Artur Avila, Carlos Gustavo Moreira
We review recent results that lead to a very precise understanding of the dynamics of typical unimodal maps from the statistical point of view. We also describe the (generalized) renormalization approach to the study of the statistical properties of typical unimodal maps.
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We consider the bifurcation structure of one-parameter families of unimodal maps whose differentiability is only $C^1$. The structure of its bifurcation diagram can be a very wild one in such case. However we prove that in a certain topological sense, the structure is the same as that of the standard family of quadratic polynomials. In the case of families of polynomials, irreducible component of the bifurcation diagram can be defined naturally by dividing by the polynomials corresponding to lower periods. We show that such a irreducible component can be defined even if the maps and the family…
A symmetric Lorenz map is obtain by ``flipping'' one of the two branches of a symmetric unimodal map. We use this to derive a Sharkovsky-like theorem for symmetric Lorenz maps, and also to find cases where the unimodal map restricted to the critical omega-limit set is conjugate to a Sturmian shift. This has connections with properties of unimodal inverse limit spaces embedded as attractors of some planar homeomorphisms.
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.
In this paper we prove C^k structure stability conjecture for unimodal maps. In other words, we shall prove that Action A maps are dense in the space of C^k unimodal maps in the C^k topology. Here k can be 1,2,...,\infty,ω.
It is shown that some topological equivalency classes of S-unimodal maps are equal to quasisymmetric conjugacy classes. This includes some infinitely renormalizable polynomials of unbounded type.
Non-existence of wandering intervals for asymmetric unimodal maps
本文证明了非对称单峰映射不存在游荡区间，解决了该动力系统中的一个基本问题。
A distortion theory is developed for $S-$unimodal maps. It will be used to get some geometric understanding of invariant Cantor sets. In particular attracting Cantor sets turn out to have Lebesgue measure zero. Furthermore the ergodic behavior of $S-$unimodal maps is classified according to a distortion property, called the Markov-property.
We study in this work different $(0, 1)$-codings of points from the unit interval $[0, 1]$ in the relation with the treatment of continuous unimodal maps.
We introduce {\it twist unimodal maps} of the interval and describe their structure. Sufficient conditions for the growth of over-rotation interval in families of maps are given.
We classify the measure theoretic attractors of general C^3 unimodal maps with quadratic critical points. The main ingredient is the decay of geometry.
Examples are given of tent maps $T$ for which there exist non-trivial sets $B \subset [0,1]$ such that $T:B \to B$ is a homeomorphism.
We show that the topological entropy is monotonic for unimodal interval maps which are obtained from the restriction of quadratic rational maps with real coefficients. This is done by ruling out the existence of certain post-critical curves in the moduli space of aforementioned maps, and confirms a conjecture made in [Fil19] based on experimental evidence.
We prove that the topological entropy of real unimodal maps depends as a Hoelder continuous function of the kneading parameter, and the local Hoelder exponent equals, up to a factor log 2, the value of the function at that point.
Given $C^2$ infinitely renormalizable unimodal maps $f$ and $g$ with a quadratic critical point and the same bounded combinatorial type, we prove that they are $C^{1+α}$ conjugate along the closure of the corresponding forward orbits of the critical points, for some $α>0$.
On ergodic optimization for unimodal maps
本文研究典型非一致扩张单峰映射的遍历优化问题，证明通用Lipschitz函数的唯一最大化测度由周期轨道支撑。
The thermodynamical formalism is studied for renormalisable maps of the interval and the natural potential $-t \log|Df|$. Multiple and indeed infinitely many phase transitions at positive $t$ can occur for some quadratic maps. All unimodal quadratic maps with positive topological entropy exhibit a phase transition in the negative spectrum.
We introduce here a classification of unimodal maps $[0, 1]\rightarrow [0, 1]$, which commute with piecewise linear surjective maps $[0, 1]\rightarrow [0, 1]$. Remind that if continuous piecewise linear unimodal map $g$ commutes with a non-constant piecewise linear map $ψ$, which is not an iteration of $g$, then $g$ is topologically conjugated with the tent map by piecewise linear conjugacy. We use the obtained classification to illustrate the mentioned fact.
We prove that the Newton quotient of the average R(t) of a lipschitzian function (with non vanishing variation) with respect to the SRB measure on a transversal family f_t of piecewise expanding unimodal maps, after an appropriated normalization, converges in distribution to the normal distribution.
