Daniel Smania
We study the dynamics of the renormalization operator for multimodal maps. In particular, we prove the exponential convergence of this operator for infinitely renormalizable maps with same bounded combinatorial type.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
In this paper, we consider the renormalization operator $\mathcal R$ for multimodal maps. We prove the renormalization operator $\mathcal R$ is a self-homeomorphism on any totally $\mathcal R$-invariant set. As a corollary, we prove the existence of the full renormalization horseshoe for multimodal maps.
We proved the so called complex bounds for multimodal, infinitely renormalizable analytic maps with bounded combinatorics: deep renormalizations have polynomial-like extensions with definite modulus. The complex bounds is the first step to extend the renormalization theory of unimodal maps to multimodal maps.
In this paper we generalize renormalization theory for analytic critical circle maps with a cubic critical point to the case of maps with an arbitrary odd critical exponent by proving a quasiconformal rigidity statement for renormalizations of such maps.
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.
This paper deals with the renormalization of symmetric bimodal maps with low smoothness. We prove the existence of the renormalization fixed point in the space $C^{1+Lip}$ symmetric bimodal maps. Moreover, we show that the topological entropy of the renormalization operator defined on the space of $C^{ 1+Lip}$ symmetric bimodal maps is infinite. Further we prove the existence of a continuum of fixed points of renormalization. Consequently, this proves the non-rigidity of the renormalization of symmetric bimodal maps.
We develop a renormalization theory for analytic homeomorphisms of the circle with two cubic critical points. We prove a renormalization hyperbolicity theorem. As a basis for the proofs, we develop complex a priori bounds for multi-critical circle maps.
We construct a renormalization operator which acts on analytic circle maps whose critical exponent $α$ is not necessarily an odd integer $2n+1$, $n\in\mathbb N$. When $α=2n+1$, our definition generalizes cylinder renormalization of analytic critical circle maps. In the case when $α$ is close to an odd integer, we prove hyperbolicity of renormalization for maps of bounded type. We use it to prove universality and $C^{1+α}$-rigidity for such maps.
We study the renormalization operator of circle homeomorphisms with a break point and show that it possesses a hyperbolic horseshoe attractor.
We use hyperbolicity of golden-mean renormalization of dissipative Hénon-like maps to prove that the boundaries of Siegel disks of sufficiently dissipative quadratic complex Hénon maps with golden-mean rotation number are topological circles. Conditionally on an appropriate renormalization hyperbolicity property, we derive the same result for Siegel disks of Hénon maps with all eventually periodic rotation numbers.
Hyperbolicity of Renormalization for bi-cubic circle maps with bounded combinatorics
本文为具有有界组合的双三次圆映射构造了带有余维二稳定叶层的重正化双曲吸引子。
Lorenz maps are maps of the unit interval with one critical point of order rho>1, and a discontinuity at that point. They appear as return maps of leafs of sections of the geometric Lorenz flow. We construct real a priori bounds for renormalizable Lorenz maps with certain monotone combinatorics, and use these bounds to show existence of periodic points of renormalization, as well as existence of Cantor attractors for dynamics of infinitely renormalizable Lorenz maps.
We study the dimension spectrum of Lyapunov exponents for multimodal maps of the interval and their generalizations. We also present related results for rational maps on the Riemann sphere.
We prove that for continuous maps on the interval, the existence of an n-cycle, implies the existence of n-1 points which interwind the original ones and are permuted by the map. We then use this combinatorial result to show that piecewise affine maps (with no zero slope) cannot be infinitely renormalizable.
A Lorenz map is a Poincaré map for a three-dimensional Lorenz flow. We describe the theory of renormalization for Lorenz maps with a critical point and prove that a restriction of the renormalization operator acting on such maps has a hyperbolic fixed point. The proof is computer assisted and we include a detailed exposition on how to make rigorous estimates using a computer as well as the implementation of the estimates.
We demonstrate the existence of a global attractor A with a Cantor set structure for the renormalization of critical circle mappings. The set A is invariant under a generalized renormalization transformation, whose action on A is conjugate to the two-sided shift.
A numerical algorithm to compute the topological entropy of multimodal maps is proposed. This algorithm results from a closed formula containing the so-called min-max symbols, which are closely related to the kneading symbols. Furthermore, it simplifies a previous algorithm, also based on min-max symbols, which was originally proposed for twice differentiable multimodal maps. The new algorithm has been benchmarked against the old one with a number of multimodal maps, the results being reported in the paper. In particular, the comparison is favorable to the new algorithm, except in the unimodal…
In this note, we extend the renormalization horseshoe we have recently constructed with N. Goncharuk for analytic diffeomorphisms of the circle to their small two-dimensional perturbations. As one consequence, Herman rings with rotation numbers of bounded type survive on a codimension one set of parameters under small two-dimensional perturbations.
