We consider a partially hyperbolic set $K$ on a Riemannian manifold $M$ whose tangent space splits as $T_K M=E^{cu}\oplus E^{s}$, for which the centre-unstable direction $E^{cu}$ expands non-uniformly on some local unstable disk. We show that under these assumptions $f$ induces a Gibbs-Markov structure. Moreover, the decay of the return time function can be controlled in terms of the time typical points need to achieve some uniform expanding behavior in the centre-unstable direction. As an application of the main result we obtain certain rates for decay of correlations, large deviations, an almost sure invariance principle and the validity of the Central Limit Theorem.
In this paper we study the decay of correlations for Gibbs measures associated to codimension one Axiom A attractors for flows. We prove that a codimension one Axiom A attractors whose strong stable foliation is $C^{1+α}$ either have exponential decay of correlations with respect to all Gibbs measures associated to Hölder continuous potentials or their stable and unstable bundles are jointly integrable. As a consequence, there exist $C^1$-open sets of $C^3$-vector fields generating Axiom A flows having attractors so that: (i) mix exponentially with respect to equilibrium states associated with…
In this article we will describe a new construction for Gibbs measures for hyperbolic attractors generalizing the original construction of Sinai, Bowen and Ruelle of SRB measures. The classical construction of the SRB measure is based on pushing forward the normalized volume on a piece of unstable manifold. By modifying the density at each step appropriately we show that the resulting measure is a prescribed Gibbs measure. This contrasts with, and complements, the construction of Climenhaga-Pesin-Zelerowicz who replace the volume on the unstable manifold by a fixed reference measure. Moreover,…
We consider piecewise expanding maps of the interval with finitely many branches of monotonicity and show that they are generically combinatorially stable, i.e., the number of ergodic attractors and their corresponding mixing periods do not change under small perturbations of the map. Our methods provide a topological description of the attractor and, in particular, give an elementary proof of the density of periodic orbits.
We prove that given any closed $n$-manifold $M^n$, $n\geq 4$, there is an A-flow $f^t$ on $M^n$ such that the non-wandering set $NW(f^t)$ consists of 2-dimensional expanding attractor (the both, orientable and non-orientable) and trivial basic sets. For 3-manifolds, we prove that given any closed 3-manifold $M^3$, there is an A-flow $f^t$ on $M^3$ such that the non-wandering set $NW(f^t)$ consists of a non-orientable 2-dimensional expanding attractor and trivial basic sets. Moreover, there is a nonsingular A-flow $f^t$ on a 3-sphere such that the non-wandering set $NW(f^t)$ consists of an orie…
We study partially hyperbolic sets $K$ on a Riemannian manifold $M$ whose tangent space splits as $T_K M=E^{cu}\oplus E^{s}$, for which the center-unstable direction $E^{cu}$ is non-uniformly expanding on some local unstable disk. We prove that the (stretched) exponential decay of recurrence times for an induced scheme can be deduced under the assumption of (stretched) exponential decay of the time that typical points need to achieve some uniform expanding in the center-unstable direction. This extends a result by Alves and Pinheiro to the (stretched) exponential case. As an application of our…
We consider the local dimension spectrum of a weak Gibbs measure on a C^1 non-uniformly hyperbolic system of Manneville- Pomeau type. We present the spectrum in three ways: using invariant measures, uniformly hyperbolic ergodic measures and equilibrium states. We are also proving analyticity of the spectrum under additional assumptions. All three presentations are well known for smooth uniformly hyperbolic systems.
In this paper we define distance expanding random dynamical systems. We develop the appropriate thermodynamic formalism of such systems. We obtain in particular the existence and uniqueness of invariant Gibbs states, the appropriate pressure function and exponentially fast convergence of iterates of Perron-Frobenius operators resulting, in particular, in an exponential decay of correlations. We also obtain the formula for the derivative of the expected value of the pressure function. Next, we define and investigate in detail conformal random expanding repellers. Applying the developed machiner…
We establish bounds for the measure of deviation sets associated to continuous observables with respect to not necessarily invariant weak Gibbs measures. Under some mild assumptions, we obtain upper and lower bounds for the measure of deviation sets of some non-uniformly expanding maps, including quadratic maps and robust multidimensional non-uniformly expanding local diffeomorphisms. For that purpose, a measure theoretical weak form of specification is introduced and proved to hold for the robust classes of multidimensional nonuniformly expanding local diffeomorphisms and Viana maps.
