Hyperbolicity of the renormalization operator for critical $\mathcal{C}^{r}$ circle mappings

2006 ◽  
Vol 26 (02) ◽  
pp. 585
Author(s):  
ÉTIENNE VOUTAZ
Keyword(s):  
Renormalization Operator ◽  
Circle Mappings

Asymptotic rigidity of scaling ratios for critical circle mappings

1999 ◽  
Vol 19 (4) ◽  
pp. 995-1035 ◽  
Author(s):  
EDSON DE FARIA
Keyword(s):  
Critical Point ◽  
Surgical Techniques ◽  
Careful Analysis ◽  
The Family ◽  
Critical Circle ◽  
Renormalization Operator ◽  
Irrational Rotation Number ◽  
Circle Mappings ◽  
Commuting Pairs ◽  
Unimodal Mappings

Let $f$ be a smooth homeomorphism of the circle having one cubic-exponent critical point and irrational rotation number of bounded combinatorial type. Using certain pull-back and quasiconformal surgical techniques, we prove that the scaling ratios of $f$ about the critical point are asymptotically independent of $f$. This settles in particular the golden mean universality conjecture. We introduce the notion of a holomorphic commuting pair, a complex dynamical system that, in the analytic case, represents an extension of $f$ to the complex plane and behaves somewhat as a quadratic-like mapping. We define a suitable renormalization operator that acts on such objects. Through careful analysis of the family of entire mappings given by $z\mapsto z+\theta -(1/2\pi)\sin{2\pi z}$, $\theta$ real, we construct examples of holomorphic commuting pairs, from which certain necessary limit set pre-rigidity results are extracted. The rigidity problem for $f$ is thereby reduced to one of renormalization convergence. We handle this last problem by means of Teichmüller extremal methods made available through the recent work of Sullivan on Riemann surface laminations and renormalization of unimodal mappings.

scholarly journals Chaotic period doubling

2009 ◽  
Vol 29 (2) ◽  
pp. 381-418 ◽  
Author(s):  
V. V. M. S. CHANDRAMOULI ◽  
M. MARTENS ◽  
W. DE MELO ◽  
C. P. TRESSER
Keyword(s):  
Fixed Point ◽  
Asymptotic Behavior ◽  
Chaotic Dynamics ◽  
A Priori ◽  
Period Doubling ◽  
Small Scale ◽  
Unimodal Maps ◽  
One Dimensional ◽  
Renormalization Operator ◽  
Renormalization Theory

AbstractThe period doubling renormalization operator was introduced by Feigenbaum and by Coullet and Tresser in the 1970s to study the asymptotic small-scale geometry of the attractor of one-dimensional systems that are at the transition from simple to chaotic dynamics. This geometry turns out not to depend on the choice of the map under rather mild smoothness conditions. The existence of a unique renormalization fixed point that is also hyperbolic among generic smooth-enough maps plays a crucial role in the corresponding renormalization theory. The uniqueness and hyperbolicity of the renormalization fixed point were first shown in the holomorphic context, by means that generalize to other renormalization operators. It was then proved that, in the space ofC2+αunimodal maps, forα>0, the period doubling renormalization fixed point is hyperbolic as well. In this paper we study what happens when one approaches from below the minimal smoothness thresholds for the uniqueness and for the hyperbolicity of the period doubling renormalization generic fixed point. Indeed, our main result states that in the space ofC2unimodal maps the analytic fixed point is not hyperbolic and that the same remains true when adding enough smoothness to geta prioribounds. In this smoother class, calledC2+∣⋅∣, the failure of hyperbolicity is tamer than inC2. Things get much worse with just a bit less smoothness thanC2, as then even the uniqueness is lost and other asymptotic behavior becomes possible. We show that the period doubling renormalization operator acting on the space ofC1+Lipunimodal maps has infinite topological entropy.

scholarly journals Rigidity of critical circle mappings II

1999 ◽  
Vol 13 (2) ◽  
pp. 343-370 ◽  
Author(s):  
Edson de Faria ◽  
Welington de Melo
Keyword(s):  
Critical Circle ◽  
Circle Mappings

The Attractor of Fibonacci-like Renormalization Operator

2020 ◽  
Vol 36 (11) ◽  
pp. 1256-1278
Author(s):  
Hao Yang Ji ◽  
Si Min Li
Keyword(s):  
Renormalization Operator

scholarly journals Local study of a renormalization operator for 1D maps under quasiperiodic forcing

2016 ◽  
Vol 9 (4) ◽  
pp. 1171-1188
Author(s):  
Àngel Jorba ◽  
◽  
Pau Rabassa ◽  
Joan Carles Tatjer ◽  
◽  
...  
Keyword(s):  
Local Study ◽  
Renormalization Operator

scholarly journals Lyapunov exponent of random dynamical systems on the circle

2021 ◽  
pp. 1-28
Author(s):  
DOMINIQUE MALICET
Keyword(s):  
Lyapunov Exponent ◽  
Dynamical Systems ◽  
Taylor Expansion ◽  
Diophantine Condition ◽  
Quantitative Version ◽  
Rotation Numbers ◽  
Diophantine Approximations ◽  
Simultaneous Diophantine Approximations ◽  
Random Products ◽  
Circle Mappings

Abstract We consider products of an independent and identically distributed sequence in a set $\{f_1,\ldots ,f_m\}$ of orientation-preserving diffeomorphisms of the circle. We can naturally associate a Lyapunov exponent $\lambda $ . Under few assumptions, it is known that $\lambda \leq 0$ and that the equality holds if and only if $f_1,\ldots ,f_m$ are simultaneously conjugated to rotations. In this paper, we state a quantitative version of this fact in the case where $f_1,\ldots ,f_m$ are $C^k$ perturbations of rotations with rotation numbers $\rho (f_1),\ldots ,\rho (f_m)$ satisfying a simultaneous diophantine condition in the sense of Moser [On commuting circle mappings and simultaneous diophantine approximations. Math. Z.205(1) (1990), 105–121]: we give a precise estimate of $\lambda $ (Taylor expansion) and we prove that there exist a diffeomorphism g and rotations $r_i$ such that $\mbox {dist}(gf_ig^{-1},r_i)\ll |\lambda |^{{1}/{2}}$ for $i=1,\ldots , m$ . We also state analogous results for random products of $2\times 2$ matrices, without any diophantine condition.

A NOTE ON THE UNIMODAL FEIGENBAUM'S MAPS

2009 ◽  
Vol 23 (14) ◽  
pp. 3101-3111
Author(s):  
GUIFENG HUANG ◽  
LIDONG WANG ◽  
GONGFU LIAO
Keyword(s):  
Fixed Point ◽  
Hausdorff Dimension ◽  
Periodic Points ◽  
Limit Set ◽  
Almost Periodic ◽  
Limit Sets ◽  
Shift Map ◽  
Renormalization Operator

We mainly investigate the likely limit sets and the kneading sequences of unimodal Feigenbaum's maps (Feigenbaum's map can be regarded as the fixed point of the renormalization operator [Formula: see text], where λ is to be determined). First, we estimate the Hausdorff dimension of the likely limit set for the unimodal Feigenbaum's map and then for every decimal s ∈ (0, 1), we construct a unimodal Feigenbaum's map which has a likely limit set with Hausdorff dimension s. Second, we prove that the kneading sequences of unimodal Feigenbaum's maps are uniformly almost periodic points of the shift map but not periodic ones.

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