Renormalization for Unimodal Maps with Non-integer Exponents

Gorbovickis, Igors; Yampolsky, Michael

doi:10.1007/s40598-018-0089-y

Cited by 7 publications

(8 citation statements)

References 9 publications

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“…The renormalization theory we develop here is done by obtaining large bounds. This is quite different from the renormalization theory obtained for real analytic unimodal maps, [53,43,44,32,3,12], see also [50,35,14]. It would be interesting to tie these approaches together.…”

Section: Open Questionsmentioning

confidence: 66%

Asymmetric unimodal maps with non-universal period-doubling scaling laws

Kozlovski,

van Strien

2019

Preprint

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We consider a family of strongly-asymmetric unimodal maps {f t } t∈[0,1] of the formwhere we assume that β > 1. We show that such a family contains a Feigenbaum-Coullet-Tresser 2 ∞ map, and develop a renormalization theory for these maps. The scalings of the renormalization intervals of the 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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Section: Open Questionsmentioning

confidence: 66%

Asymmetric unimodal maps with non-universal period-doubling scaling laws

Kozlovski,

van Strien

2019

Preprint

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“…The renormalization theory we develop here is done by obtaining large bounds. This is quite different from the renormalization theory obtained for real analytic unimodal maps, [3,14,36,48,49,60], see also [15,39,57]. Most of these results build on complex bounds and quasi-symmetric rigidity.…”

Section: Monotonicity Of Bifurcationsmentioning

confidence: 72%

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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“…A survey of four decades of research in the area is provided by [23]. More recently, Gorbovickis and Yampolsky [24] have broadened the reach to certain maps with non-integer critical exponent.…”

Section: Introduction a Backgroundmentioning

confidence: 99%

Rigorous computer-assisted bounds on the period doubling renormalisation fixed point and eigenfunctions in maps with critical point of degree 4

Burbanks,

Osbaldestin,

Thurlby

2020

Preprint

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We gain tight rigorous bounds on the renormalisation fixed point for period doubling in families of unimodal maps with degree 4 critical point. We prove that the fixed point is hyperbolic and use a contraction mapping argument to bound essential eigenfunctions and eigenvalues for the linearisation and for the scaling of additive noise. We find analytic extensions of the fixed point function to larger domains. We use multi-precision arithmetic with rigorous directed rounding to bound operations in a space of analytic functions yielding tight bounds on power series and universal constants.

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Renormalization for Unimodal Maps with Non-integer Exponents

Abstract: We define an analytic setting for renormalization of unimodal maps with an arbitrary critical exponent. We prove the global Hyperbolicity of Renormalization conjecture for unimodal maps of bounded type with a critical exponent which is sufficiently close to an even integer.

Cited by 7 publications

References 9 publications

Asymmetric unimodal maps with non-universal period-doubling scaling laws

Asymmetric unimodal maps with non-universal period-doubling scaling laws

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

Rigorous computer-assisted bounds on the period doubling renormalisation fixed point and eigenfunctions in maps with critical point of degree 4

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