E.Colli M.L.doNascimento E.Vargas Metricaspectsofthedynamicsofcoveringmapsofthecircle

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Introduction Main result Tools and Ideas

Metric aspects of the dynamics of covering maps of the circle

E. Colli ¹ M. L. do Nascimento ² E. Vargas ³

1

Departamento de Matemática Aplicada Universidade de São Paulo

2

Centro de Ciências Exatas e Naturais Universidade Federal do Pará

3

Departamento de Matemática

Universidade de São Paulo

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O ^UTLINE

1 I NTRODUCTION Set up

Topolgical aspects Metric aspects

2 M AIN RESULT Statement

Strong recurrence History

3 T OOLS AND I DEAS

Difference equation I

Difference equation II

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Set up

Topolgical aspects Metric aspects

C RITICAL COVERING MAPS ON S ¹

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8

1.0

Class C ¹ , degree d ≥ 2, critical point c of order ` > 1;

Class C ³ and Negative Schwarzian on S ¹ \ {c};

There is no wandering interval.

(4)

Set up

C RITICAL COVERING MAPS ON S ¹

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8

1.0

Class C ¹ , degree d ≥ 2, critical

point c of order ` > 1;

Class C ³ and Negative Schwarzian on S ¹ \ {c};

There is no wandering interval.

(5)

Set up

C RITICAL COVERING MAPS ON S ¹

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8

1.0

Class C ¹ , degree d ≥ 2, critical

point c of order ` > 1;

Class C ³ and Negative Schwarzian on S ¹ \ {c};

There is no wandering interval.

(6)

Set up

C RITICAL COVERING MAPS ON S ¹

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8

1.0

Class C ¹ , degree d ≥ 2, critical point c of order ` > 1;

Class C ³ and Negative Schwarzian on S ¹ \ {c};

There is no wandering interval.

(7)

Set up

C RITICAL COVERING MAPS ON S ¹

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8

1.0

Class C ¹ , degree d ≥ 2, critical point c of order ` > 1;

Class C ³ and Negative Schwarzian on S ¹ \ {c};

There is no wandering interval.

(8)

Set up

C RITICAL COVERING MAPS ON S ¹

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8

1.0

Class C ¹ , degree d ≥ 2, critical point c of order ` > 1;

Class C ³ and Negative Schwarzian on S ¹ \ {c};

There is no wandering interval.

(9)

Set up

C RITICAL COVERING MAPS ON S ¹

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8

1.0

Class C ¹ , degree d ≥ 2, critical point c of order ` > 1;

Class C ³ and Negative Schwarzian on S ¹ \ {c};

There is no wandering interval.

(10)

Set up

C RITICAL COVERING MAPS ON S ¹

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8

1.0

Class C ¹ , degree d ≥ 2, critical point c of order ` > 1;

Class C ³ and Negative Schwarzian on S ¹ \ {c};

There is no wandering interval.

(11)

Set up

C RITICAL COVERING MAPS ON S ¹

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8

1.0

Class C ¹ , degree d ≥ 2, critical point c of order ` > 1;

Class C ³ and Negative Schwarzian on S ¹ \ {c};

There is no wandering interval.

(12)

Set up

C RITICAL COVERING MAPS ON S ¹

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8

1.0

Class C ¹ , degree d ≥ 2, critical point c of order ` > 1;

Class C ³ and Negative Schwarzian on S ¹ \ {c};

There is no wandering interval.

(13)

Set up

T ^{HE MAP} R(z ) = 2z ^MOD 1

0.0 0.2 0.4 0.6 0.8 1.0

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Set up

Q ^UESTIONS

Regularity of conjugacy;

Invariant measures Attractors;

Decay of geometry;

Renormalization;

Growth of Df ⁿ (c ^f ).

(15)

Set up

Q ^UESTIONS

Regularity of conjugacy;

Invariant measures Attractors;

Decay of geometry;

Renormalization;

Growth of Df ⁿ (c ^f ).

(16)

Set up

Q ^UESTIONS

Regularity of conjugacy;

Invariant measures Attractors;

Decay of geometry;

Renormalization;

Growth of Df ⁿ (c ^f ).

(17)

Set up

Q ^UESTIONS

Regularity of conjugacy;

Invariant measures Attractors;

Decay of geometry;

Renormalization;

Growth of Df ⁿ (c ^f ).

(18)

Set up

Q ^UESTIONS

Regularity of conjugacy;

Invariant measures Attractors;

Decay of geometry;

Renormalization;

Growth of Df ⁿ (c ^f ).

(19)

Set up

Q ^UESTIONS

Regularity of conjugacy;

Invariant measures Attractors;

Decay of geometry;

Renormalization;

Growth of Df ⁿ (c ^f ).

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Statement Strong recurrence History

G ROWTH OF DERIVATIVES

T HEOREM

Let f be as above, µ ⁺ := ¹⁺

√ 1+4`

2` and ϕ := ¹⁺

√ 5 2 . If in addition f has the Fibonacci combinatorics then:

1

If 1 < ` < 2 then ^{log log} ^Df _n

^sn

^(c

^f

⁾ → log µ ⁺ > 0.

2

If 1 < ` < 2 then ^{log log} _log ^Df _k

^k

^(c

^f

⁾ → ^log _log ^µ _ϕ

⁺

< 1.

3

If ` = 2 then 0 < K ≤ ^log ^Df _n

^sn

^(c

^f

⁾ ≤ K ⁻¹ .

