Vilton Pinheiro
We construct SRB measures for endomorphisms satisfying conditions far weaker than the non-uniformly expansion. As a consequence, the definition of non-uniformly expanding map can be weakened. We also prove the existence of an absolutely continuous invariant measure for local diffeomorphisms, only assuming the existence of hyperbolic times for Lebesgue almost all point of the manifold.
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We prove that a partially hyperbolic attractor for a $C^1$ vector field with two dimensional center supports an SRB measure. In addition, we show that if the vector field is $C^2$, and the center bundle admits the sectional expanding condition w.r.t. any Gibbs $u$-state, then the attractor can only support finitely many SRB/physical measures whose basins cover Lebesgue almost all points of the topological basin. The proof of these results has to deal with the difficulties which do not occur in the case of diffeomorphisms.
Behavior of the SRB Entropy Functional in Families of Hyperbolic Attractors and Expanding Maps
本文综述均匀双曲系统族开分支上SRB熵泛函的研究现状，探讨其极值、梯度流及边界附近行为，涉及Anosov系统等多类双曲系统。
For any C1 expanding map f of the circle we study the equilibrium states for the potential -log |f'|. We formulate a C1 generalization of Pesin's Entropy Formula that holds for all the SRB measures if they exist, and for all the (necessarily existing) SRB-like measures. In the C1-generic case Pesin's Entropy Formula holds for a unique SRB measure which is not absolutely continuous with respect to Lebesgue. The result also stands in the non generic case for which no SRB measure exists.
We consider partially hyperbolic \( C^{1+} \) diffeomorphisms of compact Riemannian manifolds of arbitrary dimension which admit a partially hyperbolic tangent bundle decomposition \( E^s\oplus E^{cu} \). Assuming the existence of a set of positive Lebesgue measure on which \( f \) satisfies a weak nonuniform expansivity assumption in the centre~unstable direction, we prove that there exists at most a finite number of transitive attractors each of which supports an SRB measure. As part of our argument, we prove that each attractor admits a Gibbs-Markov-Young geometric structure with integrable…
SRB measures for mostly expanding partially hyperbolic diffeomorphisms via the variational approach
本研究采用变分方法，证明了主要扩张的部分双曲$\u2102^1$微分同胚及一类控制分裂情形下SRB测度的存在性，并在$\u2102^{1+α}$正则性下验证了盆覆盖性质。
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…
We prove that a $C^1$ hyperbolic map whose differential is regular enough has an SRB measure. The precise regularity condition is weaker than H{ö}lder and was mentionned by various authors through the developement of expanding and uniformly hyperbolic dynamics.
We consider classes of dynamical systems admitting Markov induced maps. Under general assumptions, which in particular guarantee the existence of SRB measures, we prove that the entropy of the SRB measure varies continuously with the dynamics. We apply our result to a vast class of non-uniformly expanding maps of a compact manifold and prove the continuity of the entropy of the SRB measure. In particular, we show that the SRB entropy of Viana maps varies continuously with the map.
We study the existence of SRB measures of C 2 diffeomorphisms for attractors whose bundles admit Holder continuous invariant (non-dominated) splittings. We prove the existence when one subbundle has the non-uniform expanding property on a set with positive Lebesgue measure and the other subbundle admits non-positive Lyapunov exponents on a total probability set.
