Generic expanding maps without absolutely continuous invariant $σ$-finite measure

Absolutely Continuous Invariant Measure for Generalized Horseshoe Maps

In this paper, we study the SRB measures of generalized horseshoe map. We prove that under the conditions of transversality and fatness, the SRB measure is actually absolutely continuous with respect to the Lebesgue measure.

Non-existence of absolutely continuous invariant probabilities for exponential maps

We show that for entire maps of the form $z \mapsto λ\exp(z)$ such that the orbit of zero is bounded and such that Lebesgue almost every point is transitive, no absolutely continuous invariant probability measure can exist. This answers a long-standing open problem.

Wandering intervals and absolutely continuous invariant probability measures of interval maps

For piecewise $C^1$ interval maps possibly containing critical points and discontinuities with negative Schwarzian derivative, under two summability conditions on the growth of the derivative and recurrence along critical orbits, we prove the nonexistence of wandering intervals, the existence of absolutely continuous invariant measures, and the bounded backward contraction property. The proofs are based on the method of proving the existence of absolutely continuous invariant measures of unimodal map, developed by Nowicki and van Strien.

Multidimensional expanding maps with singularities: a pedestrian approach

I provide a proof of the existence of absolutely continuous invariant measures (and study their statistical properties) for multidimensional piecewise expanding systems with not necessarily bounded derivative or distortion. The proof uses basic properties of multidimensional BV functions.

Continuous homomorphisms on piecewise absolutely continuous maps of $\mathbb{S}^{1}$

Let $IET(\mathbb{S}^{1})$ be the group of interval exchange transformation of $\mathbb{S}^{1}$ and $\mathcal{AC}_{+}(\mathbb{S}^{1})$ be the group of absolutely continuous preserving orientation bijection with inverse absolutely continuous. We denote by $\mathcal{ACI}$ the group generated by $IET(\mathbb{S}^{1})$ and $\mathcal{AC}_{+}(\mathbb{S}^{1})$. Given a suitable distance on $\mathcal{ACI}$, we classify all continuous homomorphisms $ρ:\mathbb{R} \to \mathcal{ACI}$. More precisely, $ρ$ is conjugated to a continuous homomorphism $\hatρ:\mathbb{R} \to \mathcal{AC}_{+}(\uplus_{i}(\mathbb{S}^…

There are no $σ$-finite absolutely continuous invariant measures for multicritical circle maps

It is well-known that every multicritical circle map without periodic orbits admits a unique invariant Borel probability measure which is purely singular with respect to Lebesgue measure. Can such a map leave invariant an infinite, $σ$-finite invariant measure which is absolutely continuous with respect to Lebesgue measure? In this paper, using an old criterion due to Katznelson, we show that the answer to this question is no.

Strong statistical stability of non-uniformly expanding maps

We consider families of transformations in multidimensional Riemannian manifolds with non-uniformly expanding behavior. We give sufficient conditions for the continuous variation (in the $L^1$-norm) of the densities of absolutely continuous (with respect to the Lebesgue measure) invariant probability measures for those transformations.

Absolutely Continuous Invariant Measures of Piecewise Linear Lorenz Maps

Consider piecewise linear Lorenz maps on $[0, 1]$ of the following form \[ f_{a,b,c}(x)= {ll} ax+1-ac & x \in [0, c) b(x-c) & x \in (c, 1].\] We prove that $f_{a,b,c}$ admits an absolutely continuous invariant probability measure (acim) $μ$ with respect to the Lebesgue measure if and only if $f_{a,b,c}(0) \le f_{a,b,c}(1)$, i.e. $ac+(1-c)b \ge 1$. The acim is unique and ergodic unless $f_{a,b,c}$ is conjugate to a rational rotation. The equivalence between the acim and the Lebesgue measure is also fully investigated via the renormalization theory.