This paper is devoted to the study of dimension theory, in particular multifractal analysis, for multimodal maps. We describe the Lyapunov spectrum, generalising previous results by Todd. We also study the multifractal spectrum of pointwise dimension. The lack of regularity of the thermodynamic formalism for this class of maps is reflected in the phase transitions of the spectra.
Inou and Shishikura provided a class of maps that is invariant by near-parabolic renormalization, and that has proved extremely useful in the study of the dynamics of quadratic polynomials. We provide here another construction, using more general arguments. This will allow to extend the range of applications to unicritical polynomials of all degrees.
Area-preserving maps have been observed to undergo a universal period-doubling cascade, analogous to the famous Feigenbaum-Coullet-Tresser period doubling cascade in one-dimensional dynamics. A renormalization approach has been used by Eckmann, Koch and Wittwer in a computer-assisted proof of existence of a conservative renormalization fixed point. Furthermore, it has been shown by Gaidashev, Johnson and Martens that infinitely renormalizable maps in a neighborhood of this fixed point admit invariant Cantor sets with vanishing Lyapunov exponents on which dynamics for any two maps is smoothly c…
Renormalization of circle maps and smoothness of Arnold tongues
本文研究重整化算子在包含三次临界圆周映射与圆周微分同胚的 Banach 流形上的全局行为，并据此证明了无理 Arnold 舌的光滑性。
Hyperbolicity of renormalization of critical quasicircle maps
本文研究了临界准圆映射的重正化超几何性，扩展了Yampolsky的结果，通过冠状重正化算子证明了其超几何固定点性质。
We show that a strengthened version of the Collet-Eckmann condition for multimodal maps is topologically invariant. In particular, if f is non-uniformly expanding and the critical points are generic with respect to the absolutely continuous invariant measure, then any map g topologically conjugate to f is also non-uniformly expanding.
We prove exponential contraction of renormalization along hybrid classes of infinitely renormalizable unimodal maps (with arbitrary combinatorics), in any even degree $d$. We then conclude that orbits of renormalization are asymptotic to the full renormalization horseshoe, which we construct. Our argument for exponential contraction is based on a precompactness property of the renormalization operator ("beau bounds"), which is leveraged in the abstract analysis of holomorphic iteration. Besides greater generality, it yields a unified approach to all combinatorics and degrees: there is no need …
For each piecewise linear Lorenz map that expand on average, we show that it admits a dichotomy: it is either periodic renormalizable or prime. As a result, such a map is conjugate to a $β$-transformation.
In \cite{Mil}, Milnor posed the {\em Monotonicity Conjecture} that the set of parameters within a family of real multimodal polynomial interval maps, for which the topological entropy is constant, is connected. This conjecture was proved for quadratic by Milnor & Thurston \cite{MT} and for cubic maps by Milnor & Tresser, see \cite{MTr} and also \cite{DGMT}. In this paper we will prove the general case.
For a two parameter family of two-dimensional piecewise linear maps and for every natural number $ n $ we prove not only the existence of intervals of parameters for which the respective maps are $ n $ times renormalizable but also we show the existence of intervals of parameters where the coexistence of at least $ 2^n $ strange attractors takes place. This family of maps contains the two-dimensional extension of the classical one-dimensional family of tent maps.
We propose an extension of the one dimensional (doubling) renormalization operator to the case of maps on the cylinder. The kind of maps considered are commonly referred as quasi-periodic forced one dimensional maps. We prove that the fixed point of the one dimensional renormalization operator extends to a fixed point of the quasi-periodic forced renormalization operator. We also prove that the operator is differentiable around the fixed point and we study its derivative. Then we consider a two parametric family of quasi-periodically forced maps which is a unimodal one dimensional map with a f…
In a previous work by the authors the one dimensional (doubling) renormalization operator was extended to the case of quasi-periodically forced one dimensional maps. The theory was used to explain different self-similarity and universality observed numerically in the parameter space of the Forced Logistic Maps. The extension proposed was not complete in the sense that we assumed a total of four conjectures to be true. In this paper we present numerical support for these conjectures. We also discuss the applicability of this theory to the Forced Logistic Map.
