Computational Homology in Topological Dynamics

Transcription

1 1 Computational Homology in Topological Dynamics ACAT School, Bologna, Italy May 26, 2012 Marian Mrozek Jagiellonian University, Kraków

2 Weather forecasts for Galway, Ireland from Weather Underground 2

3 Weather forecasts for Galway, Ireland from Weather Underground 2

4 Outline 3 Part I Dynamical systems Rigorous numerics of dynamical systems Homological invariants of dynamical systems Computing homological invariants Part II Homology algorithms for subsets of R d Homology algorithms for maps of subsets of R d Applications

5 Outline 4 Dynamical systems Rigorous numerics of dynamical systems Homological invariants of dynamical systems Computing homological invariants Homology algorithms for subsets of R d Homology algorithms for maps of subsets of R d Applications

6 Main sources: Differential equations Iterates of maps Time series Dynamical systems 5

7 Dynamical systems 6

8 Dynamical systems 7 X a compact subset of R d (in general: a topological space).

9 Dynamical systems 7 X a compact subset of R d (in general: a topological space). T {R, R +, Z, Z + } time (continuous if T {R, R + }, discrete if T {Z, Z + })

10 Dynamical systems 7 X a compact subset of R d (in general: a topological space). T {R, R +, Z, Z + } time (continuous if T {R, R + }, discrete if T {Z, Z + }) A (semi)dynamical system is a continuous map ϕ : X T X such that for any x X and s, t T ϕ(ϕ(x, t), s) = ϕ(x, s + t) ϕ(x, 0) = x

11 Dynamical systems 7 X a compact subset of R d (in general: a topological space). T {R, R +, Z, Z + } time (continuous if T {R, R + }, discrete if T {Z, Z + }) A (semi)dynamical system is a continuous map ϕ : X T X such that for any x X and s, t T ϕ(ϕ(x, t), s) = ϕ(x, s + t) ϕ(x, 0) = x T = R a flow

12 Dynamical systems 7 X a compact subset of R d (in general: a topological space). T {R, R +, Z, Z + } time (continuous if T {R, R + }, discrete if T {Z, Z + }) A (semi)dynamical system is a continuous map ϕ : X T X such that for any x X and s, t T ϕ(ϕ(x, t), s) = ϕ(x, s + t) ϕ(x, 0) = x T = R a flow T = Z + a discrete semidynamical system (dsds)

13 Dynamical systems 7 X a compact subset of R d (in general: a topological space). T {R, R +, Z, Z + } time (continuous if T {R, R + }, discrete if T {Z, Z + }) A (semi)dynamical system is a continuous map ϕ : X T X such that for any x X and s, t T ϕ(ϕ(x, t), s) = ϕ(x, s + t) ϕ(x, 0) = x T = R a flow T = Z + a discrete semidynamical system (dsds) ϕ t : X x ϕ(x, t) X t-translation map

14 Dynamical systems 7 X a compact subset of R d (in general: a topological space). T {R, R +, Z, Z + } time (continuous if T {R, R + }, discrete if T {Z, Z + }) A (semi)dynamical system is a continuous map ϕ : X T X such that for any x X and s, t T ϕ(ϕ(x, t), s) = ϕ(x, s + t) ϕ(x, 0) = x T = R a flow T = Z + a discrete semidynamical system (dsds) ϕ t : X x ϕ(x, t) X t-translation map f := ϕ 1 the generator (for discrete time only) identified with sds

15 Jules Henri Poincaré 8 Henri Poincaré,

16 Invariant sets 9 the trajectory (orbit) of x X ϕ(x) := { ϕ(x, t) t T }

17 Invariant sets 9 the trajectory (orbit) of x X ϕ(x) := { ϕ(x, t) t T } x X is stationary iff ϕ(x) = x

18 Invariant sets 9 the trajectory (orbit) of x X ϕ(x) := { ϕ(x, t) t T } x X is stationary iff ϕ(x) = x x X is periodic iff there exists a t T + such that ϕ(x, t) = x

19 Invariant sets 9 the trajectory (orbit) of x X ϕ(x) := { ϕ(x, t) t T } x X is stationary iff ϕ(x) = x x X is periodic iff there exists a t T + such that ϕ(x, t) = x the invariant part of N X: Inv(N, ϕ) := { x N ϕ(x) N }

20 Invariant sets 9 the trajectory (orbit) of x X ϕ(x) := { ϕ(x, t) t T } x X is stationary iff ϕ(x) = x x X is periodic iff there exists a t T + such that ϕ(x, t) = x the invariant part of N X: Inv(N, ϕ) := { x N ϕ(x) N } A X is invariant if Inv(A, ϕ) = A

21 Invariant sets 9 the trajectory (orbit) of x X ϕ(x) := { ϕ(x, t) t T } x X is stationary iff ϕ(x) = x x X is periodic iff there exists a t T + such that ϕ(x, t) = x the invariant part of N X: Inv(N, ϕ) := { x N ϕ(x) N } A X is invariant if Inv(A, ϕ) = A alpha and omega limit sets α(x) := { y X t n s.t. y = lim ϕ(x, t n ) } ω(x) := { y X t n + s.t. y = lim ϕ(x, t n ) }

22 Invariant sets 9 the trajectory (orbit) of x X ϕ(x) := { ϕ(x, t) t T } x X is stationary iff ϕ(x) = x x X is periodic iff there exists a t T + such that ϕ(x, t) = x the invariant part of N X: Inv(N, ϕ) := { x N ϕ(x) N } A X is invariant if Inv(A, ϕ) = A alpha and omega limit sets α(x) := { y X t n s.t. y = lim ϕ(x, t n ) } ω(x) := { y X t n + s.t. y = lim ϕ(x, t n ) } Limit sets are invariant.

