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.
In this paper, we study the Hausdorff dimension of the generalized intrinsic level set with respect to the given ergodic meausre in a class of non-uniformly hyperbolic interval maps with finitely many branches.
For a Markov map of an interval or the circle with countably many branches and finitely many neutral periodic points, we establish conditional variational formulas for the mixed multifractal spectra of Birkhoff averages of countably many observables, in terms of the Hausdorff dimension of invariant probability measures. Using our results, we are able to exhibit new fractal-geometric results for backward continued fraction expansions of real numbers, answering in particular a question of Pollicott. Moreover, we establish formulas for multi-cusp winding spectra for the Bowen-Series maps associat…
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.
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…
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 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 …
Geometric pressure for multimodal maps of the interval
本文是作者S. Smirnov等人在黎曼球面上的有理映射迭代的背景下：非一致双曲性、几何压力和尼斯诱导方案方法的几个概念的等价性，导致了代数形式主义的结果。我们研究了广义多峰映射，即R中紧致区间的有限并到R中的光滑映射f具有非平坦临界点，使得在其极大前向不变集K上f是拓扑传递的，且具有正的拓扑熵.证明了f的几个非一致双曲性概念|_K是等价的（包括周期轨道上的一致双曲性、TCE和K双曲排斥中的所有周期轨道、Lyapunov双曲性和拉回的指数收缩）。证明了几何压力P（t）的几种定义，即映射f的压力|_K和电位- t log| F'|，给出相同的值（包括周期轨道上的压力、“树”压力、变分压力和共形压力）。最后，我们证明，所有的周期轨道在K是双曲排斥，函数P（t）是真正的解析之间的“凝聚”和“冻结”参数的t，并为每个这样的t存在唯一的平衡（和共形）措施，满足强统计性质。
Instituto de Matemática e Estatística, Universidade de São Paulo（圣保罗大学数学与统计学院）；Instituto de Matemática Pura e Aplicada（纯数学与应用数学研究所）；Shenzhen International Center for Mathematics, SUSTech（南方科技大学深圳国际数学中心）
We consider Milnor's "tower algorithm" in the space of piecewise monotone maps, an iterative algorithm on the space of metrics which unifies, on the one hand, Thurston's iterative scheme which converges to holomorphic models, and, on the other hand, the theory of piecewise linear models coming from kneading theory. We prove that the algorithm converges for unimodal maps of high entropy, and provide examples where it does not converge for unimodal maps of low entropy and multimodal maps of higher degree.
Metric stability for random walks (with applications in renormalization theory)
考虑确定性随机游动F：I x Z -> I x Z，定义为F（x，n）=（f（x），K（x）+n），其中f是区间I和K：I->Z上的扩展马尔可夫映射。本文研究了当|n|趋于无穷大时，与F拓扑共轭的G（x，n）=（f_n（x），L（x，n）+n）型扰动类的遍历性（如常返性和瞬时性）、几何性和多重分形性的普适性（稳定性）.我们给出了这些结果在研究区间上的（广义）无限可重整映射之间的共轭的正则性和一维映射的野吸引子的存在性中的应用。