We realize a dynamical decomposition for a post-critically finite rational map which admits a combinatorial decomposition. We split the Riemann sphere into two completely invariant subsets. One is a subset of the Julia set consisting of uncountably many Jordan curve components. Most of them are wandering. The other consists of components that are pullbacks of finitely many renormalizations, together with possibly uncountably many points. The quotient action on the decomposed pieces is encoded by a dendrite dynamical system. We also introduce a surgery procedure to produce post-critically finit…
Renormalization in Lorenz maps -- completely invariant sets and periodic orbits
本文研究扩张Lorenz映射的动力学，利用解析与符号方法揭示周期点、完全不变集与重整化的联系，提出检测可恢复重整化的算法，并区分单侧与双侧原像，完善了该领域的动力学描述。
In this paper we give a combinatorial description of the renormlization limits of infinitely renormalizable unimodal maps with {\it essentially bounded} combinatorics admitting quadratic-like complex extensions. As an application we construct a natural analogue of the period-doubling fixed point. Dynamical hairiness is also proven for maps in this class. These results are proven by analyzing {\it parabolic towers}: sequences of maps related either by renormalization or by {\it parabolic renormalization}.
A general ansatz in Renormalization Theory, already established in many important situations, states that exponential convergence of renormalization orbits implies that topological conjugacies are actually smooth (when restricted to the attractors of the original systems). In this paper we establish this principle for a large class of bicritical circle maps, which are $C^3$ circle homeomorphisms with irrational rotation number and exactly two (non-flat) critical points. The proof presented here is an adaptation, to the bicritical setting, of the one given by de Faria and de Melo for the case o…
Let $f$ and $g$ be smooth multimodal maps with no periodic attractors and no neutral points. If a topological conjugacy $h$ between $f$ and $g$ is $C^{1}$ at a point in the nearby expanding set of $f$, then $h$ is a smooth diffeomorphism in the basin of attraction of a renormalization interval of $f$. In particular, if $f:I \to I$ and $g:J \to J$ are $C^r$ unimodal maps and $h$ is $C^{1}$ at a boundary of $I$ then $h$ is $C^r$ in $I$.
In this work we study the renormalization operator acting on piecewise smooth homeomorphisms on the circle, that turns out to be essentially the study of Rauzy-Veech renormalizations of generalized interval exchanges maps with genus one. In particular we show that renormalizations of such maps with zero mean nonlinearity and satisfying certain smoothness and combinatorial assumptions converges to the set of piecewise affine interval exchange maps.
Hyperbolicity of renormalization for maps with multiple breaks
本文在断点小且旋转数有界类型的条件下，为多重断点圆周映射构造双曲马蹄，并证明其刚性类为解析子流形。
Given a multimodal interval map $f:I \to I$ and a Hölder potential $ϕ:I \to \mathbb{R}$, we study the dimension spectrum for equilibrium states of $ϕ$. The main tool here is inducing schemes, used to overcome the presence of critical points. The key issue is to show that enough points are `seen' by a class of inducing schemes. We also compute the Lyapunov spectrum. We obtain the strongest results when $f$ is a Collet-Eckmann map, but our analysis also holds for maps satisfying much weaker growth conditions.
Renormalizations can be considered as building blocks of complex dynamical systems. This phenomenon has been widely studied for iterations of polynomials of one complex variable. Concerning non-polynomial hyperbolic rational maps, a recent work of Cui-Tan shows that these maps can be decomposed into postcritically finite renormalization pieces. The main purpose of the present work is to perform the surgery one step deeper. Based on Thurston's idea of decompositions along multicurves, we introduce a key notion of Cantor multicurves (a stable multicurve generating infinitely many homotopic curve…
In this paper we give a new prove of hyperbolicity of renormalization of critical circle maps using the formalism of almost-commuting pairs. We extend renormalization to two-dimensional dissipative maps of the annulus which are small perturbations of one-dimensional critical circle maps. Finally, we demontsrate that a two-dimensional map which lies in the stable set of the renormalization operator possesses an attractor which is topologically a circle. Such a circle is critical: the dynamics on it is topologically, but not smoothly, conjugate to a rigid rotation.
