 1 w 2... w n is the word defined by a (usually unique) (1 + asc(ˆπ) + ɛ(ˆπ))-segmentation of ˆπ. j=1 (If ɛ(ˆπ) = 1, take ω to be the smallest with respect to < alt.) 13

32 Distributions Plots of B p (ˆπ) (top) and B p (ˆπ) (bottom) for π C n and n = 3, 4, 5, 6. 14

33 Distributions Theorem: The distribution of B p (ˆπ) = 1 + des(ˆπ) (resp. B p (ˆπ) = 1 + asc(ˆπ) + ɛ(ˆπ)) is asymptotically normal with mean µ = n+1 2 and variance σ 2 = n Why? Descents in cycles are normal, (Fulman) Plots of B p (ˆπ) (top) and B p (ˆπ) (bottom) for π C n and n = 3, 4, 5, 6. 14

34 Distributions Theorem: The distribution of B p (ˆπ) = 1 + des(ˆπ) (resp. B p (ˆπ) = 1 + asc(ˆπ) + ɛ(ˆπ)) is asymptotically normal with mean µ = n+1 2 and variance σ 2 = n Why? Descents in cycles are normal, (Fulman). Conjecture: The distribution of B p (ˆπ) (resp. B p (ˆπ)) is asymptotically normal with mean µ = n 2 and variance σ2 = n Plots of B p (ˆπ) (top) and B p (ˆπ) (bottom) for π C n and n = 3, 4, 5, 6. 14

35 References Remember that we started with patterns realized by any point in the interval? Come talk to me here or at FPSAC about it: Patterns of Negative Shifts and Signed Shifts. [1] K. Archer and S. Elizalde, Cyclic permutations realized by signed shifts, Journal Combinatorics 5 (2014), [2] C. Bandt, G. Keller and B. Pompe, Entropy of interval maps via permutations, Nonlinearity 15 (2002), [3] S. Baldwin, Generalizations of a theorem of Sarkovskii on orbits of continuous real-valued functions Discrete Mathematics 76 (1987), [4] J. Fulman, The Distribution of Descents in Fixed Conjugacy Classes of the Symmetric Groups, Journal of Combinatorial Theory 84 (1998),

36 Forcing Order For a family of interval maps F, a cycle ˆπ forces a cycle ˆτ if, for any f F, if ˆπ AlCyc(f ) then ˆτ AlCyc(f ) as well The forcing relations for continuous maps, (Baldwin). 21 1