4

If ` > 2 then Df ^s

ⁿ

(c ^f ) is bounded.

5

If 1 < ` ≤ 2 then ω(c) (which is a minimal

Cantor set) has Hausdorff dimension zero.

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G ROWTH OF DERIVATIVES

T HEOREM

Let f be as above, µ ⁺ := ¹⁺

√ 1+4`

2` and ϕ := ¹⁺

√ 5 2 . If in addition f has the Fibonacci combinatorics then:

1

If 1 < ` < 2 then ^{log log} ^Df _n

^sn

^(c

^f

⁾ → log µ ⁺ > 0.

2

If 1 < ` < 2 then ^{log log} _log ^Df _k

^k

^(c

^f

⁾ → ^log _log ^µ _ϕ

⁺

< 1.

3

If ` = 2 then 0 < K ≤ ^log ^Df _n

^sn

^(c

^f

⁾ ≤ K ⁻¹ .

4

If ` > 2 then Df ^s

ⁿ

(c ^f ) is bounded.

5

If 1 < ` ≤ 2 then ω(c) (which is a minimal

Cantor set) has Hausdorff dimension zero.

(22)

G ROWTH OF DERIVATIVES

T HEOREM

Let f be as above, µ ⁺ := ¹⁺

√ 1+4`

2` and ϕ := ¹⁺

√ 5 2 . If in addition f has the Fibonacci combinatorics then:

1

If 1 < ` < 2 then ^{log log} ^Df _n

^sn

^(c

^f

⁾ → log µ ⁺ > 0.

2

If 1 < ` < 2 then ^{log log} _log ^Df _k

^k

^(c

^f

⁾ → ^log _log ^µ _ϕ

⁺

< 1.

3

If ` = 2 then 0 < K ≤ ^log ^Df _n

^sn

^(c

^f

⁾ ≤ K ⁻¹ .

4

If ` > 2 then Df ^s

ⁿ

(c ^f ) is bounded.

5

If 1 < ` ≤ 2 then ω(c) (which is a minimal

Cantor set) has Hausdorff dimension zero.

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G ROWTH OF DERIVATIVES

T HEOREM

Let f be as above, µ ⁺ := ¹⁺

√ 1+4`

2` and ϕ := ¹⁺

√ 5 2 . If in addition f has the Fibonacci combinatorics then:

1

If 1 < ` < 2 then ^{log log} ^Df _n

^sn

^(c

^f

⁾ → log µ ⁺ > 0.

2

If 1 < ` < 2 then ^{log log} _log ^Df _k

^k

^(c

^f

⁾ → ^log _log ^µ _ϕ

⁺

< 1.

3

If ` = 2 then 0 < K ≤ ^log ^Df _n

^sn

^(c

^f

⁾ ≤ K ⁻¹ .

4

If ` > 2 then Df ^s

ⁿ

(c ^f ) is bounded.

5

If 1 < ` ≤ 2 then ω(c) (which is a minimal

Cantor set) has Hausdorff dimension zero.

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G ROWTH OF DERIVATIVES

T HEOREM

Let f be as above, µ ⁺ := ¹⁺

√ 1+4`

2` and ϕ := ¹⁺

√ 5 2 . If in addition f has the Fibonacci combinatorics then:

1

If 1 < ` < 2 then ^{log log} ^Df _n

^sn

^(c

^f

⁾ → log µ ⁺ > 0.

2

If 1 < ` < 2 then ^{log log} _log ^Df _k

^k

^(c

^f

⁾ → ^log _log ^µ _ϕ

⁺

< 1.

3

If ` = 2 then 0 < K ≤ ^log ^Df _n

^sn

^(c

^f

⁾ ≤ K ⁻¹ .

4

If ` > 2 then Df ^s

ⁿ

(c ^f ) is bounded.

5

If 1 < ` ≤ 2 then ω(c) (which is a minimal

Cantor set) has Hausdorff dimension zero.

(25)

G ROWTH OF DERIVATIVES

T HEOREM

Let f be as above, µ ⁺ := ¹⁺

√ 1+4`

2` and ϕ := ¹⁺

√ 5 2 . If in addition f has the Fibonacci combinatorics then:

1

If 1 < ` < 2 then ^{log log} ^Df _n

^sn

^(c

^f

⁾ → log µ ⁺ > 0.

2

If 1 < ` < 2 then ^{log log} _log ^Df _k

^k

^(c

^f

⁾ → ^log _log ^µ _ϕ

⁺

< 1.

3

If ` = 2 then 0 < K ≤ ^log ^Df _n

^sn

^(c

^f

⁾ ≤ K ⁻¹ .

4

If ` > 2 then Df ^s

ⁿ

(c ^f ) is bounded.

5

If 1 < ` ≤ 2 then ω(c) (which is a minimal

Cantor set) has Hausdorff dimension zero.

(26)

G ROWTH OF DERIVATIVES

T HEOREM

Let f be as above, µ ⁺ := ¹⁺

√ 1+4`

2` and ϕ := ¹⁺

√ 5 2 . If in addition f has the Fibonacci combinatorics then:

1

If 1 < ` < 2 then ^{log log} ^Df _n

^sn

^(c

^f

⁾ → log µ ⁺ > 0.

2

If 1 < ` < 2 then ^{log log} _log ^Df _k

^k

^(c

^f

⁾ → ^log _log ^µ _ϕ

⁺

< 1.