23 Invariant sets and limit sets 10 Some invariant sets and limit sets.

24 Asymptotic dynamics 11 The main goal of the theory of dynamical systems is the understanding of the asymptotic behaviour of the trajectories, i.e. the number and structure of limit sets as well as their mutual relations

25 Asymptotic dynamics 11 The main goal of the theory of dynamical systems is the understanding of the asymptotic behaviour of the trajectories, i.e. the number and structure of limit sets as well as their mutual relations Up to the half of the 20th century the dominating opinion was that the a limit set may be a stationary point or the trajectory of a periodic point.

26 Asymptotic dynamics 11 The main goal of the theory of dynamical systems is the understanding of the asymptotic behaviour of the trajectories, i.e. the number and structure of limit sets as well as their mutual relations Up to the half of the 20th century the dominating opinion was that the a limit set may be a stationary point or the trajectory of a periodic point. The computers significantly contributed to the realization that the asymptotic behaviour may be much more complicated (chaotic).

27 Edward Lorenz 12 during the 1950s became skeptical of the appropriateness of the mathematical models used in meteorology Edward Lorenz

28 Edward Lorenz 12 during the 1950s became skeptical of the appropriateness of the mathematical models used in meteorology in 1963 published the famous paper: Deterministic Nonperiodic Flow the Lorenz equations: ẋ = σ(y x) ẏ = Rx y xz ż = xy bz Edward Lorenz

29 Main contributors to the discovery of deterministic chaos 13 Henri Poincaré, 1890 Mary Cartwright and John Littlewood, 1940 s Andrey Kolmogorov and Yakov Sinai, 1950 s Edward Lorenz, early 1960 s Oleksandr Sharkovsky, 1964 Stephen Smale, 1967 Tien-Yien Li and James A. Yorke 1975

30 Symbolic dynamics 14 Rotation by 120 degrees

31 Symbolic dynamics 15 Mapping to sequences of symbols

32 Shift dynamics 16 Consider Σ k := {0, 1, 2,... k 1} Z as a metric space with the metric 1 δ α(i),β(i) d(α, β) :=, 2 i i= where δ mn stands for the Kronecker delta.

33 Shift dynamics 16 Consider Σ k := {0, 1, 2,... k 1} Z as a metric space with the metric 1 δ α(i),β(i) d(α, β) :=, 2 i i= where δ mn stands for the Kronecker delta. A full shift on k symbols is the discrete dynamical system generated on Σ k by σ : Σ k α σ(α) := (n α(n 1)) Σ k.

34 Shift dynamics 16 Consider Σ k := {0, 1, 2,... k 1} Z as a metric space with the metric 1 δ α(i),β(i) d(α, β) :=, 2 i i= where δ mn stands for the Kronecker delta. A full shift on k symbols is the discrete dynamical system generated on Σ k by σ : Σ k α σ(α) := (n α(n 1)) Σ k. Features: Plenty of periodic points: Every finite sequence of symbols is in one-toone correspondence with a periodic point of σ Sensitive dependence on initial conditions: trajectories diverge exponentially fast

35 Smale horseshoe (1967) 17

36 Smale horseshoe (1967) 18

37 Smale horseshoe (1967) 19

38 Smale horseshoe (1967) 20

39 Smale horseshoe (1967) 21

40 Smale horseshoe (1967) 22

41 Smale horseshoe (1967) 23

42 Smale horseshoe (1967) 24

43 Smale horseshoe (1967) 25 Theorem. (Smale, 1967) Let N denote the square part of the domain of the horseshoe map h. Then there exists a homeomorphism ρ : Inv(N, h) Σ 2 such that σρ = ρh.

44 Lorenz equations around the origin 26

45 Lorenz equations around the origin 27

46 Lorenz equations around the origin 28

47 Poincaré map in the Lorenz equations 29

48 Problem 30 Is horseshoe dynamics present in the Lorenz system?

49 Problem 30 Is horseshoe dynamics present in the Lorenz system? Or maybe it is only present in the numerical scheme?

50 Problem 30 Is horseshoe dynamics present in the Lorenz system? Or maybe it is only present in the numerical scheme? Or maybe the chaotic behaviour is only the consequence of the rounding errors?

51 References 31 E.N. Lorenz, Deterministic Nonperiodic Flow, Journal of the Atmospheric Science (1963). R.L. Adler, A.G. Konheim, M.H.McAndrew, Topological entropy, Transactions of the American Mathematical Society (1965). S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. (1967). C. Sparrow, The Lorenz Equations: Bifurcations, Chaos and Strange Attractors, Springer-Verlag (1982).

52 Outline 32 Dynamical systems Rigorous numerics of dynamical systems Homological invariants of dynamical systems Computing homological invariants Homology algorithms for subsets of R d Homology algorithms for maps of subsets of R d Applications

53 Ghost solutions 33 Consider the equation z = (αi z )z, z C The only periodic trajectory of this equation is the stationary point at the origin.