Absolutely Continuous Spectrum for Parabolic Flows/Maps

We provide an abstract framework for the study of certain spectral properties of parabolic systems; specifically, we determine under which general conditions to expect the presence of absolutely continuous spectral measures. We use these general conditions to derive results for spectral properties of time-changes of unipotent flows on homogeneous spaces of semisimple groups regarding absolutely continuous spectrum as well as maximal spectral type; the time-changes of the horocycle flow are special cases of this general category of flows. In addition we use the general conditions to derive spec…

长分支斯特姆单峰映射的绝对连续不变测度与Collet-Eckmann条件

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。

Higher-dimensional attractors with absolutely continuous invariant probability

Consider a dynamical system $T:\mathbb{T}\times \mathbb{R}^{d} \rightarrow \mathbb{T}\times \mathbb{R}^{d} $ 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 $\mathbb{R}^d$ and $f$ is in $C^2(\mathbb{T},\mathbb{R}^d)$. We prove that if $T$ is volume expanding and $u\geq d$, then for every $E$ there exists an open set $\mathcal{U}$ of pairs $(C,f)$ for which the corresponding dynamic $T$ admits an absolutely continuous invariant probability. A geometrical characteristic of transversality between self-intersections …

Yet another induction scheme for non-uniformly expanding transformations

We introduce a new induction scheme for non-uniformly expanding maps $f$ of compact Riemannian manifolds, proving that the existence of a Gibbs-Markov-Young structure is a necessary condition for $f$ to preserve an absolutely continuous probability with all its Lyapunov exponents positive.

Topological invariance of generic non-uniformly expanding multimodal maps

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.

SRB measures for weakly expanding maps

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.

Absolutely Periodic Billiard Orbits of Arbitrarily High Order

We show that for any natural number n, the set of domains containing absolutely periodic orbits of order n are dense in the set of bounded strictly convex domains with smooth boundary. The proof that such an orbit exists is an extension to billiard maps of the results of a paper by Gonchenko, Shilnikov, and Turaev, where it is proved that such maps are dense in Newhouse domains in regions of real-analytic area-preserving two-dimensional maps. Our result is a step toward disproving a conjecture that no absolutely periodic billiard orbits of infinite order exist in Euclidean billiards and is als…

C^r 一维映射的双曲绝对连续不变测度

Hyperbolic absolutely continuous invariant measures for C^r one-dimensional maps

本文证明，对 r>1 的 C^r 区间或圆周映射，若在正测度集上 Lyapunov 指数超过阈值，则存在关于 Lebesgue 测度绝对连续的双曲遍历不变测度，且其吸引域几乎处处覆盖该集合。

$W$-like maps with various instabilities of acim's

This paper generalizes the results of [13] and then provides an interesting example. We construct a family of $W$-like maps $\{W_a\}$ with a turning fixed point having slope $s_1$ on one side and $-s_2$ on the other. Each $W_a$ has an absolutely continuous invariant measure $μ_a$. Depending on whether $\frac{1}{s_1}+\frac{1}{s_2}$ is larger, equal or smaller than 1, we show that the limit of $μ_a$ is a singular measure, a combination of singular and absolutely continuous measure or an absolutely continuous measure, respectively. It is known that the invariant density of a single piecewise expa…

Non-uniform hyperbolicity and existence of absolutely continuous invariant measures

We prove that for certain partially hyperbolic skew-products, non-uniform hyperbolicity along the leaves implies existence of a finite number of ergodic absolutely continuous invariant probability measures which describe the asymptotics of almost every point. The main technical tool is an extension for sequences of maps of a result of de Melo and van Strien relating hyperbolicity to recurrence properties of orbits. As a consequence of our main result, we also obtain a partial extension of Keller's theorem guaranteeing the existence of absolutely continuous invariant measures for non-uniformly …

具有高阶临界集的非一致扩张映射的不变测度

Invariant Measures of Non-Uniformly Expanding Maps with Higher Order Critical Set

本文研究含高阶临界点的 Viana 型非一致扩张映射，证明其绝对连续不变测度的存在唯一性，并由此推出相关性超多项式衰减。

On the liftability of expanding stationary measures

We consider random perturbations of a topologically transitive local diffeomorphism of a Riemannian manifold. We show that if an absolutely continuous ergodic stationary measures is expanding (all Lyapunov exponents positive), then there is a random Gibbs-Markov-Young structure which can be used to lift that measure. We also prove that if the original map admits a finite number of expanding invariant measures then the stationary measures of a sufficiently small stochastic perturbation are expanding.