We prove existence of equilibrium states with special properties for a class of distance expanding local homeomorphisms on compact metric spaces and continuous potentials. Moreover, we formulate a C$^1$ generalization of Pesin's Entropy Formula: all ergodic weak-SRB-like measures satisfy Pesin's Entropy Formula for $C^1$ non-uniformly expanding maps. We show that for weak-expanding maps such that $\Leb$-a.e $x$ has positive frequency of hyperbolic times, then all the necessarily existing ergodic weak-SRB-like measures satisfy Pesin's Entropy Formula and are equilibrium states for the potential…
We consider random perturbations of discrete-time dynamical systems. We give sufficient conditions for the stochastic stability of certain classes of maps, in a strong sense. This improves the main result in J. F. Alves, V. Araujo, Random perturbations of non-uniformly expanding maps, Asterisque 286 (2003), 25--62, where it was proved the convergence of the stationary measures of the random process to the SRB measure of the initial system in the weak* topology. Here, under slightly weaker assumptions on the random perturbations, we obtain a stronger version of stochastic stability: convergence…
We consider polygonal billiards with collisions contracting the reflection angle towards the normal to the boundary of the table. In previous work, we proved that such billiards has a finite number of ergodic SRB measures supported on hyperbolic generalized attractors. Here we study the relation of these measures with the ergodic absolutely continuous invariant probability measures (acips) of the slap map, the 1-dimensional map obtained from the billiard map when the angle of reflection is always equal to zero. Our main result states that for a generic polygon, if the reflection law has a Lips…
We provide a general framework to study differentiability of SRB measures for one dimensional non-uniformly expanding maps. Our technique is based on inducing the non-uniformly expanding system to a uniformly expanding one, and on showing how the linear response formula of the non-uniformly expanding system is inherited from the linear response formula of the induced one. We apply this general technique to interval maps with a neutral fixed point (Pomeau-Manneville maps) to prove differentiability of the corresponding SRB measure. Our work covers systems that admit a finite SRB measure and it …
SRB Measures for $C^{1+\mathrm{Dini}}$ Diffeomorphisms
研究\(C^{1 + \mathrm{Dini}}\)微分同胚，证明其SRB测度刻画，当支配丛有正Lyapunov指数时条件测度绝对连续性与部分Pesin熵公式等价，得到部分双曲大多扩张吸引子的SRB测度，还指出\(C^1\)范畴不成立。
Fourier Decay of SRB Measures for Skew Products
研究实解析斜积系统的自然不变测度，证明其SRB测度和最大熵测度要么吸引子为水平图，要么具有多项式傅里叶衰减，无需横截性假设。
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.
Entropy formula for C1 expanding maps
研究C1扩张动力系统上的伪物理测度，证明其满足Pesin熵公式，并举例说明圆周和二维紧致流形上的C1非C1+Hölder扩张映射及其伪物理测度。
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…
We consider open sets of maps in a manifold $M$ exhibiting non-uniform expanding behaviour in some domain $S\subset M$. Assuming that there is a forward invariant region containing $S$ where each map has a unique SRB measure, we prove that under general uniformity conditions, the SRB measure varies continuously in the $L^1$-norm with the map. As a main application we show that the open class of maps introduced in [V] fits to this situation, thus proving that the SRB measures constructed in [A] vary continuously with the map.
In this paper we address the existence and ergodicity of non-hyperbolic attracting sets for a certain class of smooth endomorphisms on the solid torus. Such systems allow a formulation as a skew product system defined by planar diffeomorphisms have contraction on average which forced by any expanding circle map. These attractors are invariant graphs of upper semicontinuous maps which support exactly one $SRB$ measure. In our approach, these skew product systems arising from iterated function systems generated by a finitely many weak contractive diffeomorphisms. Under some conditions including …
We show that a class of higher-dimensional hyperbolic endomorphisms admit absolutely continuous invariant probabilities whose density are regular. The maps we consider are given by $T(x,y) = (E (x), C(y) + f(x) )$, where $E$ is a linear expanding map of $\mathbb{T}$, $C$ is a linear contracting map of $\mathbb{R}^d$, $f$ is in $C^r(\mathbb{T}^u,\mathbb{R}^d)$ and $r \geq 2$. We prove that if $|(\det C)(\det E)| \|C^{-1}\|^{-2s}>1$ for some $s<r-(\frac{u+d}{2}+1)$ and $T$ satisfies a certain transversality condition, then the density of the SRB measure of $T$ is contained in the Sobolev space $…
We give both sufficient conditions and necessary conditions for the stochastic stability of non-uniformly expanding maps either with or without critical sets. We also show that the number of probability measures describing the statistical asymptotic behaviour of random orbits is bounded by the number of SRB measures if the noise level is small enough. As an application of these results we prove the stochastic stability of certain classes of non-uniformly expanding maps introduced in \cite{V} and \cite{ABV}.