54 Ghost solutions 33 Consider the equation z = (αi z )z, z C The only periodic trajectory of this equation is the stationary point at the origin. Consider its Euler discretization Φ h (z) := z(1 + h(αi z )) For every h > 0 this discretization has invariant circles of radius r ± := 1 ± 1 h 2 α 2 h

55 Disappearing Smale s horseshoe 34 The logistic equation y = y(1 y) may be solved explicitly and it clearly does not exhibit chaotic behaviour However, Koçak and Hale (1991) prove that the two step numerical scheme ( ) ( 1 λ y1 Φ h,λ := 1+λ y 2 + 2λ 1+λ y ) 1 + 2hy 1 (1 y 2 ) y 2 contains an invariant subset conjugate to a horseshoe for every h > 0. y 1

56 Fatal consequences of numerical errors 35 On 25th of February 1991 the Patriot missile system failed to intercept an Iraqi Scud and 28 American soldiers were killed.

57 Fatal consequences of numerical errors 35 On 25th of February 1991 the Patriot missile system failed to intercept an Iraqi Scud and 28 American soldiers were killed. Ariane 5 s first test flight on 4 June 1996 failed, with the rocket self-destructing 37 seconds after launch because of a malfunction in the control software, resulting in a loss of more than $370 million.

58 Fatal consequences of numerical errors 35 On 25th of February 1991 the Patriot missile system failed to intercept an Iraqi Scud and 28 American soldiers were killed. Ariane 5 s first test flight on 4 June 1996 failed, with the rocket self-destructing 37 seconds after launch because of a malfunction in the control software, resulting in a loss of more than $370 million. In both cases the failures were attributed to numerical errors.

59 Interval arithmetic 36 ˆR R fixed, finite set of representable numbers

60 Interval arithmetic 36 ˆR R fixed, finite set of representable numbers representable intervals: I := {[a, b] a, b ˆR, a b}

61 Interval arithmetic 36 ˆR R fixed, finite set of representable numbers representable intervals: I := {[a, b] a, b ˆR, a b} For {+,,, /} and I, J I denote by I J the smallest representable interval that contains {a b a I, b J}.

62 Interval arithmetic 36 ˆR R fixed, finite set of representable numbers representable intervals: I := {[a, b] a, b ˆR, a b} For {+,,, /} and I, J I denote by I J the smallest representable interval that contains {a b a I, b J}. Given the endpoints of I and J, one can easily construct the endpoints of I J.

63 Interval arithmetic 36 ˆR R fixed, finite set of representable numbers representable intervals: I := {[a, b] a, b ˆR, a b} For {+,,, /} and I, J I denote by I J the smallest representable interval that contains {a b a I, b J}. Given the endpoints of I and J, one can easily construct the endpoints of I J. first proposed by M. Warmus in 1956 rediscovered by R.E. Moore in 1959

64 The simplest topological tool: Darboux property 37

65 Discretization in time 38

66 Discretization in time 39

67 Numerical Analysis of Dynamical Systems 40

68 Numerical Analysis of Dynamical Systems 41

69 Discretization in space 42

70 Discretization in space 43

71 Discretization in space 44

72 Discretization in space 45

73 Discretization in space 46

74 Discretization in space 47

75 Discretization in space 48 A combinatorial multivalued map F : X X is a combinatorial enclosure of f : X X if for every Q X f(q) int F(Q).

76 Discretization in space 48 A combinatorial multivalued map F : X X is a combinatorial enclosure of f : X X if for every Q X f(q) int F(Q). In this case we say that f is a selector of F.

77 Numerical Analysis of Dynamical Systems 49

78 Numerical Analysis of Dynamical Systems 50

79 Numerical Analysis of Dynamical Systems 51

80 Numerical Analysis of Dynamical Systems 52

81 Numerical Analysis of Dynamical Systems 53

82 Numerical Analysis of Dynamical Systems 54

83 Numerical Analysis of Dynamical Systems 55

84 Chaos in Lorenz equations 56 Theorem. (1995) Consider the Lorenz equations ẋ = σ(y x) ẏ = Rx y xz ż = xy bz and put P := {(x, y, z) R 3 z = 53}. For all parameter values in a sufficiently small neighborhood of (σ, R, b) = (45, 54, 10), there exists a Poincaré section N P such that the Poincaré map g induced by (1) is Lipschitz and well defined. Furthermore, there exists a d N and a continuous surjection ρ : Inv(N, g) Σ 2 such that ρ g d = σ ρ where σ : Σ 2 Σ 2 is the full shift dynamics on two symbols.

85 Rigorous numerics of dynamical systems 57 Goal: use the outcome of numerical simulations to draw rigorous conclusions about the behaviour of the original dynamical system.

86 Rigorous numerics of dynamical systems 57 Goal: use the outcome of numerical simulations to draw rigorous conclusions about the behaviour of the original dynamical system. Needed: (1) exact bounds for the errors resulting from the time discretization and space discretization

87 Rigorous numerics of dynamical systems 57 Goal: use the outcome of numerical simulations to draw rigorous conclusions about the behaviour of the original dynamical system. Needed: (1) exact bounds for the errors resulting from the time discretization and space discretization (2) a method to draw conclusions about the original dynamical system from the outcome of numerical simulations

88 Interval arithmetic 58 ˆR R fixed, finite set of representable numbers

89 Interval arithmetic 58 ˆR R fixed, finite set of representable numbers representable intervals: I := {[a, b] a, b ˆR, a b}

90 Interval arithmetic 58 ˆR R fixed, finite set of representable numbers representable intervals: I := {[a, b] a, b ˆR, a b} For {+,,, /} and I, J I denote by I J the smallest representable interval that contains {a b a I, b J}.