The first result of the paper (Theorem 1.1) is an explicit construction of unimodal maps that are semiconjugate, on the post-critical set, to the circle rotation by an arbitrary irrational angle $θ\in(3/5,2/3)$. Our construction is a generalization of the construction by Milnor and Lyubich [LM] of the Fibonacci unimodal maps semi-conjugate to the circle rotation by the golden ratio. Generalizing a theorem by Milnor and Lyubich for the Fibonacci map, we prove that the Hausdorff dimension of the post-critical set of our unimodal maps is $0$, provided the denominators of the continued fraction of…
We investigate the set of Galois conjugates of growth rates of superattracting real quadratic polynomials, following W. Thurston. In particular, we prove that the closure of this set is path-connected and locally connected.
We prove that almost every non-regular real quadratic map is Collet-Eckmann and has polynomial recurrence of the critical orbit (proving a conjecture by Sinai). It follows that typical quadratic maps have excellent ergodic properties, as exponential decay of correlations (Keller and Nowicki, Young) and stochastic stability in the strong sense (Baladi and Viana). This is an important step to get the same results for more general families of unimodal maps.
We prove that there is a residual set of families of smooth or analytic unimodal maps with quadratic critical point and negative Schwarzian derivative such that almost every non-regular parameter is Collet-Eckmann with subexponential recurrence of the critical orbit. Those conditions lead to a detailed and robust statistical description of the dynamics. This proves the Palis conjecture in this setting.
Maximal attractors for perturbations of unimodal maps near a homoclinic tangency
本文证明单峰映射族的小平面扰动在同宿切附近参数处存在极大吸引子，该结果适用于具特定双曲性质的一般单峰映射，推广了相关学者针对类Hénon族等的已有结果。
We obtain estimates relating the phase space and the parameter space of analytic families of unimodal maps. Using those estimates, we show that typical analytic unimodal maps admit a quasiquadratic renormalization. This reduces the study of the statistical properties of typical unimodal maps to the quasiquadratic case which had been studied in \cite {AM2}. The estimates proved here correspond exactly to the Phase-Parameter relation proved in \cite {AM} in the quadratic case, and allows one to obtain sharp estimates on the dynamics of typical unimodal maps which were available only in the quadr…
For a non-flat $C^3$ unimodal map with a Cantor attractor, we show that for any open cover $\mathcal U$ of this attractor, the complexity function $p(\mathcal U, n)$ is of order $n\log n$. In the appendix, we construct a non-renormalizable map with a Cantor attractor for which $p(\mathcal{U}, n)$ is bounded from above for any open cover $\mathcal{U}$.
In the space of C^k piecewise expanding unimodal maps, k>=1, we characterize the C^1 smooth families of maps where the topological dynamics does not change (the "smooth deformations") as the families tangent to a continuous distribution of codimension-one subspaces (the "horizontal" directions) in that space. Furthermore such codimension-one subspaces are defined as the kernels of an explicit class of linear functionals. As a consequence we show the existence of C^{k-1+Lip} deformations tangent to every given C^k horizontal direction, for k>=2.
Absolutely continuous invariant measures and the Collet-Eckmann condition for long-branched Sturmian unimodal maps
该研究聚焦长分支斯特姆单峰映射，通过揉类与特定洛伦兹构造的关联，计算切割共切割时间，建立绝对连续不变测度判据，还证明几乎所有旋转角下无有限阶S-单峰实现满足Collet-Eckmann条件。
Let $g:\, [0, 1]\rightarrow [0, 1]$ be piecewise linear unimodal map. We say that $g$ has non-trivial piecewise linear commutator, if there is a continuous piecewise linear $ψ:\, [0, 1]\rightarrow [0, 1]$ such that $g\circ ψ= ψ\circ g$, and, moreover, $ψ$ is neither an iteration of $g$, not a constant map. We prove that if $g$ has a non-trivial piecewise linear commutator, then $g$ is topologically conjugated with the tent map by a piecewise linear conjugacy.
We investigate the flexibility of the entropy (topological and metric) for the class of piecewise expanding unimodal maps. We show that the only restrictions for the values of the topological and metric entropies in this class are that both are positive, the topological entropy is at most $\log 2$, and by the Variational Principle, the metric entropy is not larger than the topological entropy. In order to have a better control on the metric entropy, we work mainly with topologically mixing piecewise expanding skew tent maps, for which there are only 2 different slopes. For those maps, there is…
Let $f : [0, 1] \to [0, 1]$ be a piecewise expanding unimodal map of class $C^{k+1}$, with $k \geq 1$, and $μ= ρdx$ the (unique) SRB measure associated to it. We study the regularity of $ρ$. In particular, points $\mathcal{N}$ where $ρ$ is not differentiable has zero Hausdorff dimension, but is uncountable if the critical orbit of $f$ is dense. This improves on a work of Szewc (1984). We also obtain results about higher orders of differentiability of $ρ$ in the sense of Whitney.