Topological entropy is a measure of complex dynamics. In this regard, multimodal maps play an important role when it comes to study low-dimensional chaotic dynamics or explain some features of higher dimensional complex dynamics with conceptually simple models. In the first part of this paper an analytical formula for the topological entropy of twice differentiable multimodal maps is derived, and some basic properties are studied. This expression involves the so-called min-max symbols, which are closely related to the kneading symbols. Furthermore, its proof leads to a numerical algorithm that…
This paper is devoted to the study of the thermodynamic formalism for a class of real multimodal maps. This class contains, but it is larger than, Collet-Eckmann. For a map in this class, we prove existence and uniqueness of equilibrium states for the geometric potentials $-t \log|Df|$, for the largest possible interval of parameters $t$. We also study the regularity and convexity properties of the pressure function, completely characterising the first order phase transitions. Results concerning the existence of absolutely continuous invariant measures with respect to the Lebesgue measure are …
We define a hyperbolic renormalizations suitable for maps of small determinant, with uniform bounds for large periods. The techniques involve an improvement of the celebrated Palis-Takens renormalization and normal forms (fibered linearizations). These techniques are useful to study the dynamics of Hénon like maps and the geometry of their parameter space.
We describe a 2 parameter family of polygon exchange transformations parameterized by points in a square. Whenever the two parameters are irrational, the polygon exchange has periodic orbits of arbitrarily large period. We show that for almost all parameters, the polygon exchange map has the property that almost every point is periodic. However, there is a dense set of irrational parameters for which this fails. By choosing parameters carefully, the measure of non-periodic points can be made arbitrarily close to full measure. These results are powered by a notion of renormalization which holds…
In this paper, we explore the period tripling and period quintupling renormalizations below $C^2$ class of unimodal maps. We show that for a given proper scaling data there exists a renormalization fixed point on the space of piece-wise affine maps which are infinitely renormalizable. Furthermore, we show that this renormalization fixed point is extended to a $C^{1+Lip}$ unimodal map, considering the period tripling and period quintupling combinatorics. Moreover, we show that there exists a continuum of fixed points of renormalizations by considering a small variation on the scaling data. Fina…
On Regular Hénon-like Renormalization
本文发展了非微扰耗散 Hénon-like 映射的重正化理论，通过引入 Pesin 理论控制高维小尺度几何，证明其收敛至一维吸引子且条件有限时间可验证，并揭示了无限可重正化映射的正则单临界性质。
We show that in any family of stunted sawtooth maps, the set of maps whose set of periods is the set of all powers of 2 has no interior point, i.e., the combinatorial description of the boundary of chaos coincides with the topological description. We also show that, under mild assumptions, smooth multimodal maps whose set of periods is the set of all powers of 2 are infinitely renormalizable.
The renormalization of a quadratic-like map is studied. The three-dimensional Yoccoz puzzle for an infinitely renormalizable quadratic-like map is discussed. For an unbranched quadratic-like map having the {\sl a priori} complex bounds, the local connectivity of its Julia set is proved by using the three-dimensional Yoccoz puzzle. The generalized version of Sullivan's sector theorem is discussed and is used to prove his result that the Feigenbaum quadratic polynomial has the {\sl a priori} complex bounds and is unbranched. A dense subset on the boundary of the Mandelbrot set is constructed so …
We consider multimodal maps with holes and study the evolution of the open systems with respect to equilibrium states for both geometric and Hölder potentials. For small holes, we show that a large class of initial distributions share the same escape rate and converge to a unique absolutely continuous conditionally invariant measure; we also prove a variational principle connecting the escape rate to the pressure on the survivor set, with no conditions on the placement of the hole. Finally, introducing a weak condition on the centre of the hole, we prove scaling limits for the escape rate for …
We study highly dissipative Hénon maps $$ F_{c,b}: (x,y) \mapsto (c-x^2-by, x) $$ with zero entropy. They form a region $Π$ in the parameter plane bounded on the left by the curve $W$ of infinitely renormalizable maps. We prove that Morse-Smale maps are dense in $Π$, but there exist infinitely many different topological types of such maps (even away from $W$). We also prove that in the infinitely renormalizable case, the average Jacobian $b_F$ on the attracting Cantor set $\OO_F$ is a topological invariant. These results come from the analysis of the heteroclinic web of the saddle periodic poi…
Some invariant classes for parabolic renormalization in the multicritical case
本文推广Shishikura关于抛物重整化稳定映射类的结果，将其扩展到与有限Blaschke乘积共轭的更一般映射类，并引入Milnor映射方案的类比以描述重整化作用，为精细研究扰动奠定基础。
For a class of non-uniformly hyperbolic interval maps, we study rates of escape with respect to conformal measures associated with a family of geometric potentials. We establish the existence of physically relevant conditionally invariant measures and equilibrium states and prove a relation between the rate of escape and pressure with respect to these potentials. As a consequence, we obtain a Bowen formula: we express the Hausdorff dimension of the set of points which never exit through the hole in terms of the relevant pressure function. Finally, we obtain an expression for the derivative of …
In this note we construct hyperbolic fixed points for cylinder renormalization of maps with Siegel disks.