91 Interval arithmetic 58 ˆR R fixed, finite set of representable numbers representable intervals: I := {[a, b] a, b ˆR, a b} For {+,,, /} and I, J I denote by I J the smallest representable interval that contains {a b a I, b J}. Given the endpoints of I and J, one can easily construct the endpoints of I J.

92 Interval arithmetic 58 ˆR R fixed, finite set of representable numbers representable intervals: I := {[a, b] a, b ˆR, a b} For {+,,, /} and I, J I denote by I J the smallest representable interval that contains {a b a I, b J}. Given the endpoints of I and J, one can easily construct the endpoints of I J. first proposed by M. Warmus in 1956 rediscovered by R.E. Moore in 1959

93 The simplest topological tool: Darboux property 59

94 Advanced tool: topology of multivalued maps 60

95 Multivalued maps 61 Let X, Y be topological spaces. A multivalued map F : X Y from X to Y is a function F : X 2 Y from X to subsets of Y.

96 Multivalued maps 61 Let X, Y be topological spaces. A multivalued map F : X Y from X to Y is a function F : X 2 Y from X to subsets of Y. The image of A X is F (A) := F (x). x A

97 Multivalued maps 61 Let X, Y be topological spaces. A multivalued map F : X Y from X to Y is a function F : X 2 Y from X to subsets of Y. The image of A X is F (A) := x A F (x). The weak preimage of B Y under F is F 1 (B) := {x X F (x) B }.

98 Multivalued maps 61 Let X, Y be topological spaces. A multivalued map F : X Y from X to Y is a function F : X 2 Y from X to subsets of Y. The image of A X is F (A) := x A F (x). The weak preimage of B Y under F is F 1 (B) := {x X F (x) B }. F is upper semicontinuous if F 1 (B) is closed for any closed set B Y, and it is lower semicontinuous if the set F 1 (U) is open for any open set U Y.

99 Representations of rational functions 62 f : R m R n a rational function.

100 Representations of rational functions 62 f : R m R n a rational function. [f] : I m I n, the interval extension of f obtained by replacing the arithmetic operations in f with their interval counterparts

101 Representations of rational functions 62 f : R m R n a rational function. [f] : I m I n, the interval extension of f obtained by replacing the arithmetic operations in f with their interval counterparts Proposition. Assume f : R m R n is a rational function. Then for any x 1,... x n dom [f] we have f(x 1,... x n ) [f](x 1,... x n ).

102 Representations of rational functions 62 f : R m R n a rational function. [f] : I m I n, the interval extension of f obtained by replacing the arithmetic operations in f with their interval counterparts Proposition. Assume f : R m R n is a rational function. Then for any x 1,... x n dom [f] we have f(x 1,... x n ) [f](x 1,... x n ).

103 Arbitrary functions 63 f : R m R n g : R m R n a rational approximation of f such that for x D and some w I n f(x) g(x) w, then f(x) [g](x)[+]w.

104 References 64 M. Warmus, Calculus of approximations, Bull. de l Académie Polonaise des Sciences Cl. III) (1956). R.E. Moore, Interval analysis, Prentice-Hall (1966). J.K. Hale and H. Koçak, Dynamics and Bifurcations, Springer- Verlag (1991). K. Mischaikow, MM, Chaos in Lorenz equations: a computer assisted proof, Bull. Amer. Math. Soc. (N.S.) (1995). K. Mischaikow, MM, Chaos in the Lorenz equations: a computer assisted proof. Part II: details, Mathematics of Computation (1998). A.M. Stuart, A.R. Humphries, Dynamical Systems and Numerical Analysis, Cambridge Univ. Press (1998).

105 Outline 65 Dynamical systems Rigorous numerics of dynamical systems Homological invariants of dynamical systems Computing homological invariants Homology algorithms for subsets of R d Homology algorithms for maps of subsets of R d Applications

106 Ważewski Theorem 66 Tadeusz Ważewski,

107 Ważewski Theorem 67

108 Ważewski Theorem 68

109 Isolated invariant sets 69 a compact set N is an isolating neighborhood iff x bd N ϕ(x) N.

110 Isolated invariant sets 69 a compact set N is an isolating neighborhood iff x bd N ϕ(x) N. A compact set N is an isolating block iff N := {x N ɛ > 0 : ϕ(x, t) N for 0 < t < ɛ}

111 Isolated invariant sets 69 a compact set N is an isolating neighborhood iff x bd N ϕ(x) N. A compact set N is an isolating block iff N := {x N ɛ > 0 : ϕ(x, t) N for 0 < t < ɛ} N is an isolating neighborhood iff Inv(N, ϕ) int N.

112 Isolated invariant sets 69 a compact set N is an isolating neighborhood iff x bd N ϕ(x) N. A compact set N is an isolating block iff N := {x N ɛ > 0 : ϕ(x, t) N for 0 < t < ɛ} N is an isolating neighborhood iff Inv(N, ϕ) int N. A compact set S X is called an isolated invariant set if there exists an isolating neighborhood N such that S = Inv(N, ϕ).

113 Conley index 70 Theorem. (Conley and students, 1978) For every isolating neighborhood N of S there exists an isolating block M such that S M N.

114 Conley index 70 Theorem. (Conley and students, 1978) For every isolating neighborhood N of S there exists an isolating block M such that S M N. If M 1 and M 2 are two such blocks, then (M 1 /M1, [M 1 ]) and (M 2 /M2, [M 2 ]) are homotopy equivalent and, in particular, H (M 1, M1 ) = H (M 2, M2 ).