3

If ` = 2 then 0 < K ≤ ^log ^Df _n

^sn

^(c

^f

⁾ ≤ K ⁻¹ .

4

If ` > 2 then Df ^s

ⁿ

(c ^f ) is bounded.

5

If 1 < ` ≤ 2 then ω(c) (which is a minimal

Cantor set) has Hausdorff dimension zero.

(27)

G ROWTH OF DERIVATIVES

T HEOREM

Let f be as above, µ ⁺ := ¹⁺

√ 1+4`

2` and ϕ := ¹⁺

√ 5 2 . If in addition f has the Fibonacci combinatorics then:

1

If 1 < ` < 2 then ^{log log} ^Df _n

^sn

^(c

^f

⁾ → log µ ⁺ > 0.

2

If 1 < ` < 2 then ^{log log} _log ^Df _k

^k

^(c

^f

⁾ → ^log _log ^µ _ϕ

⁺

< 1.

3

If ` = 2 then 0 < K ≤ ^log ^Df _n

^sn

^(c

^f

⁾ ≤ K ⁻¹ .

4

If ` > 2 then Df ^s

ⁿ

(c ^f ) is bounded.

5

If 1 < ` ≤ 2 then ω(c) (which is a minimal

Cantor set) has Hausdorff dimension zero.

(28)

G ROWTH OF DERIVATIVES

T HEOREM

Let f be as above, µ ⁺ := ¹⁺

√ 1+4`

2` and ϕ := ¹⁺

√ 5 2 . If in addition f has the Fibonacci combinatorics then:

1

If 1 < ` < 2 then ^{log log} ^Df _n

^sn

^(c

^f

⁾ → log µ ⁺ > 0.

2

If 1 < ` < 2 then ^{log log} _log ^Df _k

^k

^(c

^f

⁾ → ^log _log ^µ _ϕ

⁺

< 1.

3

If ` = 2 then 0 < K ≤ ^log ^Df _n

^sn

^(c

^f

⁾ ≤ K ⁻¹ .

4

If ` > 2 then Df ^s

ⁿ

(c ^f ) is bounded.

5

If 1 < ` ≤ 2 then ω(c) (which is a minimal

Cantor set) has Hausdorff dimension zero.

(29)

G ROWTH OF DERIVATIVES

T HEOREM

Let f be as above, µ ⁺ := ¹⁺

√ 1+4`

2` and ϕ := ¹⁺

√ 5 2 . If in addition f has the Fibonacci combinatorics then:

1

If 1 < ` < 2 then ^{log log} ^Df _n

^sn

^(c

^f

⁾ → log µ ⁺ > 0.

2

If 1 < ` < 2 then ^{log log} _log ^Df _k

^k

^(c

^f

⁾ → ^log _log ^µ _ϕ

⁺

< 1.

3

If ` = 2 then 0 < K ≤ ^log ^Df _n

^sn

^(c

^f

⁾ ≤ K ⁻¹ .

4

If ` > 2 then Df ^s

ⁿ

(c ^f ) is bounded.

5

If 1 < ` ≤ 2 then ω(c) (which is a minimal

Cantor set) has Hausdorff dimension zero.

(30)

G ROWTH OF DERIVATIVES

T HEOREM

Let f be as above, µ ⁺ := ¹⁺

√ 1+4`

2` and ϕ := ¹⁺

√ 5 2 . If in addition f has the Fibonacci combinatorics then:

1

If 1 < ` < 2 then ^{log log} ^Df _n

^sn

^(c

^f

⁾ → log µ ⁺ > 0.

2

If 1 < ` < 2 then ^{log log} _log ^Df _k

^k

^(c

^f

⁾ → ^log _log ^µ _ϕ

⁺

< 1.

3

If ` = 2 then 0 < K ≤ ^log ^Df _n

^sn

^(c

^f

⁾ ≤ K ⁻¹ .

4

If ` > 2 then Df ^s

ⁿ

(c ^f ) is bounded.

5

If 1 < ` ≤ 2 then ω(c) (which is a minimal

Cantor set) has Hausdorff dimension zero.

(31)

G ROWTH OF DERIVATIVES

T HEOREM

Let f be as above, µ ⁺ := ¹⁺

√ 1+4`

2` and ϕ := ¹⁺

√ 5 2 . If in addition f has the Fibonacci combinatorics then:

1

If 1 < ` < 2 then ^{log log} ^Df _n

^sn

^(c

^f

⁾ → log µ ⁺ > 0.

2

If 1 < ` < 2 then ^{log log} _log ^Df _k

^k

^(c

^f

⁾ → ^log _log ^µ _ϕ

⁺

< 1.

3

If ` = 2 then 0 < K ≤ ^log ^Df _n

^sn

^(c

^f

⁾ ≤ K ⁻¹ .

4

If ` > 2 then Df ^s

ⁿ

(c ^f ) is bounded.

5

If 1 < ` ≤ 2 then ω(c) (which is a minimal

Cantor set) has Hausdorff dimension zero.

(32)

T ^HE F IBONACCI COMBINATORICS

c is recurrent;

Set I 0 := (0, 1) and I n 3 c, a domain of the first return map φ n to I n−1 ( φ n : I n → I n−1 );

If φ _n (c) = f ^s

ⁿ

(c) then s n is the Fibonacci

sequence 1, 2, 3, 5, 8, . . .

