The Classification of Critically Preperiodic Polynomials as Dynamical Systems

Bielefeld, Ben; Fisher, Yuval; Hubbard, John H.

doi:10.2307/2152709

Cited by 36 publications

(66 citation statements)

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“…Let A be a finite abelian group. Let H be a subset of A which is the disjoint union of four pairs {±h 1…”

Section: Coset Numbers and Nonseparating Setsmentioning

confidence: 99%

“…Then B ∼ = A/B ∼ = Z/4Z. There are only six possible choices for B: (1, 0) , (0, 1) , (1,2) , (2,1) , (3,1) , and (1,1) . If B = (1, 1) or B = (1, 3) , one verifies in these cases that c 1 = c 2 = c 3 = c 4 = 1.…”

Section: On Thurston Maps Of Degreementioning

confidence: 99%

“…We first show that if H contains the point (0, 0), then H is in the Hurwitz class of H 1 . Suppose that (0, 0) ∈ H. Since H contains at least one element of order 4, by a group automorphism of A, we may assume that H contains ± (1,0) If H is a nonseparating subset of A that contains an element of order 2, say τ , then H = H + τ is a nonseparating subset of A containing the point (0, 0). The above argument shows that H is in the Hurwitz class of H 1 .…”

Section: On Thurston Maps Of Degreementioning

confidence: 99%

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On NET maps: Examples and nonexistence results

Saenz

2019

Conform. Geom. Dyn.

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A Thurston map is called nearly Euclidean if its local degree at each critical point is 2 and it has exactly four postcritical points. Nearly Euclidean Thurston (NET) maps are simple generalizations of rational Lattès maps. We investigate when such a map has the property that the associated pullback map on Teichmüller space is constant. We also show that no Thurston map of degree 2 has constant pullback map.

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“…Let A be a finite abelian group. Let H be a subset of A which is the disjoint union of four pairs {±h 1…”

Section: Coset Numbers and Nonseparating Setsmentioning

confidence: 99%

Section: On Thurston Maps Of Degreementioning

confidence: 99%

Section: On Thurston Maps Of Degreementioning

confidence: 99%

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On NET maps: Examples and nonexistence results

Saenz

2019

Conform. Geom. Dyn.

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“…In the first case it is topologically conjugate to the angle doubling map h defined 2000 Mathematics Subject Classification: Primary 37F20, 37E10; Secondary 37F45. [1] r r r r r r r r by h(β) = 2β mod 1, and in the second case to the angle "antidoubling" map h defined by h(β) = −2β mod 1 (cf. Proposition 2.1).…”

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confidence: 99%

“…Such a switching behavior is typical for periodic angles of h as well as of h, the only exceptions being the fixed points, where the kneading sequence is * and the clockwise and counter-clockwise limits coincide. For 0, the fixed point of h, they are 0, and for 0, 1 3 , 2 3 , the fixed points of h, they are 01.…”

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confidence: 99%

Combinatorics of distance doubling maps

Keller

Winter²

2005

Fund. Math.

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We study the combinatorics of distance doubling maps on the circle R/Z with prototypes h(β) = 2β mod 1 and h(β) = −2β mod 1, representing the orientation preserving and orientation reversing case, respectively. In particular, we identify parts of the circle where the iterates f •n of a distance doubling map f exhibit "distance doubling behavior". The results include well known statements for h related to the structure of the Mandelbrot set M. For h they suggest some analogies to the structure of the tricorn, the "antiholomorphic Mandelbrot set". c (resp. p c (R β c) = R h(β) c). This property indicates that the structure of J c is strongly related to h, and similarly the structure of J c to h. The "antidoubling" map h plays a similar role for antiquadratic maps in the complex plane as h for quadratic maps. A dynamic ray having exactly one accumulation point z in the Julia set is said to land at z. In case the Julia set is locally connected, each dynamic ray lands and the Julia set can be considered as the quotient of T with respect to the equivalence relation that identifies angles β 1 , β 2 iff the corresponding dynamic rays land at the same point. (For a general discussion of the structure of polynomial Julia sets see [8].) Here we are only interested in the landing behavior of rays corresponding to periodic angles β ∈ T (periodic with respect to h or h). The crucial point is that all dynamic rays for periodic angles land at a periodic point of the Julia set different from 0 and that only finitely many such rays land at the same periodic point. We come back to this fact when defining bifurcation chords. Kneading sequences. In order to describe the dynamics of a distance doubling map f on T we look at the orbits of angles and how they differ. For this we use kneading sequences, first introduced by Milnor and Thurston [12] for interval maps. The idea is to divide T appropriately into two half-circles. The kneading sequence of an angle indicates in which half-circle each of its iterates lies. Definition 1.1. Let f be a distance doubling map on T and for α ∈ T let β 1 , β 2 be the two angles with f (β 1) = f (β 2) = α. From the two open halfcircles ]β 1 , β 2 [ and ]β 2 , β 1 [, taken in counterclockwise direction, let T 0 be the one containing α and T 1 the other one. (In case α = β 1 set T 0 = ]β 1 , β 2 [.) The kneading sequence of α with respect to f is the sequence s 1 s 2 s 3. .. ∈ {0, 1, * } N defined by s i =      * if f •i−1 (α) ∈ {β 1 , β 2 }, 0 if f •i−1 (α) ∈ T 0 , 1 if f •i−1 (α) ∈ T 1 ,

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