Infinitely renormalizable Hénon-like map in arbitrary finite dimension is considered. The set, $\mathcal N$ of infinitely renormalizable Hénon-like maps satisfying the certain condition is invariant under renormalization operator. The Cantor attractor of infinitely renormalizable Hénon-like map, $F$ in $\mathcal N$ has {\em unbounded geometry} almost everywhere in the parameter space of the universal number which corresponds to the average Jacobian of two dimensional map. This is an extension of the same result in $\mathcal N$ for three dimensional infinitely renormalizable Hénon-like maps.
In this paper we are concerned with quasi-periodic forced one dimensional maps. We consider a two parametric family of quasi-periodically forced maps such that the one dimensional map (before forcing) is unimodal and it has a full cascade of period doubling bifurcations. Between one period doubling and the next one it is known that there exist a parameter value where the $2^n$-periodic orbit is superatracting. In a previous work we proposed an extension of the one-dimensional (doubling) renormalization operator to the quasi-periodic case. We proved that, if the family satisfies suitable hypoth…
We study renormalizations of piecewise smooth homeomorphisms on the circle, by considering such maps as generalized interval exchange maps of genus one. Suppose that $Df$ is absolutely continuous on each interval of continuity and $D\ln{Df}\in \mathbb{L}_{p}$ for some $p>1$. We prove, that under certain combinatorial assumptions on $f_{1}$ and $f_{2}$, corresponding renormalizations approach to each other in $C^{1+L_{1}}$-norm.
For a class of polynomial maps of one variable with a parabolic fixed points and degrees bigger than $21$, the parabolic renormalization is introduced based on Fatou coordinates and horn maps, and a type of maps which are invariant under the parabolic renormalization is also given. For the small perturbation of these kinds of maps, the near-parabolic renormalization is also introduced based on the first return maps defined on the fundamental regions. As an application, we show the existence of non-renormalizable polynomial maps with degrees bigger than $21$ such that the Julia sets have positi…
Renormalization of Unicritical Diffeomorphisms of the Disk
本文引入圆盘上无限可重整化的单临界微分同胚，证明其重整化后成为 Hénon 型并超指数收敛至一维单峰映射，同时刻画稳定与中心流形几何。
We study the effects of IID random perturbations of amplitude $ε> 0$ on the asymptotic dynamics of one-parameter families $\{f_a : S^1 \to S^1, a \in [0,1]\}$ of smooth multimodal maps which "predominantly expanding", i.e., $|f'_a| \gg 1$ away from small neighborhoods of the critical set $\{ f'_a = 0 \}$. We obtain, for any $ε> 0$, a \emph{checkable, finite-time} criterion on the parameter $a$ for random perturbations of the map $f_a$ to exhibit (i) a unique stationary measure, and (ii) a positive Lyapunov exponent comparable to $\int_{S^1} \log |f_a'| \, dx$. This stands in contrast with the …
Let $f$ be an orientation preserving homeomorphisms on the circle with several break points, that is, its derivative $Df$ has jump discontinuities at these points. We study Rauzy-Veech renormalizations of piecewise smooth circle homeomorphisms, by considering such maps as generalized interval exchange maps with genus one. Suppose that $Df$ is absolutely continuous on the each interval of continuity and $D\ln{Df}\in \mathbb{L}_{p}$ for some $p>1$. We prove that, under certain combinatorial assumptions on $f$, renormalizations $R^{n}(f)$ are approximated by piecewise Möbus functions in $C^{1+L_{…
We study renormalization of highly dissipative analytic three dimensional Hénon maps $$ F(x,y,z) = (f(x) - \varepsilon(x,y,z),\ x,\ δ(x,y,z)) $$ where $ \varepsilon(x,y,z) $ is a sufficiently small perturbation of $ \varepsilon_{2d}(x,y) $. Under certain conditions, $ C^r $ single invariant surfaces each of which is tangent to the invariant plane field over the critical Cantor set exist for $ 2 \leq r < \infty $. The $ C^r $ conjugation from an invariant surface to the $ xy- $plane defines renormalization two dimensional $ C^r $ Hénon-like map. It also defines two dimensional embedded $ C^r $ …
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。