Stochastic Stability for Fiber Expanding Maps via a Perturbative Spectral Approach

We consider small perturbations of expanding maps induced by skew-product mappings whose base dynamics are not invertible necessarily. Adopting a previously developed perturbative spectral approach, we show stability of the densities of the unique absolutely continuous invariant probability measures for expanding maps under these perturbations, and upper bounds on the rate of exponential decay of fiber correlations associated to the measures as the noise level goes to zero.

On the existence of a conformal and an absolutely continuous invariant measure for transcendental entire maps

We identify a class of hyperbolic transcendental entire maps and we prove that some of its elements generate a class of potentials for which exhibit a conformal and invariant probability Gibbs measure. The methods and techniques from the thermodynamic formalism can be extended to this class of potentials. To complement this study we highlight that the dynamics of such a map on some subset of the Julia set is conjugated to the shift map over a code space with countable alphabet and the euclidean metric on the complex plane induces a metric on the symbolic space which is not compatible with the …

Typical points and families of expanding interval mappings

We study parametrised families of piecewise expanding interval mappings $T_a \colon [0,1] \to [0,1]$ with absolutely continuous invariant measures $μ_a$ and give sufficient conditions for a point $X(a)$ to be typical with respect to $(T_a, μ_a)$ for almost all parameters $a$. This is similar to a result by D. Schnellmann, but with different assumptions.

Rationality of the zeta function for Ruelle-expanding maps

We will prove that the zeta function for Ruelle-expanding maps is rational.

The almost sure invariance principle for unbounded functions of expanding maps

We consider two classes of piecewise expanding maps $T$ of $[0,1]$: a class of uniformly expanding maps for which the Perron-Frobenius operator has a spectral gap in the space of bounded variation functions, and a class of expanding maps with a neutral fixed point at zero. In both cases, we give a large class of unbounded functions $f$ for which the partial sums of $f\circ T^i$ satisfy an almost sure invariance principle. This class contains piecewise monotonic functions (with a finite number of branches) such that: - For uniformly expanding maps, they are square integrable with respect to the…

Ergodicity of polygonal slap maps

Polygonal slap maps are piecewise affine expanding maps of the interval obtained by projecting the sides of a polygon along their normals onto the perimeter of the polygon. These maps arise in the study of polygonal billiards with non-specular reflections laws. We study the absolutely continuous invariant probabilities of the slap maps for several polygons, including regular polygons and triangles. We also present a general method for constructing polygons with slap maps having more than one ergodic absolutely continuous invariant probability.

Some Remarks on Expanding Maps

In this work we treat a famous topic in Ergodic Theory and Dynamical Systems: uniformly expanding maps. We relate regularity of expanding maps and conjugacies with Lyapunov exponents, metric and topological entropies for expanding maps of the circle. In Theorem C, we present a result of rigidity of expanding maps on the torus $\mathbb{T}^2,$ after we naturally generalize it in higher dimensions. Here, we present techniques involving ordinary differential equations to study rigidity problems of expanding maps. Along of this work we provide another proof of a Shub-Sullivan's Theorem, concerning …

Smooth rigidity for very non-algebraic expanding maps

We show that the space of expanding maps contains an open and dense set where smooth conjugacy classes of expanding maps are determined by the values of the Jacobians of return maps at periodic points.

Iterated function systems of affine expanding and contracting maps on the unit interval

We analyze the two-point motions of iterated function systems on the unit interval generated by expanding and contracting affine maps, where the expansion and contraction rates are determined by a pair $(M,N)$ of integers. This dynamics depends on the Lyapunov exponent. For a negative Lyapunov exponent we establish synchronization, meaning convergence of orbits with different initial points. For a vanishing Lyapunov exponent we establish intermittency, where orbits are close for a set of iterates of full density, but are intermittently apart. For a positive Lyapunov exponent we show the existe…

On stochastic stability of non-uniformly expanding interval maps

We study the expanding properties of random perturbations of regular interval maps satisfying the summability condition of exponent one. Under very general conditions on the interval maps and perturbation types, we prove strong stochastic stability.