115 Conley index 70 Theorem. (Conley and students, 1978) For every isolating neighborhood N of S there exists an isolating block M such that S M N. If M 1 and M 2 are two such blocks, then (M 1 /M1, [M 1 ]) and (M 2 /M2, [M 2 ]) are homotopy equivalent and, in particular, H (M 1, M1 ) = H (M 2, M2 ). The cohomological Conley index of S and N is Con (N, ϕ) := Con (S, ϕ) := H (M, M ).

116 Charles Conley 71 Charles Conley

117 An example 72

118 An example 73

119 Main properties 74 Theorem. (Conley and students, 1978) Ważewski property: Con (N, ϕ) 0 Inv(N, ϕ). Hopf property: If χ(con (N, ϕ)) 0 then there exists an x N such that ϕ(x) = {x}. Additivity: If S = S 1 S 2 and S 1 S 2 then Con (S, ϕ) = Con (S 1, ϕ) Con (S 2, ϕ). Homotopy invariance: If N is an isolating neighborhood for a family of flows ϕ t continuously depending on t then Con (N, ϕ 0 ) = Con (N, ϕ 1 ).

120 Discrete case 75 Let f : X X be the generator of ϕ, i.e. f(x) := ϕ(x, 1).

121 Discrete case 75 Let f : X X be the generator of ϕ, i.e. f(x) := ϕ(x, 1). A function σ : Z N is a solution to f through x in N X if σ(0) = x and f(σ(n)) = σ(n + 1) for all n Z.

122 Discrete case 75 Let f : X X be the generator of ϕ, i.e. f(x) := ϕ(x, 1). A function σ : Z N is a solution to f through x in N X if σ(0) = x and f(σ(n)) = σ(n + 1) for all n Z. invariant part of N X: Inv(N, f) := { x N σ : Z Na solution to f through x }

123 Discrete case 75 Let f : X X be the generator of ϕ, i.e. f(x) := ϕ(x, 1). A function σ : Z N is a solution to f through x in N X if σ(0) = x and f(σ(n)) = σ(n + 1) for all n Z. invariant part of N X: Inv(N, f) := { x N σ : Z Na solution to f through x } A compact set S X is called an isolated invariant set for f if there exists a compact neighborhood N of S such that S = Inv(N, ϕ) int N.

124 Discrete case 75 Let f : X X be the generator of ϕ, i.e. f(x) := ϕ(x, 1). A function σ : Z N is a solution to f through x in N X if σ(0) = x and f(σ(n)) = σ(n + 1) for all n Z. invariant part of N X: Inv(N, f) := { x N σ : Z Na solution to f through x } A compact set S X is called an isolated invariant set for f if there exists a compact neighborhood N of S such that S = Inv(N, ϕ) int N. Then, N is called an isolating neighborhood (for S).

125 Index pairs 76 A pair of compact sets P = (P 1, P 2 ) is called an index pair for f and an isolated invariant set S iff (i) (positive relative invariance) f(p 2 ) P 1 P 2 (ii) (exit set) P 1 cl(f(p 1 ) \ P 1 ) P 2 (iii) (isolation) S = Inv(cl(P 1 \ P 2 ), f) int(p 1 \ P 2 )

126 Index pairs 76 A pair of compact sets P = (P 1, P 2 ) is called an index pair for f and an isolated invariant set S iff (i) (positive relative invariance) f(p 2 ) P 1 P 2 (ii) (exit set) P 1 cl(f(p 1 ) \ P 1 ) P 2 (iii) (isolation) S = Inv(cl(P 1 \ P 2 ), f) int(p 1 \ P 2 )

127 Index pairs 77 Proposition. If pair P = (P 1, P 2 ) of compact subsets of an isolating neighborhood N satisfies f(p 2 ) P 1 P 2 P 1 \ f 1 (N) P 2 Inv N int(p 1 \P 2 ). then P is an index pair for f and Inv(N, f). H (P 1, P 2 ) is not an invariant.

128 E - a category Leray Functor 78

129 Leray Functor 78 E - a category the category of endomorphisms of E: Objects: pairs (E, e), where A E and e E(E, E) Morphisms: ψ (E 1,, e 1 ) (E 2, e 2 ) iff ψ E(E 1, E 2 ) and ψe 1 = e 2 ψ

130 Leray Functor 78 E - a category the category of endomorphisms of E: Objects: pairs (E, e), where A E and e E(E, E) Morphisms: ψ (E 1,, e 1 ) (E 2, e 2 ) iff ψ E(E 1, E 2 ) and ψe 1 = e 2 ψ the category of automorphisms of E - full subcategory of Endo(E) consisting of pairs (A, a) Endo(E) such that a E(A, A) is an automorphism

131 Leray Functor 78 E - a category the category of endomorphisms of E: Objects: pairs (E, e), where A E and e E(E, E) Morphisms: ψ (E 1,, e 1 ) (E 2, e 2 ) iff ψ E(E 1, E 2 ) and ψe 1 = e 2 ψ the category of automorphisms of E - full subcategory of Endo(E) consisting of pairs (A, a) Endo(E) such that a E(A, A) is an automorphism V 0 - the category of finitely dimensional vector spaces over R

132 Leray Functor 78 E - a category the category of endomorphisms of E: Objects: pairs (E, e), where A E and e E(E, E) Morphisms: ψ (E 1,, e 1 ) (E 2, e 2 ) iff ψ E(E 1, E 2 ) and ψe 1 = e 2 ψ the category of automorphisms of E - full subcategory of Endo(E) consisting of pairs (A, a) Endo(E) such that a E(A, A) is an automorphism V 0 - the category of finitely dimensional vector spaces over R the generalized kernel of α V 0 (V, V ) is gker α := ker α n n N

