Quadratic-Like Dynamics of Cubic Polynomials

Blokh, Alexander; Oversteegen, Lex G.; Ptacek, Ross; Timorin, Vladlen

doi:10.1007/s00220-015-2559-6

Cited by 16 publications

(29 citation statements)

References 19 publications

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“…The set Ar f * (z b * ) of arguments of external rays of f * landing at z b * consists of at least two angles. Since all maps in U are quasi-symmetrically conjugate, it is easy to see (e.g., by Lemma 3.5 [BOPT13b]) that all maps f λ,b ∈ U have repelling periodic cutpoints z b corresponding to z b * . By Lemma 2.1…”

Section: The Proof Of Inclusion Phdmentioning

confidence: 99%

The main cubioid

et al. 2014

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Section: The Proof Of Inclusion Phdmentioning

confidence: 99%

The main cubioid

et al. 2014

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“…As mentioned in section (2), we will impose a technical condition on the complex coefficients, a 1 and a 0 of P a that |a 1 | < 1, |a 0 | < 1. An imminently stricter condition may also be required to make the corresponding Julia set hyperbolic, see [4,3].…”

Section: Cubic Polynomialsmentioning

confidence: 99%

The dependence of lyapunov exponents of polynomials on their coefficients

Sridharan¹,

Tiwari²

2018

Journal of Computational Dynamics

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In this paper, we consider the family of hyperbolic quadratic polynomials parametrised by a complex constant; namely P c (z) = z 2 + c with |c| < 1 and the family of hyperbolic cubic polynomials parametrised by two complex constants; namely P (a 1 ,a 0 ) (z) = z 3 + a 1 z + a 0 with |a i | < 1, restricted on their respective Julia sets. We compute the Lyapunov characteristic exponents for these polynomial maps over corresponding Julia sets, with respect to various Bernoulli measures and obtain results pertaining to the dependence of the behaviour of these exponents on the parameters describing the polynomial map. We achieve this using the theory of thermodynamic formalism, the pressure function in particular.

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“…Theorem 4.11 (Theorem B [BOPT15]). Let P : C → C be a polynomial, and Y ⊂ C be a full P -invariant continuum.…”

Section: 2mentioning

confidence: 99%

“…Proof. By [BOPT15], P | K * : K * → K * is a polynomial-like map. Hence the result follows from Theorem B.…”

Section: 2mentioning

confidence: 99%

Non-degenerate locally connected models for plane continua and Julia sets

Blokh¹,

Oversteegen²,

Timorin

2017

Discrete &Amp; Continuous Dynamical Systems - A

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Suppose that a X is an unshielded plane continuum (i.e., X coincides with the boundary of the unbounded complementary component of X). Then there exists a finest monotone map m : X → L, where L is a locally connected continuum (i.e., m −1 (y) is connected for each y ∈ L, and any monotone map ϕ : X → L ′ onto a locally connected continuum is a composition ϕ = ϕ ′ • m where ϕ ′ : L → L ′ is monotone). Such finest locally connected model L of X is easier to understand because L is locally connected (in particular it can be described by a picture) and represents the finest but still understandable decomposition of X into possibly complicated but pairwise disjoint fibers (point-preimages) of m. However, in some cases (i.e., in case X is indecomposable) L is a singleton. In this paper we provide sufficient conditions for the existence of a non-degenerate model depending on the existence of certain subcontinua of X and apply these results to the connected Julia sets of polynomials.

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Quadratic-Like Dynamics of Cubic Polynomials

Cited by 16 publications

References 19 publications

The main cubioid

The main cubioid

The dependence of lyapunov exponents of polynomials on their coefficients

Non-degenerate locally connected models for plane continua and Julia sets

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