Let $T : Λ\to Λ$ be an expanding map on a Cantor set. For each suitably normalized Hölder continuous potential, we construct a spectral triple from which one may recover the associated Gibbs measure as a noncommutative measure.
The path closure of figure eight knot complement space, $N_0$, supports a natural DA (derived from Anosov) expanding attractor. Using this attractor, Franks-Williams constructed the first example of non-transitive Anosov flow on the manifold $M_0$ obtained by gluing two copies of $N_0$ through identity map along their boundaries, named by Franks-Williams manifold. In this paper, our main goal is to classify expanding attractors on $N_0$ and non-transitive Anosov flows on $M_0$. We prove that, up to orbit-equivalence, the DA expanding attractor is the unique expanding attractor supported by $N_…
We prove a result motivated by Williams's classification of expanding attractors and the Franks-Newhouse Theorem on codimension-1 Anosov diffeomorphisms: If a mixing hyperbolic attractor has 1-dimensional unstable manifolds then it is either is expanding or is homeomorphic to a compact abelian group (a toral solenoid); in the latter case the dynamics is conjugate to a group automorphism. As a corollary we obtain a classification of all 2-dimensional basic sets in 3-manifolds. Furthermore we classify all hyperbolic attractors in 3-manifolds in terms of the classically studied examples, answerin…
We consider a class of non-conformal expanding maps on the $d$-dimensional torus. For an equilibrium measure of an Hölder potential, we prove an analogue of the Central Limit Theorem for the fluctuations of the logarithm of the measure of balls as the radius goes to zero. An unexpected consequence is that when the measure is not absolutely continuous, then half of the balls of radius $\eps$ have a measure smaller than $\eps^δ$ and half of them have a measure larger than $\eps^δ$, where $δ$ is the Hausdorff dimension of the measure. We first show that the problem is equivalent to the study of t…
Chaotic attractors in the two-dimensional border-collision normal form (a piecewise-linear map) can persist throughout open regions of parameter space. Such robust chaos has been established rigorously in some parameter regimes. Here we provide formal results for robust chaos in the original parameter regime of [S. Banerjee, J.A. Yorke, C. Grebogi, Robust Chaos, Phys. Rev. Lett. 80(14):3049--3052, 1998]. We first construct a trapping region in phase space to prove the existence of a topological attractor. We then construct an invariant expanding cone in tangent space to prove that tangent vect…
If there exists a diffeomorphism $f$ on a closed, orientable $n$-manifold $M$ such that the non-wandering set $Ω(f)$ consists of finitely many orientable $(\pm)$ attractors derived from expanding maps, then $M$ must be a rational homology sphere; moreover all those attractors are of topological dimension $n-2$. Expanding maps are expanding on (co)homologies.
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.
In this work, we study ergodic properties of certain partially hyperbolic attractors whose central direction has a neutral behavior, the main feature is a condition of transversality between unstable leaves when projected by the stable holonomy. We prove that partial hyperbolic attractors satisfying conditions of transversality between unstable leaves via the stable holonomy, neutrality in the central direction and regularity of the stable foliation admits a finite number of physical measures, coinciding with the ergodic u-Gibbs States, whose union of the basins has full Lebesgue measure. More…
An {\em attractor} is a transitive set of a flow to which all positive orbit close to it converges. An attractor is {\em singular-hyperbolic} if it has singularities (all hyperbolic) and is partially hyperbolic with volume expanding central direction \cite{MPP}. The geometric Lorenz attractor \cite{GW} is an example of a singular-hyperbolic attractor with topological dimension $\geq 2$. We shall prove that {\em all} singular-hyperbolic attractors on compact 3-manifolds have topological dimension $\geq 2$. The proof uses the methods in \cite{MP}.
A topological classification of many classes of dynamical systems with regular dynamics in low dimensions is often reduced to combinatorial invariants. In dimension 3 combinatorial invariants are proved to be insufficient even for simplest Morse-Smale diffeomorphisms. The complete topological invariant for the systems with a single saddle point on the 3-sphere is the embedding of the homotopy non-trivial knot into the manifold $\mathbb S^2\times\mathbb S^1$. If a diffeomorphism has several saddle points their unstable separatrices form arcs frames in the basin of the sink and circles frame in …
It is well known that topological classification of dynamical systems with hyperbolic dynamics is significantly defined by dynamics on nonwandering set. F. Przytycki generalized axiom $A$ for smooth endomorphisms that was previously introduced by S. Smale for diffeomorphisms and proved spectral decomposition theorem which claims that nonwandering set of an $A$-endomorphism is a union of a finite number basic sets. In present paper the criterion for a basic sets of an $A$-endomorphism to be an attractor is given. Moreover, dynamics on basic sets of codimension one is studied. It is shown, that …
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…
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.