133 Leray Functor 78 E - a category the category of endomorphisms of E: Objects: pairs (E, e), where A E and e E(E, E) Morphisms: ψ (E 1,, e 1 ) (E 2, e 2 ) iff ψ E(E 1, E 2 ) and ψe 1 = e 2 ψ the category of automorphisms of E - full subcategory of Endo(E) consisting of pairs (A, a) Endo(E) such that a E(A, A) is an automorphism V 0 - the category of finitely dimensional vector spaces over R the generalized kernel of α V 0 (V, V ) is gker α := ker α n n N the Leray functor L : Endo(V 0 ) Auto(V 0 ): L(V, v) := (V/ gker v, v )

134 Leray Functor 78 E - a category the category of endomorphisms of E: Objects: pairs (E, e), where A E and e E(E, E) Morphisms: ψ (E 1,, e 1 ) (E 2, e 2 ) iff ψ E(E 1, E 2 ) and ψe 1 = e 2 ψ the category of automorphisms of E - full subcategory of Endo(E) consisting of pairs (A, a) Endo(E) such that a E(A, A) is an automorphism V 0 - the category of finitely dimensional vector spaces over R the generalized kernel of α V 0 (V, V ) is gker α := ker α n n N the Leray functor L : Endo(V 0 ) Auto(V 0 ): L(V, v) := (V/ gker v, v )

135 Index quadruples and index maps 79 A quadruple P = (P 1, P 2, P 1, P 2 ) is an index quadruple for f and S if (P 1, P 2 ) is an index pair for f and S and ( P 1, P 2 ) is a topological pair such that the map f P : (P 1, P 2 ) x f(x) ( P 1, P 2 ) ι P : (P 1, P 2 ) x x ( P 1, P 2 ) are well defined and ι P is an excision (induces an isomorphism in cohomology)

136 Index quadruples and index maps 79 A quadruple P = (P 1, P 2, P 1, P 2 ) is an index quadruple for f and S if (P 1, P 2 ) is an index pair for f and S and ( P 1, P 2 ) is a topological pair such that the map f P : (P 1, P 2 ) x f(x) ( P 1, P 2 ) ι P : (P 1, P 2 ) x x ( P 1, P 2 ) are well defined and ι P is an excision (induces an isomorphism in cohomology) Given an index quadruple, we define the index map as the composition I P := H (f P P) H (ι P ) 1

137 The Conley index for discrete dynamical systems 80 Theorem. (MM,1990,2005) For every isolating neighborhood N of f there exists an index quadruple P such that Inv(N, f) P 1 P 1 N. Moreover, if P and Q are two such quadruples, then L(H (P 1, P 2 ), I P ) = L(H (Q 1, Q 2 ), I Q ).

138 The Conley index for discrete dynamical systems 80 Theorem. (MM,1990,2005) For every isolating neighborhood N of f there exists an index quadruple P such that Inv(N, f) P 1 P 1 N. Moreover, if P and Q are two such quadruples, then L(H (P 1, P 2 ), I P ) = L(H (Q 1, Q 2 ), I Q ). The Conley index of f in N is (CH (N, f), χ(n, f)) := L(H (P 1, P 2 ), I P ).

139 81 J.W. Robbin, D. Salamon, shape theory, inverse limit functor MM, cohomology, Leray functor A. Szymczak, homotopy, Szymczak functor (most general) J. Franks, D. Richeson, a reformulation of Szymczak construction in terms of shift equivalence

140 An example 82

141 An example 83

142 Main properties 84 Theorem. Ważewski property (J.W. Robbin, D. Salamon, 1988): Con (N, f) 0 Inv(N, f). Lefschetz property (MM, 1989): If Λ(χ(N, f)) 0 then there exists an x N such that f(x) = x. Additivity:(J.W. Robbin, D. Salamon, 1988): If S = S 1 S 2 and S 1 S 2 then Con (S, f) = Con (S 1, f) Con (S 2, f). Homotopy invariance:(j.w. Robbin, D. Salamon, 1988): If N is an isolating neighborhood for a family of flows f t continuously depending on t then Con (N, f 0 ) = Con (N, f 1 ).

143 Discrete vs. continuous case. 85 Theorem. (MM, 1990) Let ϕ : X R X be a flow and for t R let ϕ t : X X be the map defined by ϕ t (x) := ϕ(x, t). If S X is a compact set, then the following conditions are equivalent. (i) S is an isolated invariant set with respect to ϕ, (ii) S is an isolated invariant set with respect to ϕ t for all t 0, (iii) S is an isolated invariant set with respect to ϕ t for some t 0. Moreover, if one of the above conditions is satisfied, then for any t 0 χ(s, ϕ t ) = id, Con (S, ϕ t ) = Con (S, ϕ).