(33)

T ^HE F IBONACCI COMBINATORICS

c is recurrent;

Set I 0 := (0, 1) and I n 3 c, a domain of the first return map φ n to I n−1 ( φ n : I n → I n−1 );

If φ _n (c) = f ^s

ⁿ

(c) then s n is the Fibonacci

sequence 1, 2, 3, 5, 8, . . .

(34)

T ^HE F IBONACCI COMBINATORICS

c is recurrent;

Set I 0 := (0, 1) and I n 3 c, a domain of the first return map φ n to I n−1 ( φ n : I n → I n−1 );

If φ _n (c) = f ^s

ⁿ

(c) then s n is the Fibonacci

sequence 1, 2, 3, 5, 8, . . .

(35)

T ^HE F IBONACCI COMBINATORICS

c is recurrent;

Set I 0 := (0, 1) and I n 3 c, a domain of the first return map φ n to I n−1 ( φ n : I n → I n−1 );

If φ _n (c) = f ^s

ⁿ

(c) then s n is the Fibonacci

sequence 1, 2, 3, 5, 8, . . .

(36)

T ^HE F IBONACCI COMBINATORICS

c is recurrent;

Set I 0 := (0, 1) and I n 3 c, a domain of the first return map φ n to I n−1 ( φ n : I n → I n−1 );

If φ _n (c) = f ^s

ⁿ

(c) then s n is the Fibonacci

sequence 1, 2, 3, 5, 8, . . .

(37)

T ^HE F IBONACCI COMBINATORICS

c is recurrent;

Set I 0 := (0, 1) and I n 3 c, a domain of the first return map φ n to I n−1 ( φ n : I n → I n−1 );

If φ _n (c) = f ^s

ⁿ

(c) then s n is the Fibonacci

sequence 1, 2, 3, 5, 8, . . .

(38)

T ^HE F IBONACCI COMBINATORICS

c is recurrent;

Set I 0 := (0, 1) and I n 3 c, a domain of the first return map φ _n to I n−1 ( φ _n : I n → I n−1 );

If φ n (c) = f ^s

ⁿ

(c) then s n is the Fibonacci sequence 1, 2, 3, 5, 8, . . .

c S_n

S_{n−1}

S_n

S_{n+1}

c

(39)

T ^HE F IBONACCI COMBINATORICS

c is recurrent;

Set I 0 := (0, 1) and I n 3 c, a domain of the first return map φ _n to I n−1 ( φ _n : I n → I n−1 );

If φ n (c) = f ^s

ⁿ

(c) then s n is the Fibonacci sequence 1, 2, 3, 5, 8, . . .

c S_n

S_{n−1}

S_n

S_{n+1}

c

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O THER DYNAMICS

0.0 0.2 0.4 0.6 0.8 1.0

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O THER DYNAMICS

0.0 0.2 0.4 0.6 0.8 1.0

(42)

O THER DYNAMICS

0.0 0.2 0.4 0.6 0.8 1.0

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O THER DYNAMICS

0.0 0.2 0.4 0.6 0.8 1.0

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Difference equation I Difference equation II

C ^{ROSS RATIO}

Df ^s

ⁿ

⁻¹ (c _s ^f

_n+1

) = ^|J _|j

ⁿ

^|

n

| ≥ _|L ^|L

ⁿ

^||R

ⁿ

^|

n

∪R

n

|

|l

n

∪r

n

|

|l

n

||r

n

|

csn−1

xn csn+1

c

x^fn jn:=c^fsn+1

c^f

ln rn

0 or 1 Jn=csn+2

csn

Rn

Ln

f

f^sⁿ⁻¹

E. Colli, M. L. do Nascimento, E. Vargas School and Workshop on Dynamical Systems, July/2008

(45)

D IFFERENCE INEQUATION I

Df ^s

ⁿ

⁻¹ (c ^f _s

n+1

) = |J n |

|j _n | ≥ |L n ||R n |

|L _n ∪ R n |

|l n ∪ r n |

|l _n ||r _n |

Df ^s

ⁿ

⁻¹ (c _s ^f

_n+1

) ≥ |L _n |

|L _n ∪ R n |

d n − d n+2

d _n+1 ^f

d _n−1 ^f d _n−1 ^f − d _n+1 ^f ,

where d n := |c _s

_n

− c|.

For the next slide λ n := _d ^d

ⁿ

n+2

≥ λ > 1.

(46)

D IFFERENCE INEQUATION I

Df ^s

ⁿ

⁻¹ (c ^f _s

n+1

) = |J n |

|j _n | ≥ |L n ||R n |

|L _n ∪ R n |

|l n ∪ r n |

|l _n ||r _n |

Df ^s

ⁿ

⁻¹ (c _s ^f

_n+1

) ≥ |L _n |

|L _n ∪ R n |

d n − d n+2

d _n+1 ^f

d _n−1 ^f d _n−1 ^f − d _n+1 ^f ,

where d n := |c _s

_n

− c|.

For the next slide λ n := _d ^d

ⁿ

n+2

≥ λ > 1.

(47)

D IFFERENCE INEQUATION I

Df ^s

ⁿ

⁻¹ (c ^f _s

n+1

) = |J n |

|j _n | ≥ |L n ||R n |

|L _n ∪ R n |

|l n ∪ r n |

|l _n ||r _n |

Df ^s

ⁿ

⁻¹ (c _s ^f

_n+1

) ≥ |L _n |

|L _n ∪ R n |

d n − d n+2

d _n+1 ^f

d _n−1 ^f d _n−1 ^f − d _n+1 ^f ,

where d n := |c _s

_n

− c|.

