Tongues and bifurcations on a family of degree 4 Blaschke products - PowerPoint PPT Presentation

Tongues and bifurcations on a family of degree 4 Blaschke products Jordi Canela Institute of Mathematics Polish Academy of Sciences IMPAN — Joint work with: N´ uria Fagella and Antonio Garijo — Barcelona, 23 November 2015 Jordi Canela (IMPAN) Tongues in degree 4 Blaschke products 23 November 2015 1 / 28

The degree 4 Blaschke products 1 Tongues of the Blaschke family 2 Bifurcations around the tip of the tongues 3 Extending the tongues 4 Jordi Canela (IMPAN) Tongues in degree 4 Blaschke products 23 November 2015 2 / 28

The degree 4 Blaschke products 1 Tongues of the Blaschke family 2 Bifurcations around the tip of the tongues 3 Extending the tongues 4 Jordi Canela (IMPAN) Tongues in degree 4 Blaschke products 23 November 2015 3 / 28

Degree 4 Blaschke products We want to study the degree 4 Blaschke products B a ( z ) = z 3 z − a az . 1 − ¯ They are rational perturbations of the doubling map of the circle R 2 ( z ) = z 2 , equivalently given by θ → 2 θ (mod 1) . These products are the rational version of the double standard map: S 1 → S 1 C ∗ → C ∗ C → ˆ ˆ C e i α · z · e β/ 2( z +1 / z ) e it z 2 z − a Standard map θ → θ + α + β sin θ 1 − ¯ az e i α · z 2 · e β/ 2( z +1 / z ) e it z 3 z − a Double standard map θ → 2 θ + α + β sin θ 1 − ¯ az M. Herman, Sur la conjugaison des diff´ eomorphismes du cercle ` a des rotations , 1976 N. Fagella and C. Henriksen, Arnold Disks and the Moduli of Herman Rings of the Complex Standard Family , 2006 M. Misiurewicz and A. Rodrigues, Double standard maps , 2007 A. Dezotti, Connectedness of the Arnold tongues for double standard maps , 2010 Jordi Canela (IMPAN) Tongues in degree 4 Blaschke products 23 November 2015 4 / 28

Degree 4 Blaschke products We want to study the degree 4 Blaschke products B a ( z ) = z 3 z − a az . 1 − ¯ They are rational perturbations of the doubling map of the circle R 2 ( z ) = z 2 , equivalently given by θ → 2 θ (mod 1) . These products are the rational version of the double standard map: S 1 → S 1 C ∗ → C ∗ C → ˆ ˆ C e i α · z · e β/ 2( z +1 / z ) e it z 2 z − a Standard map θ → θ + α + β sin θ 1 − ¯ az e i α · z 2 · e β/ 2( z +1 / z ) e it z 3 z − a Double standard map θ → 2 θ + α + β sin θ 1 − ¯ az M. Herman, Sur la conjugaison des diff´ eomorphismes du cercle ` a des rotations , 1976 N. Fagella and C. Henriksen, Arnold Disks and the Moduli of Herman Rings of the Complex Standard Family , 2006 M. Misiurewicz and A. Rodrigues, Double standard maps , 2007 A. Dezotti, Connectedness of the Arnold tongues for double standard maps , 2010 Jordi Canela (IMPAN) Tongues in degree 4 Blaschke products 23 November 2015 4 / 28

Degree 4 Blaschke products We want to study the degree 4 Blaschke products B a ( z ) = z 3 z − a az . 1 − ¯ They are rational perturbations of the doubling map of the circle R 2 ( z ) = z 2 , equivalently given by θ → 2 θ (mod 1) . These products are the rational version of the double standard map: S 1 → S 1 C ∗ → C ∗ C → ˆ ˆ C e i α · z · e β/ 2( z +1 / z ) e it z 2 z − a Standard map θ → θ + α + β sin θ 1 − ¯ az e i α · z 2 · e β/ 2( z +1 / z ) e it z 3 z − a Double standard map θ → 2 θ + α + β sin θ 1 − ¯ az M. Herman, Sur la conjugaison des diff´ eomorphismes du cercle ` a des rotations , 1976 N. Fagella and C. Henriksen, Arnold Disks and the Moduli of Herman Rings of the Complex Standard Family , 2006 M. Misiurewicz and A. Rodrigues, Double standard maps , 2007 A. Dezotti, Connectedness of the Arnold tongues for double standard maps , 2010 Jordi Canela (IMPAN) Tongues in degree 4 Blaschke products 23 November 2015 4 / 28

Properties of these Blaschke products The main properties of these Blaschke products are the following: They leave S 1 invariant. They are symmetric with respect to S 1 , i.e., B a ( z ) = ( B a ( z ∗ )) ∗ , where z ∗ = 1 / z . z = 0 and z = ∞ are superattracting fixed points of local degree 3. z ∞ = 1 / a and z 0 = a are the only pole and zero respectively. They have two “free” critical points c ± (i.e., B ′ a ( c ± ) = 0) 1 � � � 2 + | a | 2 ± ( | a | 2 − 4)( | a | 2 − 1) c ± = a · 3 | a | 2 which control all possible stable dynamics other than the basins of attraction of z = 0 and z = ∞ . Jordi Canela (IMPAN) Tongues in degree 4 Blaschke products 23 November 2015 5 / 28