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.
Physical measures for asymptotically sectional expanding flows in higher co-dimensions
本文研究了高余维渐进横向扩张流中物理测度的存在性，提出了适用于高余维数的吸引子构造方法，并证明了其具有完整遍历盆地的SRB测度。
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.
Attracting measures
本文提出“吸引测度”概念，将Axiom A吸引子上SRB测度的Lebesgue绝对连续测度前推弱收敛性质抽象为拓扑框架下的结果，强调该性质与物理测度同等重要。
We study the susceptibility function Psi(z) associated to the perturbation f_t=f+tX of a piecewise expanding interval map f. The analysis is based on a spectral description of transfer operators. It gives in particular sufficient conditions which guarantee that Psi(z) is holomorphic in a disc of larger than one. Although Psi(1) is the formal derivative of the SRB measure of f_t with respect to t, we present examples satisfying our conditions so that the SRB measure is not Lipschitz.*We propose a new version of Ruelle's conjectures.* In v2, we corrected a few minor mistakes and added Conjecture…
We prove that the statistical properties of random perturbations of a nonuniformly hyperbolic diffeomorphism are described by a finite number of stationary measures. We also give necessary and sufficient conditions for the stochastic stability of such dynamical systems. We show that a certain $C^2$-open class of nonuniformly hyperbolic diffeomorphisms introduced in [Alves, J; Bonatti, C. and Viana, V., SRB measures for partially hyperbolic systems with mostly expanding central direction, Invent. Math., 140 (2000), 351-398] are stochastically stable. Our setting encompasses that of partially hy…
We introduce the Markov extension, represented schematically as a tower, to the study of dynamical systems with holes. For tower maps with small holes, we prove the existence of conditionally invariant probability measures which are absolutely continuous with respect to Lebesgue measure (abbreviated a.c.c.i.m.). We develop restrictions on the Lebesgue measure of the holes and simple conditions on the dynamics of the tower which ensure existence and uniqueness in a class of Holder continuous densities. We then use these results to study the existence and properties of a.c.c.i.m. for expanding m…
On positive Lyapunov exponents and SRB measures for partially hyperbolic systems
研究紧致黎曼流形上部分双曲系统，通过可观测量测度计算沿$E^{cu}$的最小李雅普诺夫指数，证明特定条件下$f$沿$E^{cu}$非一致扩张，得出满足条件的$C^{1 + \alpha}$微分同胚有SRB测度的结论。
Let f be a self-map of a compact manifold M, admitting an global SRB measure μ. For a continuous test function ϕon M and a constant α>0, consider the set of the initial points for which the Birkhoff time averages of the function ϕdiffer from its μ--space average by at least α. As the measure μis an SRB one, the intersection of this set with the basin of attraction of μshould have zero Lebesgue measure. The \emph{special ergodic theorem}, whenever it holds, claims that, moreover, this intersection has the Hausdorff dimension less than the dimension of M. We prove that for Lipschitz maps, the sp…
We study quadratic skew-products with parameters driven over piecewise expanding and Markov interval maps with countable many inverse branches, a generalization of the class of maps introduced by Viana. In particular we construct a class of multidimensional non-uniformly expanding attractors that exhibit both critical points and discontinuities and prove existence and uniqueness of an SRB measure with stretched-exponential decay of correlations, stretched-exponential large deviations and satisfying some limit laws. Moreover, generically such maps admit the coexistence of a dense subset of poin…