In this note we consider a collection $\cal{C}$ of one parameter families of unimodal maps of $[0,1].$ Each family in the collection has the form $\{μf\}$ where $μ\in [0,1].$ Denoting the kneading sequence of $μf$ by $K(μf)$, we will prove that for each member of $\cal{C}$, the map $μ\mapsto K(μf)$ is monotone. It then follows that for each member of $\cal{C}$ the map $μ\mapsto h(μf)$ is monotone, where $h(uf)$ is the topological entropy of $μf.$ For interest, $μf(x)=4μx(1-x)$ and $μf(x)=μ\sin(πx)$ are shown to belong to $\cal{C}.$ This extends the work of Masato Tsujii [1].
We study the parameter space of unicritical polynomials $f_c:z\mapsto z^d+c$. For complex parameters, we prove that for Lebesgue almost every $c$, the map $f_c$ is either hyperbolic or infinitely renormalizable. For real parameters, we prove that for Lebesgue almost every $c$, the map $f_c$ is either hyperbolic, or Collet-Eckmann, or infinitely renormalizable. These results are based on controlling the spacing between consecutive elements in the ``principal nest'' of parapuzzle pieces.
Towers and Bratteli-Vershik systems in Fibonacci-like unimodal maps
本文研究了斐波那契-like单峰映射中通过塔结构构造康托尔集，并推导了其唯一不变测度的显式公式。
In this paper we construct a correspondence between the parameter spaces of two families of one-dimensional dynamical systems, the alpha-continued fraction transformations T_alpha and unimodal maps. This correspondence identifies bifurcation parameters in the two families, and allows one to transfer topological and metric properties from one setting to the other. As an application, we recover results about the real slice of the Mandelbrot set, and the set of univoque numbers.
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 …
In this work, we relate the geometry of chaotic attractors of typical analytic unimodal maps to the behavior of the critical orbit. Our main result is an explicit formula relating the combinatorics of the critical orbit with the exponents of periodic orbits. This connection between topological and smooth invariants is obtained through an analysis of the physical measure. Since the exponents of periodic orbits form a complete set of smooth invariants in this setting, we have ``typical geometric rigidity'' of the dynamics of such chaotic attractors. This unexpected result implies that the lamina…
In this paper we initiate the study of the arithmetical properties of a set numbers which encode the dynamics of unimodal maps in a universal way along with that of the corresponding topological zeta function. Here we are concerned in particular with the Feigenbaum bifurcation.
Let $f_t:[0,1] \to [0,1]$ be a family of piecewise expanding unimodal maps with a common critical point that is dense for almost all $t \in [a,b]$. If $μ_t$ is the corresponding SRB measure for $f_t$, we study the regularity of $Γ(t)=\int ϕdμ_t$ when assuming that the family is transversal to the topological classes of these maps, more precisely, we prove that if $J_t(c)=\sum_{k=0}^{\infty} \frac{v_t(f_t^k(c))}{Df_t^k(f_t(c))} \neq 0$ for all $t$, where $v_t(x)=\partial_s f_s(x)|_{s=t}$, then $Γ(t)$ is not Lipschitz for almost all $t\in [a,b]$. Furthermore, we give the exact modulus of continu…
While the forward trajectory of a point in a discrete dynamical system is always unique, in general a point can have infinitely many backward trajectories. The union of the limit points of all backward trajectories through $x$ was called by M.~Hero the "special $α$-limit" ($sα$-limit for short) of $x$. In this article we show that there is a hierarchy of $sα$-limits of points under iterations of a S-unimodal map: the size of the $sα$-limit of a point increases monotonically as the point gets closer and closer to the attractor. The $sα$-limit of any point of the attractor is the whole non-wande…
Let $\{f_t\colon I\to I\}$ be a family of unimodal maps with topological entropies $h(f_t)>\frac12\log 2$, and ${\widehat{f}}_t\colon{\widehat{I}}_t\to{\widehat{I}}_t$ be their natural extensions, where ${\widehat{I}}_t=\varprojlim(I,f_t)$. Subject to some regularity conditions, which are satisfied by tent maps and quadratic maps, we give a complete description of the prime ends of the Barge-Martin embeddings of ${\widehat{I}}_t$ into the sphere. We also construct a family $\{χ_t\colon S^2\to S^2\}$ of sphere homeomorphisms with the property that each $χ_t$ is a factor of ${\widehat{f}}_t$, by…