Properties of these Blaschke products The main properties of these Blaschke products are the following: They leave S 1 invariant. They are symmetric with respect to S 1 , i.e., B a ( z ) = ( B a ( z ∗ )) ∗ , where z ∗ = 1 / z . z = 0 and z = ∞ are superattracting fixed points of local degree 3. z ∞ = 1 / a and z 0 = a are the only pole and zero respectively. They have two “free” critical points c ± (i.e., B ′ a ( c ± ) = 0) 1 � � � 2 + | a | 2 ± ( | a | 2 − 4)( | a | 2 − 1) c ± = a · 3 | a | 2 which control all possible stable dynamics other than the basins of attraction of z = 0 and z = ∞ . Jordi Canela (IMPAN) Tongues in degree 4 Blaschke products 23 November 2015 5 / 28

Properties of these Blaschke products The main properties of these Blaschke products are the following: They leave S 1 invariant. They are symmetric with respect to S 1 , i.e., B a ( z ) = ( B a ( z ∗ )) ∗ , where z ∗ = 1 / z . z = 0 and z = ∞ are superattracting fixed points of local degree 3. z ∞ = 1 / a and z 0 = a are the only pole and zero respectively. They have two “free” critical points c ± (i.e., B ′ a ( c ± ) = 0) 1 � � � 2 + | a | 2 ± ( | a | 2 − 4)( | a | 2 − 1) c ± = a · 3 | a | 2 which control all possible stable dynamics other than the basins of attraction of z = 0 and z = ∞ . Jordi Canela (IMPAN) Tongues in degree 4 Blaschke products 23 November 2015 5 / 28

Properties of these Blaschke products The main properties of these Blaschke products are the following: They leave S 1 invariant. They are symmetric with respect to S 1 , i.e., B a ( z ) = ( B a ( z ∗ )) ∗ , where z ∗ = 1 / z . z = 0 and z = ∞ are superattracting fixed points of local degree 3. z ∞ = 1 / a and z 0 = a are the only pole and zero respectively. They have two “free” critical points c ± (i.e., B ′ a ( c ± ) = 0) 1 � � � 2 + | a | 2 ± ( | a | 2 − 4)( | a | 2 − 1) c ± = a · 3 | a | 2 which control all possible stable dynamics other than the basins of attraction of z = 0 and z = ∞ . Jordi Canela (IMPAN) Tongues in degree 4 Blaschke products 23 November 2015 5 / 28

We study the dynamics of these products depending on | a | . Case | a | > 2 z ∞ , c − ∈ D z 0 , c + ∈ C \ D B a | S 1 : S 1 → S 1 is a covering of degree 2. z 0 c + c − z ∞ c − = 1 / c + ⇒ Critical orbits are symmetric w.r.t. S 1 . Jordi Canela (IMPAN) Tongues in degree 4 Blaschke products 23 November 2015 6 / 28

c − c + c − c + Dynamical plane of B 5 / 2 . Both free critical Dynamical plane of B 4 . Each critical orbit orbits accumulate on a fixed point in S 1 . accumulates on a different attracting cycle. Jordi Canela (IMPAN) Tongues in degree 4 Blaschke products 23 November 2015 7 / 28

Case | a | = 2 c + = c − = a / 2 ∈ S 1 z ∞ ∈ D z 0 ∈ C \ D z 0 c z ∞ B a | S 1 : S 1 → S 1 is a covering of degree 2. Jordi Canela (IMPAN) Tongues in degree 4 Blaschke products 23 November 2015 8 / 28

Case 1 < | a | < 2 There are two different critical points: 1 � � � 2 + | a | 2 + i (4 − | a | 2 )( | a | 2 − 1) c + = a · = a · k 3 | a | 2 c − = a · ¯ k The critical points satisfy | c ± | = 1. The critical orbits are not symmetric. c + z 0 z ∞ c − Jordi Canela (IMPAN) Tongues in degree 4 Blaschke products 23 November 2015 9 / 28

c + c − Dynamical plane of B 3 / 2 . We see in green an Dynamical plane of B 3 / 2 i . There are no other attracting basin of a period 2 cycle. attracting basins than the ones of z = 0 and z = ∞ . Jordi Canela (IMPAN) Tongues in degree 4 Blaschke products 23 November 2015 10 / 28

c + c − Dynamical plane of the Blaschke product B 1 , 07398+0 , 5579 i . Jordi Canela (IMPAN) Tongues in degree 4 Blaschke products 23 November 2015 11 / 28

Case | a | ≤ 1 If | a | ≤ 1 then B a ( D ) = D and the dynamics is well understood. Jordi Canela (IMPAN) Tongues in degree 4 Blaschke products 23 November 2015 12 / 28

Parameter plane of B a . It has been drawn by iterating the critical point c + . Remark: B a and B ξ a are conjugate, where ξ is a third root of unity. Jordi Canela (IMPAN) Tongues in degree 4 Blaschke products 23 November 2015 13 / 28

The degree 4 Blaschke products 1 Tongues of the Blaschke family 2 Bifurcations around the tip of the tongues 3 Extending the tongues 4 Jordi Canela (IMPAN) Tongues in degree 4 Blaschke products 23 November 2015 14 / 28