144 Conley index and horseshoe dynamics 86 Given a compact set N and α {0, 1} n put n 1 N α := f i (N αi ) i=0 and for ᾱ = (α 1, α 2,... α m ) with α j {0, 1} n put m Nᾱ := N α j. j=1

145 Conley index and horseshoe dynamics 86 Given a compact set N and α {0, 1} n put n 1 N α := f i (N αi ) i=0 and for ᾱ = (α 1, α 2,... α m ) with α j {0, 1} n put m Nᾱ := N α j. j=1 Proposition. so is N α and Nᾱ If N is an isolating neighborhood for f then

146 Conley index and horseshoe dynamics 87 Theorem. (K. Mischaikow, MM, 1993) Assume N = N 0 N 1 is an isolating neighbourhood for f such that N 0 and N 1 are disjoint compact polyhedra. If for k = 0, 1 { Con n (Q, Id) if n = 1 (N k ) = 0 otherwise and χ (N 00,01,11, f), χ (N 00,10,11, f) are different from identity then there exists a d N and a continuous surjection ρ : Inv(N, f) Σ 2 such that ρ f d = σ ρ where σ : Σ 2 Σ 2 is the full shift dynamics on two symbols. Moreover, for each periodic sequence α Σ 2 there exists a periodic point x N such that ρ(x) = α.

147 An example 88

148 An example 89

149 An example 90

150 References 91 T. Ważewski, Une méthode topologique de l examen du phénomène asymptotique relativement aux équations différentielles ordinaires, Rend. Accad. Nazionale dei Lincei, Cl. Sci. fisiche, mat. e naturali (1947). C. Conley, Isolated invariant sets and the Morse index, CBMS Regional Conference Series in Math (1978). J. Robbin and D. Salamon, Dynamical systems, shape theory, and the Conley index, Ergodic Theory and Dynamical Systems (1988). MM, Leray functor and the cohomological Conley index for discrete time dynamical systems, Trans. Amer. Math. Soc. (1990). A. Szymczak, The Conley index for discrete semidynamical systems, Topology and Its Applications (1995). J. Franks and D. Richeson, Shift equivalence and the Conley index, Trans. Amer. Math. Soc. (2000).

151 Outline 92 Dynamical systems Rigorous numerics of dynamical systems Homological invariants of dynamical systems Computing homological invariants Homology algorithms for subsets of R d Homology algorithms for maps of subsets of R d Applications

152 Cubical sets 93 The set A R d is cubical if there exists a finite family A K such that A = A.

153 Cubical sets 93 The set A R d is cubical if there exists a finite family A K such that A = A. The family A is referred to as the representation of A.

154 Cubical sets 93 The set A R d is cubical if there exists a finite family A K such that A = A. The family A is referred to as the representation of A. The unique minimal representation, the minimal representation of A, is denoted by K min (A).

155 Cubical sets 93 The set A R d is cubical if there exists a finite family A K such that A = A. The family A is referred to as the representation of A. The unique minimal representation, the minimal representation of A, is denoted by K min (A). A cubical set is a full cubical set if its minimal representation consists only of full elementary cubes.

156 Cubical sets 93 The set A R d is cubical if there exists a finite family A K such that A = A. The family A is referred to as the representation of A. The unique minimal representation, the minimal representation of A, is denoted by K min (A). A cubical set is a full cubical set if its minimal representation consists only of full elementary cubes. Theorem. (Blass, Holsztyński, 1972) Every polyhedron is homeomorphic to a cubical set.

157 A cubical set in R 2 94

158 A full cubical set in R 3 95

159 Combinatorial boundary and interior 96 For A R d define o d (A) := { Q K d Q A },

160 Combinatorial boundary and interior 96 For A R d define o d (A) := { Q K d Q A }, For N K d define int N := { Q N o d (Q) N }, bd N := N \ int(n ).

161 Combinatorial boundary and interior 97

162 Combinatorial boundary and interior 98

163 Combinatorial boundary and interior 99

164 Multivalued combinatorial maps 100 X K d a finite subfamily F : X X a multivalued combinatorial map

165 Multivalued combinatorial maps 100 X K d a finite subfamily F : X X a multivalued combinatorial map The associated digraph has X as the set of vertices and an edge from P to Q iff Q F(P ).

166 Combinatorial boundary and interior 101

167 Combinatorial enclosures 102 A combinatorial multivalued map F : X X is a combinatorial enclosure of f : X X if for every Q X o d (f(q)) F(Q).

168 Combinatorial enclosures 102 A combinatorial multivalued map F : X X is a combinatorial enclosure of f : X X if for every Q X o d (f(q)) F(Q). In this case we say that f is a selector of F.

169 Combinatorial enclosures 102 A combinatorial multivalued map F : X X is a combinatorial enclosure of f : X X if for every Q X o d (f(q)) F(Q). In this case we say that f is a selector of F. If F : X X is a combinatorial enclosure of f : X X, then for every Q X f(q) int F(Q).

170 Combinatorial enclosures 103

171 Combinatorial enclosures 104

172 Combinatorial enclosures 105

173 Combinatorial enclosures 106

174 Graph of a continuous map f 107

175 Estimates of values on the grid of cubes 108

176 Multivalued representation F 109

177 Combinatorial solutions 110 Let I be an interval in Z containing 0. A solution through Q K under F is a function Γ : I K satisfying the following two properties: (1) Γ(0) = Q, (2) Γ(n + 1) F(Γ(n)) for all n such that n, n + 1 I.

178 Combinatorial solutions 110 Let I be an interval in Z containing 0. A solution through Q K under F is a function Γ : I K satisfying the following two properties: (1) Γ(0) = Q, (2) Γ(n + 1) F(Γ(n)) for all n such that n, n + 1 I. In the language of the associated digraph a solution is just a path in the digraph

179 Combinatorial invariant parts 111 Assume N K is finite. The invariant part of N under F is Inv(N, F) := { Q N there exists a full solution Γ : Z N }.