遍历保守扩张圆映射的无界扭曲的C1-通有性

C1-genericity of unbounded distortion for ergodic conservative expanding circle maps

本文证明了在圆的遍历Lebesgue保测C1扩张映射空间中，无界扭曲具有C1-通有性。

Flexibility of exponents for expanding maps on a circle

We consider a smooth expanding map g on the circle of degree 2. It is known that the Lyapunov exponent of g with respect to the unique invariant measure that is absolutely continuous with respect to the Lebesgue measure is positive and less than or equal to log 2 which, in addition, is less than or equal to the Lyapunov exponent of g with respect to the measure of maximal entropy. Moreover, the equalities only occur simultaneously. We show that these are the only restrictions on the Lyapunov exponents considered above for smooth expanding maps of degree 2.

Lower bounds for the Ruelle spectrum of analytic expanding circle maps

We prove that there exists a dense set of analytic expanding maps of the circle for which the Ruelle eigenvalues enjoy exponential lower bounds. The proof combines potential theoretic techniques and explicit calculations for the spectrum of expanding Blaschke products.

Statistical stability for multidimensional piecewise expanding maps

We present sufficient conditions for the (strong) statistical stability of some classes of multidimensional piecewise expanding maps. As a consequence we get that a certain natural two-dimensional extension of the classical one-dimensional family of tent maps is statistically stable.

A footnote on Expanding maps

I introduce Banach spaces on which it is possible to precisely characterize the spectrum of the transfer operator associated to a piecewise expanding map with Hölder weight.

Inducing schemes for multi-dimensional piecewise expanding maps

We construct inducing schemes for general multi-dimensional piecewise expanding maps where the base transformation is Gibbs-Markov and the return times have exponential tails. Such structures are a crucial tool in proving statistical properties of dynamical systems with some hyperbolicity.

Dimension theoretical study on skew product maps with coupled-expanding property

We discuss on some families of skew product maps on a square. For a kind of skew product maps with coupled-expanding property, we estimate Hausdorff dimension of its attractor. And we prove that there exists an ergodic measure with full Hausdorff dimension for this kind of skew product maps.

Historic Behaviour for Random Expanding Maps on the Circle

Takens constructed a residual subset of the state space consisting of initial points with historic behaviour for expanding maps on the circle. We prove that this statistical property of expanding maps on the circle is preserved under small random perturbations. The proof is given by establishing a random Markov partition, which follows from a random version of Shub's Theorem on topological conjugacy with the folding maps.

Accurate bounds on Lyapunov exponents for expanding maps of the interval

In this short note we describe a simple but remarkably effective method for rigorously estimating Lyapunov exponents for expanding maps of the interval. We illustrate the applicability of this method with some standard examples.

Regularity of coboundaries for non uniformly expanding Markov maps

We prove that solutions $u$ of the equation $f=u-u\circ T$ are automatically Holder continuous when $f$ is Holder continuous, and $T$ is non uniformly expanding and Markov. This result applies in particular to Young towers and to intermittent maps.

Observable Lyapunov irregular sets for planar piecewise expanding maps

For any integer $r$ with $1\leq r<\infty$, we present a one-parameter family $F_σ$ $(0<σ<1)$ of 2-dimensional piecewise $\mathcal C^r$ expanding maps such that each $F_σ$ has an observable (i.e. Lebesgue positive) Lyapunov irregular set. These maps are obtained by modifying the piecewise expanding map given in Tsujii (2000). In strong contrast to it, we also show that any Lyapunov irregular set of any 2-dimensional piecewise real analytic expanding map is not observable. This is based on the spectral analysis of piecewise expanding maps in Buzzi (2000).

Heat kernel and the rate of mixing for compact extensions of expanding maps

We consider skew extensions of expanding maps by compact Lie groups. For a class of natural invariant measures, we prove an explicit lower bound on the rate of (exponential) mixing involving topological pressure. Proof uses representation theory and some elliptic pde estimates.

Almost sure invariance principle for random piecewise expanding maps

We prove a fiberwise almost sure invariance principle for random piecewise expanding transformations in one and higher dimensions using recent developments on martingale techniques.