For the next slide λ n := _d ^d

ⁿ

n+2

≥ λ > 1.

(48)

D IFFERENCE INEQUATION I

Df ^s

ⁿ

⁻¹ (c ^f _s

n+1

) = |J n |

|j _n | ≥ |L n ||R n |

|L _n ∪ R n |

|l n ∪ r n |

|l _n ||r _n |

Df ^s

ⁿ

⁻¹ (c _s ^f

_n+1

) ≥ |L _n |

|L _n ∪ R n |

d n − d n+2

d _n+1 ^f

d _n−1 ^f d _n−1 ^f − d _n+1 ^f ,

where d n := |c _s

_n

− c|.

For the next slide λ n := _d ^d

ⁿ

n+2

≥ λ > 1.

(49)

D IFFERENCE INEQUATION I

Df ^s

ⁿ

⁻¹ (c ^f _s

n+1

) = |J n |

|j _n | ≥ |L n ||R n |

|L _n ∪ R n |

|l n ∪ r n |

|l _n ||r _n |

Df ^s

ⁿ

⁻¹ (c _s ^f

_n+1

) ≥ |L _n |

|L _n ∪ R n |

d n − d n+2

d _n+1 ^f

d _n−1 ^f d _n−1 ^f − d _n+1 ^f ,

where d n := |c _s

_n

− c|.

For the next slide λ n := _d ^d

ⁿ

n+2

≥ λ > 1.

(50)

D IFFERENCE INEQUATION I

Df ^s

ⁿ⁺¹

(c ^f ) 2

Df ^s

ⁿ

(c ^f )Df ^s

ⁿ⁻¹

(c ^f ) ≥ K n ` ² λ ^2−` _n−1 q n−1

q n

1 − λ ⁻¹ _n 1 − λ ^−` _n−2

1 − λ ⁻¹ _n−1 1 − λ ^−` _n−1

Df ^s

ⁿ⁺¹

(c ^f ) 1 − λ ⁻¹ _n

² "

Df ^s

ⁿ

(c ^f ) 1 − λ ⁻¹ _n−1

# −1 "

Df ^s

ⁿ⁻¹

(c ^f ) 1 − λ ⁻¹ _n−2

# −1

≥ Q n−1

Q n

σ n

where

σ _n := K n ` ² λ ^2−` _n−1 1 − λ ⁻¹ _n−1 1 − λ ^−` _n−1

1 − λ ⁻¹ _n−2

1 − λ ^−` _n−2 ≥ σ > 1

(51)

D IFFERENCE INEQUATION I

Df ^s

ⁿ⁺¹

(c ^f ) 2

Df ^s

ⁿ

(c ^f )Df ^s

ⁿ⁻¹

(c ^f ) ≥ K n ` ² λ ^2−` _n−1 q n−1

q n

1 − λ ⁻¹ _n 1 − λ ^−` _n−2

1 − λ ⁻¹ _n−1 1 − λ ^−` _n−1

Df ^s

ⁿ⁺¹

(c ^f ) 1 − λ ⁻¹ _n

² "

Df ^s

ⁿ

(c ^f ) 1 − λ ⁻¹ _n−1

# −1 "

Df ^s

ⁿ⁻¹

(c ^f ) 1 − λ ⁻¹ _n−2

# −1

≥ Q n−1

Q n

σ n

where

σ _n := K n ` ² λ ^2−` _n−1 1 − λ ⁻¹ _n−1 1 − λ ^−` _n−1

1 − λ ⁻¹ _n−2

1 − λ ^−` _n−2 ≥ σ > 1

(52)

D IFFERENCE INEQUATION I

Df ^s

ⁿ⁺¹

(c ^f ) 2

Df ^s

ⁿ

(c ^f )Df ^s

ⁿ⁻¹

(c ^f ) ≥ K n ` ² λ ^2−` _n−1 q n−1

q n

1 − λ ⁻¹ _n 1 − λ ^−` _n−2

1 − λ ⁻¹ _n−1 1 − λ ^−` _n−1

Df ^s

ⁿ⁺¹

(c ^f ) 1 − λ ⁻¹ _n

² "

Df ^s

ⁿ

(c ^f ) 1 − λ ⁻¹ _n−1

# −1 "

Df ^s

ⁿ⁻¹

(c ^f ) 1 − λ ⁻¹ _n−2

# −1

≥ Q n−1

Q n

σ n

where

σ _n := K n ` ² λ ^2−` _n−1 1 − λ ⁻¹ _n−1 1 − λ ^−` _n−1

1 − λ ⁻¹ _n−2

1 − λ ^−` _n−2 ≥ σ > 1

(53)

E XPONENTIAL GROWTH

X n := log

Df

^sn

(c

^f

) 1−λ

⁻¹_n−1

≥ Y n , where

Y n+1 − 1 2 Y n − 1

2 Y n−1 = 1

2 log Q n−1 − 1

2 log Q n + 1 2 log σ _n

lim inf Y n

n ≥ 1

3 lim inf log σ _n ≥ ε > 0

(54)

E XPONENTIAL GROWTH

X n := log

Df

^sn

(c

^f

) 1−λ

⁻¹_n−1

≥ Y n , where

Y n+1 − 1 2 Y n − 1

2 Y n−1 = 1

2 log Q n−1 − 1

2 log Q n + 1 2 log σ _n

lim inf Y n

n ≥ 1

3 lim inf log σ n ≥ ε > 0

(55)