180 Combinatorial invariant parts 111 Assume N K is finite. The invariant part of N under F is Inv(N, F) := { Q N there exists a full solution Γ : Z N }. The positively invariant part and the negatively invariant part of N under F are defined respectively by Inv + (N, F) := { Q N there exists a solution Γ : Z + N } Inv (N, F) := { Q N there exists a solution Γ : Z N }

181 Combinatorial invariant parts 111 Assume N K is finite. The invariant part of N under F is Inv(N, F) := { Q N there exists a full solution Γ : Z N }. The positively invariant part and the negatively invariant part of N under F are defined respectively by Inv + (N, F) := { Q N there exists a solution Γ : Z + N } Inv (N, F) := { Q N there exists a solution Γ : Z N } We have the following obvious formula Inv(N, F) = Inv (N, F) Inv + (N, F).

182 Algorithmizable formulae for invariant parts 112 Let F N : N N denote the map given by F N (Q) := F(Q) N.

183 Algorithmizable formulae for invariant parts 112 Let F N : N N denote the map given by F N (Q) := F(Q) N. There exists an integer n such that n Inv + (N, F) = F i N (N ) Inv (N, F) = i=0 n i=0 F i N (N )

184 Combinatorial Index Pairs. 113 A finite subset N of K d is an isolating neighborhood for F if Inv(N, F) int N.

185 Combinatorial Index Pairs. 113 A finite subset N of K d is an isolating neighborhood for F if Inv(N, F) int N. We say that (P 1, P 2 ) is a combinatorial index pair for F in N if P 2 P 1 N and the following three conditions are satisfied.

186 Combinatorial Index Pairs. 113 A finite subset N of K d is an isolating neighborhood for F if Inv(N, F) int N. We say that (P 1, P 2 ) is a combinatorial index pair for F in N if P 2 P 1 N and the following three conditions are satisfied. (positive relative invariance) F(P i ) N P i

187 Combinatorial Index Pairs. 113 A finite subset N of K d is an isolating neighborhood for F if Inv(N, F) int N. We say that (P 1, P 2 ) is a combinatorial index pair for F in N if P 2 P 1 N and the following three conditions are satisfied. (positive relative invariance) F(P i ) N P i (exit set) F(P 1 ) bd N P 2

188 Combinatorial Index Pairs. 113 A finite subset N of K d is an isolating neighborhood for F if Inv(N, F) int N. We say that (P 1, P 2 ) is a combinatorial index pair for F in N if P 2 P 1 N and the following three conditions are satisfied. (positive relative invariance) F(P i ) N P i (exit set) F(P 1 ) bd N P 2 (isolation) Inv(N, F) P 1 \ P 2

189 Index Pairs from Combinatorial Index Pairs. 114 Theorem. (A. Szymczak 1997, MM 1996,2006) Assume N is an isolating neighborhood for F and (P 1, P 2 ) is a combinatorial index pair for F in N. Then for any selector f of F the set N is an isolating neighborhood for f and ( P 1, P 2 ) is a index pair for f.

190 Construction of index quadruples 115 Theorem. (MM,2005) Assume N is an isolating neighborhood for F. Let P 1 := Inv (N, F), P 2 := Inv (N, F) \ Inv + (N, F). Then (P 1, P 2 ) is a combinatorial index pair for F in N and P 1 \ P 2 int N. Moreover, if P 1 := P 1 F(P 1 ), P 2 := P 2 (F(P 1 ) \ P 1 ), then for any selector f of F the quadruple ( P 1, P 1, P 1, P 2 ) is an index quadruple.

191 Positive invariant part algorithm 116 function positiveinvariantpart(set N, combinatorialmap F) F := restrictedmap(f, N); S := C := N; repeat S = S; C := evaluate(f, C); S := S C; until (S = S ); return S;

192 Positive invariant part algorithm 116 function positiveinvariantpart(set N, combinatorialmap F) F := restrictedmap(f, N); S := C := N; repeat S = S; C := evaluate(f, C); S := S C; until (S = S ); return S; Proposition. Assume the algorithm is called with a collection of cubes N and a combinatorial multivalued map F on input. Then it always stops and returns the positive invariant part of F in N.

193 Combinatorial Index Pair Algorithm 117 function combinatorialindexpair(set N, combinatorialmap F) S + := positiveinvariantpart(n, F); Finv := evaluateinverse(f); S := positiveinvariantpart(n, Finv); if S S + int(n) then P 1 := S ; P 2 := S \ S + ; P 1 := P 1 F(P 1 ); P 2 := P 2 F(P 1 ) \ P 1 ; return (P 1, P 2, P 1, P 2 ); else return F ailure ; endif;

194 Combinatorial Index Pair Algorithm 117 function combinatorialindexpair(set N, combinatorialmap F) S + := positiveinvariantpart(n, F); Finv := evaluateinverse(f); S := positiveinvariantpart(n, Finv); if S S + int(n) then P 1 := S ; P 2 := S \ S + ; P 1 := P 1 F(P 1 ); P 2 := P 2 F(P 1 ) \ P 1 ; return (P 1, P 2, P 1, P 2 ); else return F ailure ; endif; Theorem. Assume the algorithm is called with a collection of cubes N and a combinatorial enclosure of f on input. If it does not fail, then it returns representations of an index quadruple of f.

195 References 118 MM, Topological invariants, multivalued maps and computer assisted proofs in Dynamics, Computers Math. Applic. (1996). A. Szymczak, A combinatorial procedure for finding isolating neighborhoods and index pairs, Proc. Royal Soc. Edinburgh, Ser. A (1997). MM, Index Pairs Algorithms, Foundations of Computational Mathematics (2006).