Generalized beta-transformations are the class of piecewise continuous interval maps given by taking the beta-transformation $x \mapsto βx ~\pmod 1$, where $β>1$, and replacing some of the branches with branches of constant negative slope. If the orbit of 1 is finite, then the map is Markov, and we call beta (which must be an algebraic number) a generalized Parry number. We show that the Galois conjugates of such beta have modulus less than 2, and the modulus is bounded away from 2 apart from the exceptional case of conjugates lying on the real line. We give a characterization of the closure o…
A trichotomy for hitting times and escape rates for a class of unimodal maps
本文研究了一类单峰映射中击中时间与逃逸速率的三元划分，揭示了周期点在后临界轨道上的特性，并证明了广义渐近逃逸速率的性质。
We devote this work to the functional equation $ψ\circ g = g\circ ψ$, where $ψ$ is an unknown function and $g$ is piecewise linear unimodal map, which is topologically conjugated to the tent map. We will call such $ψ$ self-semiconjugations of $g$. Our the main results are the following: 1. Suppose that there is a self-semiconjugation of $g$, whose tangent at $0$ is not a power of $2$, and suppose that all the kinks of $g$ are in the complete pre-image of $0$. Then all the self-semiconjugations of $g$ are piecewise linear. 2. Suppose that all self-semiconjugations of $g$ are piecewise linear. T…
Recurrence, symbolic dynamics, and wild attractors for unimodal maps
本文通过符号动力学研究临界轨道递归性质与野Cantor吸引子的关系，构造了具有野吸引子的非重整化单峰映射族，并证明了长分支组合与持续递归的不兼容性。
For an infinitely renormalizable negative Schwarzian unimodal map $f$ with non-flat critical point, we analyze statistical properties of periodic points as the periods tend to infinity. Introducing a weight function $φ$ which is a continuous or a geometric potential $φ=-β\log|f'|$ ($β\in\mathbb R$), we establish the level-2 Large Deviation Principle for weighted periodic points. From this, we deduce that all weighted periodic points equidistribute with respect to equilibrium states for the potential $φ$. In particular, it follows that all periodic points are equidistributed with respect to mea…
We consider a family of strongly-asymmetric unimodal maps $\{f_t\}_{t\in [0,1]}$ of the form $f_t=t\cdot f$ where $f\colon [0,1]\to [0,1]$ is unimodal, $f(0)=f(1)=0$, $f(c)=1$ is of the form and $$f(x)=\left\{ \begin{array}{ll} 1-K_-|x-c|+o(|x-c|)& \mbox{ for }x<c, \\ 1-K_+|x-c|^β+ o(|x-c|^β) &\mbox{ for }x>c, \end{array}\right. $$ where we assume that $β>1$. We show that such a family contains a Feigenbaum-Coullet-Tresser $2^\infty$ map, and develop a renormalization theory for these maps. The scalings of the renormalization intervals of the $2^\infty$ map turn out to be super-exponential and…
The average R(t) of a smooth function with respect to the SRB measure of a smooth one-parameter family f_t of piecewise expanding interval maps is not always Lipschitz. We prove that if f_t is tangent to the topological class of f_0, then R(t) is differentiable at zero, and the derivative coincides with the resummation previously proposed by the first named author of the (a priori divergent) series given by Ruelle's conjecture.
Let f be a unimodal map of the interval with critical point c. If the orbit of c is not dense then most points in lim{[0,1],f} have neighborhoods that are homeomorphic with the product of a Cantor set and an open arc. The points without this property are called inhomogeneities, and the set, I, of inhomogeneities is equal to lim {ω(c),f| ω(c) }. In this paper we consider the relationship between the limit complexity of ω(c) and the limit complexity of I. We show that if ω(c) is more complicated than a finite collection of convergent sequences then I can have arbitrarily high limit complexity. W…
The tent map family is arguably the simplest 1-parametric family of maps with non-trivial dynamics and it is still an active subject of research. In recent works the second author, jointly with J. Yorke, studied the graph and backward limits of S-unimodal maps. In this article we generalize those results to tent-like unimodal maps. By tent-like here we mean maps that share fundamental properties that characterize tent maps, namely unimodal maps without wandering intervals nor attracting cycles and whose graph has a finite number of nodes.