D IFFERENCE INEQUATION II

Df ^s

ⁿ

⁻¹ (c ^f ) = ^|R _|r

ⁿ

^|

n

| ≤ ^(|L

ⁿ

^∪J _|L

ⁿ

^|)|J

ⁿ

^|

n

|

|l

n

| (|l

n

∪j

n

|)|j

n

|

csn−1

xn csn+1 c

x^fn c^fsn+1

rn:=c^f ln

0 or 1 csn+2

Rn=csn

Ln

f

jn

Jn

f^sⁿ⁻¹

E. Colli, M. L. do Nascimento, E. Vargas School and Workshop on Dynamical Systems, July/2008

(56)

D IFFERENCE INEQUATION II

C ⁻¹ λ ^2−` _n−1 ≤ Df ^s

ⁿ⁺¹

(c ^f ) 2

Df ^s

ⁿ

(c ^f )Df ^s

ⁿ⁻¹

(c ^f ) ≤ Cλ ^2−` _n−1 .

C ₁ ⁻¹ λ ^` _n−1 ≤ Df ^s

ⁿ

(c ^f ) Df ^s

ⁿ⁻¹

(c ^f ) ≤ C 1 λ ^` _n−1

X n := log Df ^s

ⁿ

(c ^f ) satisfies

−∆ ≤ X n+1 − 1

` (X n−1 + X n ) ≤ ∆

(57)

D IFFERENCE INEQUATION II

C ⁻¹ λ ^2−` _n−1 ≤ Df ^s

ⁿ⁺¹

(c ^f ) 2

Df ^s

ⁿ

(c ^f )Df ^s

ⁿ⁻¹

(c ^f ) ≤ Cλ ^2−` _n−1 .

C ₁ ⁻¹ λ ^` _n−1 ≤ Df ^s

ⁿ

(c ^f ) Df ^s

ⁿ⁻¹

(c ^f ) ≤ C 1 λ ^` _n−1

X n := log Df ^s

ⁿ

(c ^f ) satisfies

−∆ ≤ X n+1 − 1

` (X n−1 + X n ) ≤ ∆

(58)

D IFFERENCE INEQUATION II

C ⁻¹ λ ^2−` _n−1 ≤ Df ^s

ⁿ⁺¹

(c ^f ) 2

Df ^s

ⁿ

(c ^f )Df ^s

ⁿ⁻¹

(c ^f ) ≤ Cλ ^2−` _n−1 .

C ₁ ⁻¹ λ ^` _n−1 ≤ Df ^s

ⁿ

(c ^f ) Df ^s

ⁿ⁻¹

(c ^f ) ≤ C 1 λ ^` _n−1

X n := log Df ^s

ⁿ

(c ^f ) satisfies

−∆ ≤ X n+1 − 1

` (X n−1 + X n ) ≤ ∆

(59)

D IFFERENCE INEQUATION II

C ⁻¹ λ ^2−` _n−1 ≤ Df ^s

ⁿ⁺¹

(c ^f ) 2

Df ^s

ⁿ

(c ^f )Df ^s

ⁿ⁻¹

(c ^f ) ≤ Cλ ^2−` _n−1 .

C ₁ ⁻¹ λ ^` _n−1 ≤ Df ^s

ⁿ

(c ^f ) Df ^s

ⁿ⁻¹

(c ^f ) ≤ C 1 λ ^` _n−1

X n := log Df ^s

ⁿ

(c ^f ) satisfies

−∆ ≤ X n+1 − 1

` (X n−1 + X n ) ≤ ∆

(60)

D IFFERENCE INEQUATION II

C ⁻¹ λ ^2−` _n−1 ≤ Df ^s

ⁿ⁺¹

(c ^f ) 2

Df ^s

ⁿ

(c ^f )Df ^s

ⁿ⁻¹

(c ^f ) ≤ Cλ ^2−` _n−1 .

C ₁ ⁻¹ λ ^` _n−1 ≤ Df ^s

ⁿ

(c ^f ) Df ^s

ⁿ⁻¹

(c ^f ) ≤ C 1 λ ^` _n−1

X n := log Df ^s

ⁿ

(c ^f ) satisfies

−∆ ≤ X n+1 − 1

` (X n−1 + X n ) ≤ ∆

(61)

S UPEREXPONENTIAL GROWTH

Z n+1 = ¹ _` (Z n + Z n−1 ) − ∆ T (x , y ) := (y , ¹ _` (x + y ) − ∆) ; (Z n , Z n+1 ) = T (Z n−1 , Z n ) Fixed point

p := ∆

2 ` − 1 , ∆

2 ` − 1

!

, eigenvalues: µ ± = 1 ± √ 1 + 4`

2`

corresponding eigenvectors: (1, µ ± ).

X n := log Df ^s

ⁿ

(c ^f ) ≥ Z n

y

p (Z_n, Z_{n+1})

(62)

S UPEREXPONENTIAL GROWTH

Z n+1 = ¹ _` (Z n + Z n−1 ) − ∆ T (x , y ) := (y , ¹ _` (x + y ) − ∆) ; (Z n , Z n+1 ) = T (Z n−1 , Z n ) Fixed point

p := ∆

2 ` − 1 , ∆

2 ` − 1

!

