Solenoidal attractors with bounded combinatorics are shy

Smania, Daniel

doi:10.4007/annals.2020.191.1.1

Cited by 8 publications

(8 citation statements)

References 42 publications

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“…A conceptual proof was given for analytic unimodal mappings of any combinatorial type in the works of Sullivan [44] (see also [12]), McMullen [31,32], Lyubich [23,24], and Avila and Lyubich [4]. This was extended to certain smooth mappings by de Faria, de Melo and Pinto [10], and to analytic mappings with several critical points and bounded combinatorics by Smania [41,42]. Renormalization is intimately related with rigidity theory, and in many contexts, e.g.…”

Section: T Clark and M Gouveiamentioning

confidence: 99%

Hyperbolicity of renormalization for dissipative gap mappings

Clark

Gouveia²

2021

Ergod. Th. Dynam. Sys.

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A gap mapping is a discontinuous interval mapping with two strictly increasing branches that have a gap between their ranges. They are one-dimensional dynamical systems, which arise in the study of certain higher dimensional flows, for example the Lorenz flow and the Cherry flow. In this paper, we prove hyperbolicity of renormalization acting on $C^3$ dissipative gap mappings, and show that the topological conjugacy classes of infinitely renormalizable gap mappings are $C^1$ manifolds.

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Section: T Clark and M Gouveiamentioning

confidence: 99%

Hyperbolicity of renormalization for dissipative gap mappings

Clark

Gouveia²

2021

Ergod. Th. Dynam. Sys.

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“…The description of the tangent space of the space of polynomial‐like germs was first given in [34, Section 4] in the context of unicritical mappings, and [53, Sections 3 and 4] treated polynomial‐like germs with several critical points. We refer to those papers for the details.…”

Section: Spaces Of Mappingsmentioning

confidence: 99%

“…Thus proving Theorem A for

C^{3 + γ}

unimodal mappings with non‐degenerate critical points. In [50], using convergence of renormalization and rigidity, Smania proved Conjecture I for multimodal mappings with all critical points non‐degenerate and with the same

ω

‐limit set (indeed, in [53] he goes beyond this to prove hyperbolicity of renormalization for these mappings). In this paper, we remove these two conditions to prove Theorem A.…”

Section: Introductionmentioning

confidence: 99%

The boundary of chaos for interval mappings

Clark

Trejo

2020

Proc. London Math. Soc.

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A goal in the study of dynamics on the interval is to understand the transition to positive topological entropy. There is a conjecture from the 1980s that the only route to positive topological entropy is through a cascade of period doubling bifurcations. We prove this conjecture in natural families of smooth interval maps, and use it to study the structure of the boundary of mappings with positive entropy. In particular, we show that in families of mappings with a fixed number of critical points the boundary is locally connected, and for analytic mappings that it is a cellular set.

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“…For infinitely renormalizable unimodal interval maps there is a rich history, starting with the conjectures of Feigenbaum and Coullet-Tresser. Rigorous proofs were finally provided by [ 3 , 48 , 49 , 60 ], see also [ 14 , 30 , 56 – 58 ]. The weakly symmetric case is considered in [ 52 ].…”

Section: Introductionmentioning

confidence: 99%

Asymmetric Unimodal Maps with Non-universal Period-Doubling Scaling Laws

Kozlovski

Strien

2020

Commun. Math. Phys.

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We consider a family of strongly-asymmetric unimodal maps $$\{f_t\}_{t\in [0,1]}$$ { f t } t ∈ [ 0 , 1 ] of the form $$f_t=t\cdot f$$ f t = t · f where $$f:[0,1]\rightarrow [0,1]$$ f : [ 0 , 1 ] → [ 0 , 1 ] is unimodal, $$f(0)=f(1)=0$$ f ( 0 ) = f ( 1 ) = 0 , $$f(c)=1$$ f ( c ) = 1 is of the form and $$\begin{aligned} f(x)=\left\{ \begin{array}{ll} 1-K_-|x-c|+o(|x-c|)&{} \text{ for } x<c, \\ 1-K_+|x-c|^\beta + o(|x-c|^\beta ) &{} \text{ for } x>c, \end{array}\right. \end{aligned}$$ f ( x ) = 1 - K - | x - c | + o ( | x - c | ) for x < c , 1 - K + | x - c | β + o ( | x - c | β ) for x > c , where we assume that $$\beta >1$$ β > 1 . We show that such a family contains a Feigenbaum–Coullet–Tresser $$2^\infty $$ 2 ∞ map, and develop a renormalization theory for these maps. The scalings of the renormalization intervals of the $$2^\infty $$ 2 ∞ map turn out to be super-exponential and non-universal (i.e. to depend on the map) and the scaling-law is different for odd and even steps of the renormalization. The conjugacy between the attracting Cantor sets of two such maps is smooth if and only if some invariant is satisfied. We also show that the Feigenbaum–Coullet–Tresser map does not have wandering intervals, but surprisingly we were only able to prove this using our rather detailed scaling results.

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Solenoidal attractors with bounded combinatorics are shy

Abstract: We show that in a generic finite-dimensional real-analytic family of real-analytic multimodal maps, the subset of parameters on which the corresponding map has a solenoidal attractor with bounded combinatorics is a set with zero Lebesgue measure.

Cited by 8 publications

References 42 publications

Hyperbolicity of renormalization for dissipative gap mappings

Hyperbolicity of renormalization for dissipative gap mappings

The boundary of chaos for interval mappings

Asymmetric Unimodal Maps with Non-universal Period-Doubling Scaling Laws

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