Quasi-symmetric conjugacy for circle maps with a flat interval

Palmisano, Liviana

doi:10.1017/etds.2017.36

Cited by 8 publications

(14 citation statements)

References 20 publications

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“…Still within the realm of circle maps, other interesting (partial) rigidity results have been obtained. For instance, in a recent paper [21], Palmisano proved that C 2 weakly order-preserving circle maps with a flat interval are quasi-symmetrically rigid in their non-wandering sets, provided their rotation number is of bounded type and a certain bounded geometry hypothesis is satisfied.…”

Section: Introductionmentioning

confidence: 99%

Real bounds and quasisymmetric rigidity of multicritical circle maps

Estevez

Faria

2018

Trans. Amer. Math. Soc.

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Abstract. Let f, g : S 1 → S 1 be two C 3 critical homeomorphisms of the circle with the same irrational rotation number and the same (finite) number of critical points, all of which are assumed to be non-flat, of power-law type. In this paper we prove that if h : S 1 → S 1 is a topological conjugacy between f and g and h maps the critical points of f to the critical points of g, then h is quasisymmetric. When the power-law exponents at all critical points are integers, this result is a special case of a general theorem recently proved by T. Clark and S. van Strien [5]. However, unlike the proof given in [5], which relies on heavy complex-analytic machinery, our proof uses purely realvariable methods, and is valid for non-integer critical exponents as well. We do not require h to preserve the power-law exponents at corresponding critical points.

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Section: Introductionmentioning

confidence: 99%

Real bounds and quasisymmetric rigidity of multicritical circle maps

Estevez

Faria

2018

Trans. Amer. Math. Soc.

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“…This can happen even for bounded combinatorics, and return maps can degenerate [44,64] see also [43,45]. For circle maps with plateaus see for example [16,40,42,53,54,[61][62][63]. Here it is also natural to explore the role of the orders of the critical points at the boundary points of a plateau [a, b].…”

Section: Informal Summary Of the The Results In This Papermentioning

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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“…Both maps have a priori bounds [9]. A priori bounds imply that f and g are quasi-symmetrically conjugate on their attractors [28]. However, f and g are not smoothly conjugate since their critical exponents differ.…”

Section: 3mentioning

confidence: 99%

The rigidity conjecture

Martens

Palmisano

Winckler

2018

Indagationes Mathematicae

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A central question in dynamics is whether the topology of a system determines its geometry. This is known as rigidity. Under mild topological conditions rigidity holds for many classical cases, including: Kleinian groups, circle diffeomorphisms, unimodal interval maps, critical circle maps, and circle maps with a break point. More recent developments show that under similar topological conditions, rigidity does not hold for slightly more general systems. In this paper we state a conjecture which describes how topological classes are organized into rigidity classes. arXiv:1612.08939v2 [math.DS]

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Quasi-symmetric conjugacy for circle maps with a flat interval

Cited by 8 publications

References 20 publications

Real bounds and quasisymmetric rigidity of multicritical circle maps

Real bounds and quasisymmetric rigidity of multicritical circle maps

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

The rigidity conjecture

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