$J$-Stability of immediately expanding rational maps in non-Archimedean dynamical systems

The aim of this paper is to show $J$-stability of immediately expanding rational maps over an algebraically closed, complete, and non-Archimedean field, which is an analogue of R. Mane, P. Sad, and D. Sullivan's theorem of $J$-stability in complex dynamical systems.

Repellers for non-uniformly expanding maps with singular or critical points

Given an ergodic measure with positive entropy and only positive Lyapunov exponents, its dynamical quantifiers can be approximated by means of quantifiers of some family of uniformly expanding repellers. Here non-uniformly expanding maps are studied that are $C^{1+β}$ smooth outside a set of possibly critical or singular points.

Periodic points of Ruelle-expanding maps

We prove that, for a Ruelle-expanding map, the zeta function is rational and the topological entropy is equal to the exponential growth rate of the periodic points.

Analytic skew products of quadratic polynomials over circle expanding maps

We prove that a Viana map with an arbitrarily non-constant real analytic coupling function admits two positive Lyapunov exponents almost everywhere.

Equilibrium States for Non-uniformly expanding maps

We construct equilibrium states, including measures of maximal entropy, for a large (open) class of non-uniformly expanding maps on compact manifolds. Moreover, we study uniqueness of these equilibrium states, as well as some of their ergodic properties.

beta-变换的随机动力系统的绝对连续不变测度

Absolutely continuous invariant measures for random dynamical systems of beta-transformations

本文研究了beta-变换的随机动力系统，在平均扩张等条件下给出了唯一平稳测度密度函数的显式公式，并证明了有限映射情形下密度函数的解析性。

扩张区间映射的有效内在遍历性

Effective intrinsic ergodicity for expanding interval maps

将Einsidler等人关于内在遍历性的结果推广到扩张映射（包括β-变换）的简单情形，证明了有效内在遍历性。

Differentiating the absolutely continuous invariant measure of an interval map f with respect to f

Let the map $f:[-1,1]\to[-1,1]$ have a.c.i.m. $ρ$ (absolutely continuous $f$-invariant measure with respect to Lebesgue). Let $δρ$ be the change of $ρ$ corresponding to a perturbation $X=δf\circ f^{-1}$ of $f$. Formally we have, for differentiable $A$, $$ δρ(A)=\sum_{n=0}^\infty\intρ(dx) X(x){d\over dx}A(f^nx) $$ but this expression does not converge in general. For $f$ real-analytic and Markovian in the sense of covering $(-1,1)$ $m$ times, and assuming an {\it analytic expanding} condition, we show that $$λ\mapstoΨ(λ)=\sum_{n=0}^\inftyλ^n \intρ(dx) X(x){d\over dx}A(f^nx) $$ is meromorphic in…

On stochastic stability of expanding circle maps with neutral fixed points

It is well-known that the Manneville-Pomeau map with a parabolic fixed point of the form $x\mapsto x+x^{1+α} \mod 1$ is stochastically stable for $α\ge 1$ and the limiting measure is the Dirac measure at the fixed point. In this paper we show that if $α\in (0,1)$ then it is also stochastically stable. Indeed, the stationary measure of the random map converges strongly to the absolutely continuous invariant measure for the deterministic system as the noise tends to zero.

Nonuniformly expanding 1d maps with logarithmic singularities

For a certain parametrized family of maps on the circle with critical points and logarithmic singularities where derivatives blow up to infinity, we construct a positive measure set of parameters corresponding to maps which exhibit nonuniformly expanding behavior. This implies the existence of "chaotic" dynamics in dissipative homoclinic tangles in periodically perturbed differential equations.

Large deviation principle for Benedicks-Carleson quadratic maps

Since the pioneering works of Jakobson and Benedicks & Carleson and others, it has been known that a positive measure set of quadratic maps admit invariant probability measures absolutely continuous with respect to Lebesgue. These measures allow one to statistically predict the asymptotic fate of Lebesgue almost every initial condition. Estimating fluctuations of empirical distributions before they settle to equilibrium requires a fairly good control over large parts of the phase space. We use the sub-exponential slow recurrence condition of Benedicks & Carleson to build induced Markov maps of…