, eigenvalues: µ ± = 1 ± √ 1 + 4`

2`

corresponding eigenvectors: (1, µ ± ).

X n := log Df ^s

ⁿ

(c ^f ) ≥ Z n

y

p (Z_n, Z_{n+1})

(63)

S UPEREXPONENTIAL GROWTH

Z n+1 = ¹ _` (Z n + Z n−1 ) − ∆ T (x , y ) := (y , ¹ _` (x + y ) − ∆) ; (Z n , Z n+1 ) = T (Z n−1 , Z n ) Fixed point

p := ∆

2 ` − 1 , ∆

2 ` − 1

!

, eigenvalues: µ ± = 1 ± √ 1 + 4`

2`

corresponding eigenvectors: (1, µ ± ).

X n := log Df ^s

ⁿ

(c ^f ) ≥ Z n

y

p (Z_n, Z_{n+1})

(64)

S UPEREXPONENTIAL GROWTH

Z n+1 = ¹ _` (Z n + Z n−1 ) − ∆ T (x , y ) := (y , ¹ _` (x + y ) − ∆) ; (Z n , Z n+1 ) = T (Z n−1 , Z n ) Fixed point

p := ∆

2 ` − 1 , ∆

2 ` − 1

!

, eigenvalues: µ ± = 1 ± √ 1 + 4`

2`

corresponding eigenvectors: (1, µ ± ).

X n := log Df ^s

ⁿ

(c ^f ) ≥ Z n

y

p (Z_n, Z_{n+1})

(65)

S UPEREXPONENTIAL GROWTH

Z n+1 = ¹ _` (Z n + Z n−1 ) − ∆ T (x , y ) := (y , ¹ _` (x + y ) − ∆) ; (Z n , Z n+1 ) = T (Z n−1 , Z n ) Fixed point

p := ∆

2 ` − 1 , ∆

2 ` − 1

!

, eigenvalues: µ ± = 1 ± √ 1 + 4`

2`

corresponding eigenvectors: (1, µ ± ).

X n := log Df ^s

ⁿ

(c ^f ) ≥ Z n

y

p (Z_n, Z_{n+1})

(66)

S UPEREXPONENTIAL GROWTH

Z n+1 = ¹ _` (Z n + Z n−1 ) − ∆ T (x , y ) := (y , ¹ _` (x + y ) − ∆) ; (Z n , Z n+1 ) = T (Z n−1 , Z n ) Fixed point

p := ∆

2 ` − 1 , ∆

2 ` − 1

!

, eigenvalues: µ ± = 1 ± √ 1 + 4`

2`

corresponding eigenvectors: (1, µ ± ).

X n := log Df ^s

ⁿ

(c ^f ) ≥ Z n

y

p (Z_n, Z_{n+1})

(67)

S UPEREXPONENTIAL GROWTH

Z n+1 = ¹ _` (Z n + Z n−1 ) − ∆ T (x , y ) := (y , ¹ _` (x + y ) − ∆) ; (Z n , Z n+1 ) = T (Z n−1 , Z n ) Fixed point

p := ∆

2 ` − 1 , ∆

2 ` − 1

!

, eigenvalues: µ ± = 1 ± √ 1 + 4`

2`

corresponding eigenvectors: (1, µ ± ).

X n := log Df ^s

ⁿ

(c ^f ) ≥ Z n

y

p (Z_n, Z_{n+1})

(68)

E.Colli M.L.doNascimento E.Vargas Metricaspectsofthedynamicsofcoveringmapsofthecircle

Metric aspects of the dynamics of covering maps of the circle

E. Colli 1 M. L. do Nascimento 2 E. Vargas 3

Departamento de Matemática Aplicada Universidade de São Paulo

Centro de Ciências Exatas e Naturais Universidade Federal do Pará

Departamento de Matemática

Universidade de São Paulo

O UTLINE

1 I NTRODUCTION Set up

Topolgical aspects Metric aspects

2 M AIN RESULT Statement

Strong recurrence History

3 T OOLS AND I DEAS

Difference equation I

Difference equation II

C RITICAL COVERING MAPS ON S 1

Class C 1 , degree d ≥ 2, critical point c of order ` > 1;

Class C 3 and Negative Schwarzian on S 1 \ {c};

There is no wandering interval.

C RITICAL COVERING MAPS ON S 1

Class C 1 , degree d ≥ 2, critical

point c of order ` > 1;

Class C 3 and Negative Schwarzian on S 1 \ {c};

There is no wandering interval.

C RITICAL COVERING MAPS ON S 1

Class C 1 , degree d ≥ 2, critical

point c of order ` > 1;

Class C 3 and Negative Schwarzian on S 1 \ {c};

There is no wandering interval.

C RITICAL COVERING MAPS ON S 1

Class C 1 , degree d ≥ 2, critical point c of order ` > 1;

Class C 3 and Negative Schwarzian on S 1 \ {c};

There is no wandering interval.

C RITICAL COVERING MAPS ON S 1

Class C 1 , degree d ≥ 2, critical point c of order ` > 1;

Class C 3 and Negative Schwarzian on S 1 \ {c};

There is no wandering interval.

C RITICAL COVERING MAPS ON S 1

Class C 1 , degree d ≥ 2, critical point c of order ` > 1;

Class C 3 and Negative Schwarzian on S 1 \ {c};

There is no wandering interval.

C RITICAL COVERING MAPS ON S 1

Class C 1 , degree d ≥ 2, critical point c of order ` > 1;

Class C 3 and Negative Schwarzian on S 1 \ {c};

There is no wandering interval.