Physical measures for mostly sectional expanding flows
针对C2向量场的部分双曲吸引集，证明其支撑遍历物理/SRB测度的充要条件为捕获区域在正勒贝格测度子集上存在非一致截面扩张，最多支撑有限个此类测度且为中心不稳定方向吉布斯态，推广微分同胚结果至连续时间系统并给出应用实例。
Intermittent--synchronization in non-weakly coupled piecewise linear expanding map lattice: a geometric-combinatorics method
本文针对非弱耦合双节点分段线性扩张映射格，提出几何-组合方法，给出绝对连续不变测度唯一性与间歇同步的充分必要条件。
Lyapunov spectrum rigidity and simultaneous linearization for random Anosov diffeomorphisms
本文研究了随机阿诺索夫微分同胚的李雅普诺夫谱刚性，证明了在特定条件下系统可同时线性化，并探讨了正矩阵随机游走的正李雅普诺夫指数刚性。
This paper discusses possible approaches to the escape rate in infinite lattices of weakly coupled maps with uniformly expanding repeller. It is proved that computed-via-volume rates of spatially periodic approximations grow linearly with the period size, suggesting normalized escape rate as the appropriate notion for the infinite system. The proof relies on symbolic dynamics and is based on the control of cumulative effects of perturbations within cylinder sets. A piecewise affine diffusive example is presented that exhibits monotonic decay of the escape rate with coupling intensity.
Let f be a polynomial or a rational function which has r summable critical points. We prove that there exists an r-dimensional manifold in an appropriate space containing f such that for every smooth curve in it through f, the ratio between parameter and dynamical derivatives along forward iterates of at least one these summable points tends to a non-zero number.
Finiteness of physical measures for diffeomorphisms with multi 1-D centers
该研究针对具有多1维中心的部分双曲$C^2$微分同胚，在Gibbs $u$-态双曲性的条件下，证明了其遍历物理（SRB）测度具有有限性，推进了部分双曲系统的测度理论研究。
We prove that any $C^{1+BV}$ degree $d \geq 2$ circle covering $h$ having all periodic orbits weakly expanding, is conjugate in the same smoothness class to a metrically expanding map. We use this to connect the space of parabolic external maps (coming from the theory of parabolic-like maps) to metrically expanding circle coverings.
Stochastic stability for weakly hyperbolic contracting Lorenz maps
研究满足指数1可和条件的随机扰动收缩洛伦兹映射的扩张性质，证明在强意义下随机稳定性，即平稳测度密度收敛到未扰动映射物理测度密度的L1范数。
We discuss sufficient conditions that guarantee the existence of asymptotic expansions for the Central Limit Theorem for weakly dependent random variables including observations arising from sufficiently chaotic dynamical systems like piece-wise expanding maps, and strongly ergodic Markov chains. We primarily use spectral techniques to obtain the results.
Non-uniform Berry-Esseen theorems for weakly dependent random variables
本文研究弱依赖序列的非均匀Berry-Esseen估计，涵盖均匀椭圆非齐马克诺夫链、随机时间变化的弱混沌动力系统、随机矩阵乘积及局部统计等类别。
We study a system of all-to-all weakly coupled uniformly expanding circle maps in the thermodynamic limit. The state of the system is described by a probability measure and its evolution is given by the action of a nonlinear operator, also called a self-consistent transfer operator. We prove that when the coupling is sufficiently small, the system has a unique stable state that satisfies a linear response formula when varying the coupling strength.
Conditional entropy realization and approximation by uniquely ergodic measures
本文引入条件极小-中间熵性质，证明在拓扑扩张映射、可数马尔可夫移位和非均匀符号系统中，满足给定熵和势积分的遍历极小测度在相应不变测度集中稠密。
Spectral Triples on Thermodynamic Formalism and Dixmier Trace Representations of Gibbs: theory and examples
本文推广了Perron-Frobenius-Ruelle定理与Sharp的谱三元组构造，研究了扩张映射的非交换期望，统一分析了流形、分形与Dyson模型，并给出了zeta函数非交换期望表示的实例。
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