An $S$-unimodal map $f$ with flat critical point satisfying the Misiurewicz condition displays a freezing phase transition in positive spectrum. We analyze statistical properties of the equilibrium state $μ_t$ for the potential $-t\log|Df|$, as well as how the phase transition slows down the rate of decay of correlations. We show that $μ_t$ has exponential decay of correlations for all inverse temperature $t$ contained in the positive entropy phase $(t^-,t^+)$. If the critical point is not too flat, then the freezing point $t^+$ is equal to $1$, and the absolutely continuous invariant probabil…
We give two new proofs that the SRB measure of a C^2 path f_t of unimodal piecewise expanding C^3 maps is differentiable at 0 if f_t is tangent to the topological class of f_0. The arguments are more conceptual than the one in our previous paper, but require proving Holder continuity of the infinitesimal conjugacy (a new result, of independent interest) and using spaces of bounded p-variation. The first new proof gives differentiability of higher order if f_t is smooth enough and stays in the topological class of f_0 and if the observable smooth enough (a new result). In addition, this proof d…
Bifurcations of the $A_{n}$ type in Arnold's classification, in non-autonomous $p$-periodic difference equations generated by parameter depending families with $p$ maps, are studied. It is proved that the conditions of degeneracy, non-degeneracy and unfolding are invariant relative to cyclic order of compositions for any natural number $n$. The main tool for the proofs is local topological conjugacy. Invariance results are essential to the proper definition of the bifurcations $A_{n}$, and lower codimension bifurcations associated, using all the possible cyclic compositions of the fiber famili…
We study bifurcations of cubic homoclinic tangencies in two-dimensional symplectic maps. We distinguish two types of cubic homoclinic tangencies, and each type gives different first return maps derived to diverse conservative cubic Hénon maps with quite different bifurcation diagrams. In this way, we establish the structure of bifurcations of periodic orbits in two parameter general unfoldings generalizing to the conservative case the results previously obtained for the dissipative case. We also consider the problem of 1:4 resonance for the conservative cubic Hénon maps.
The claim in Theorem 7.1 for dense postscritical orbits is that there exists a sequence tn (not for all sequences).
Homeomorphisms allowing us to prove topological equivalences between one-parameter families of maps undergoing the same bifurcation are constructed in this paper. This provides a solution for a classical problem in bifurcation theory that was set out three decades ago and remained unexpectedly unpublished until now
Existence of the Lyapunov exponent for $S$-unimodal maps
研究S-单峰映射李雅普诺夫指数，证明非平坦临界点的该映射对几乎每个勒贝格点李雅普诺夫指数存在且为常数λ_T，给出其为0的充要条件，还得出相关特殊映射的结论。
We derive an algorithm to determine recursively the lap number (minimal number of monotone pieces) of the iterates of unimodal maps of an interval with free end-points. The algorithm is obtained by the sign analysis of the itineraries of the critical point and of the boundary points of the interval map. We apply this algorithm to the estimation of the growth number and the topological entropy of maps with direct and reverse bifurcations.
We strengthen the standard bifurcation theorems for saddle-node, transcritical, pitchfork, and period-doubling bifurcations of maps. Our new formulation involves adding one or two extra terms to the standard truncated normal forms with coefficients determined by algebraic equations. These extended normal forms are differentiably conjugate to the original maps on basins of attraction and repulsion of fixed points or periodic orbits. This reflects common assumptions about the additional information in normal forms despite standard bifurcation theorems being formulated only in terms of topologica…
We study bifurcations of non-orientable area-preserving maps with quadratic homoclinic tangencies. We study the case when the maps are given on non-orientable two-dimensional surfaces. We consider one and two parameter general unfoldings and establish results related to the emergence of elliptic periodic orbits.
In a previous note [Ru] the susceptibility function was analyzed for some examples of maps of the interval. The purpose of the present note is to give a concise treatment of the general unimodal Markovian case (assuming $f$ real analytic). We hope that it will similarly be possible to analyze maps satisfying the Collet-Eckmann condition. Eventually, as explained in [Ru], application of a theorem of Whitney [Wh] should prove differentiability of the map $f\mapstoρ_f$ restricted to a suitable set.
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