C RITICAL COVERING MAPS ON S 1

Class C 1 , degree d ≥ 2, critical point c of order ` > 1;

Class C 3 and Negative Schwarzian on S 1 \ {c};

There is no wandering interval.

C RITICAL COVERING MAPS ON S 1

Class C 1 , degree d ≥ 2, critical point c of order ` > 1;

Class C 3 and Negative Schwarzian on S 1 \ {c};

There is no wandering interval.

C RITICAL COVERING MAPS ON S 1

Class C 1 , degree d ≥ 2, critical point c of order ` > 1;

Class C 3 and Negative Schwarzian on S 1 \ {c};

There is no wandering interval.

T HE MAP R(z ) = 2z MOD 1

Q UESTIONS

Regularity of conjugacy;

Invariant measures Attractors;

Decay of geometry;

Renormalization;

Growth of Df n (c f ).

Q UESTIONS

Regularity of conjugacy;

Invariant measures Attractors;

Decay of geometry;

Renormalization;

Growth of Df n (c f ).

Q UESTIONS

Regularity of conjugacy;

Invariant measures Attractors;

Decay of geometry;

Renormalization;

Growth of Df n (c f ).

Q UESTIONS

Regularity of conjugacy;

Invariant measures Attractors;

Decay of geometry;

E. Colli ¹ M. L. do Nascimento ² E. Vargas ³

O ^UTLINE

C RITICAL COVERING MAPS ON S ¹

Class C ¹ , degree d ≥ 2, critical point c of order ` > 1;

Class C ³ and Negative Schwarzian on S ¹ \ {c};

C RITICAL COVERING MAPS ON S ¹

Class C ¹ , degree d ≥ 2, critical

Class C ³ and Negative Schwarzian on S ¹ \ {c};

C RITICAL COVERING MAPS ON S ¹

Class C ¹ , degree d ≥ 2, critical

Class C ³ and Negative Schwarzian on S ¹ \ {c};

C RITICAL COVERING MAPS ON S ¹

Class C ¹ , degree d ≥ 2, critical point c of order ` > 1;

Class C ³ and Negative Schwarzian on S ¹ \ {c};

C RITICAL COVERING MAPS ON S ¹

Class C ¹ , degree d ≥ 2, critical point c of order ` > 1;

Class C ³ and Negative Schwarzian on S ¹ \ {c};

C RITICAL COVERING MAPS ON S ¹

Class C ¹ , degree d ≥ 2, critical point c of order ` > 1;

Class C ³ and Negative Schwarzian on S ¹ \ {c};

C RITICAL COVERING MAPS ON S ¹

Class C ¹ , degree d ≥ 2, critical point c of order ` > 1;

Class C ³ and Negative Schwarzian on S ¹ \ {c};

C RITICAL COVERING MAPS ON S ¹

Class C ¹ , degree d ≥ 2, critical point c of order ` > 1;

Class C ³ and Negative Schwarzian on S ¹ \ {c};

C RITICAL COVERING MAPS ON S ¹

Class C ¹ , degree d ≥ 2, critical point c of order ` > 1;

Class C ³ and Negative Schwarzian on S ¹ \ {c};

C RITICAL COVERING MAPS ON S ¹

Class C ¹ , degree d ≥ 2, critical point c of order ` > 1;

Class C ³ and Negative Schwarzian on S ¹ \ {c};

T ^{HE MAP} R(z ) = 2z ^MOD 1

Q ^UESTIONS

Growth of Df ⁿ (c ^f ).

Q ^UESTIONS

Growth of Df ⁿ (c ^f ).

Q ^UESTIONS

Growth of Df ⁿ (c ^f ).

Q ^UESTIONS

Growth of Df ⁿ (c ^f ).

Q ^UESTIONS

Growth of Df ⁿ (c ^f ).

Q ^UESTIONS

Growth of Df ⁿ (c ^f ).

Let f be as above, µ ⁺ := ¹⁺

2` and ϕ := ¹⁺

If 1 < ` < 2 then ^{log log} ^Df _n

^(c

⁾ → log µ ⁺ > 0.

If 1 < ` < 2 then ^{log log} _log ^Df _k

^(c

⁾ → ^log _log ^µ _ϕ

If ` = 2 then 0 < K ≤ ^log ^Df _n

^(c

⁾ ≤ K ⁻¹ .

If ` > 2 then Df ^s

(c ^f ) is bounded.

Let f be as above, µ ⁺ := ¹⁺

2` and ϕ := ¹⁺

If 1 < ` < 2 then ^{log log} ^Df _n

^(c

⁾ → log µ ⁺ > 0.

If 1 < ` < 2 then ^{log log} _log ^Df _k

^(c

⁾ → ^log _log ^µ _ϕ

If ` = 2 then 0 < K ≤ ^log ^Df _n

^(c

⁾ ≤ K ⁻¹ .

If ` > 2 then Df ^s

(c ^f ) is bounded.

Let f be as above, µ ⁺ := ¹⁺

2` and ϕ := ¹⁺

If 1 < ` < 2 then ^{log log} ^Df _n

^(c

⁾ → log µ ⁺ > 0.

If 1 < ` < 2 then ^{log log} _log ^Df _k

^(c

⁾ → ^log _log ^µ _ϕ

If ` = 2 then 0 < K ≤ ^log ^Df _n