In this paper, we study the limit measures of the empirical measures of Lebesgue almost every point in the basin of a partially hyperbolic attractor. They are strongly related to a notion named Gibbs u-state, which can be defined in a large class of diffeomorphisms with less regularity and which is the same as Pesin-Sinai's notion for partially hyperbolic attractors of $C^{1+α}$ diffeomorphisms. In particular, we prove that for partially hyperbolic $C^{1+α}$ diffeomorphisms with one-dimensional center, and for Lebesgue almost every point: (1) the center Lyapunov exponent is well defined, but (…
We prove the stochastic stability of an open class of partially hyperbolic diffeomorphisms, each of which admits two centers $E^c_1$ and $E^c_2$ such that any Gibbs $u$-state admits only positive (resp. negative) Lyapunov exponents along $E^c_1$ (resp. $E^c_2$).
A {\em singular hyperbolic attractor} for flows is a partially hyperbolic attractor with singularities (hyperbolic ones) and volume expanding central direction \cite{mpp1}. The geometric Lorenz attractor \cite{gw} is an example of a singular hyperbolic attractor. In this paper we study the perturbations of singular hyperbolic attractors for three-dimensional flows. It is proved that any attractor obtained from such perturbations contains a singularity. So, there is an upper bound for the number of attractors obtained from such perturbations. Furthermore, every three-dimensional flow $C^r$ clos…
We establish two results under which the topology of a hyperbolic set constrains ambient dynamics. First if a set is a compact, transitive, expanding hyperbolic attractor of codimension 1 for some diffeomorphism, then it is a union of transitive, expanding attractors (or contracting repellers) of codimension 1 for any diffeomorphism such that it is hyperbolic. Secondly, if a set is a nonwandering, locally maximal, compact hyperbolic set for a surface diffeomorphism, then it is locally maximal for any diffeomorphism for which it is hyperbolic.
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 …
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.
We prove that a singular-hyperbolic attractor of a 3-dimensional flow is chaotic, in two strong different senses. Firstly, the flow is expansive: if two points remain close for all times, possibly with time reparametrization, then their orbits coincide. Secondly, there exists a physical (or Sinai-Ruelle-Bowen) measure supported on the attractor whose ergodic basin covers a full Lebesgue (volume) measure subset of the topological basin of attraction. Moreover this measure has absolutely continuous conditional measures along the center-unstable direction, is a $u$-Gibbs state and an equilibrium …
Skew product systems with monotone one-dimensional fibre maps driven by piecewise expanding Markov interval maps may show the phenomenon of intermingled basins. To quantify the degree of intermingledness the uncertainty exponent and the stability index were suggested by various authors and characterized (partially). Here we present an approach to evaluate/estimate these two quantities rigorously using thermodynamic formalism for the driving Markov map.
We study how physical measures vary with the underlying dynamics in the open class of $C^r$, $r>1$, strong partially hyperbolic diffeomorphisms for which the central Lyapunov exponents of every Gibbs $u$-state is positive. If transitive, such a diffeomorphism has a unique physical measure that persists and varies continuously with the dynamics. A main ingredient in the proof is a new Pliss-like Lemma which, under the right circumstances, yields frequency of hyperbolic times close to one. Another novelty is the introduction of a new characterization of Gibbs $cu$-states. Both of these may be of…
We prove a new result in the area of hitting time statistics. Currently, there is a lot of papers showing that the first entry times into cylinders or balls are often faster than the Birkhoff's Ergodic Theorem would suggest. We provide an opposite counterpart to these results by proving that the hitting times into shrinking balls are also often much larger than these theorems would suggest, by showing that for many dynamical systems $$ \displaystyle \limsup_{r\to 0} τ_{B(y,r)}(x)μ(B(y,r))=+\infty, $$ for an appropriately large, at least of full measure, set of points $y$ and $x$. We first do t…
In some maps the existence of an attractor with a positive Lyapunov exponent can be proved by constructing a trapping region in phase space and an invariant expanding cone in tangent space. If this approach fails it may be possible to adapt the strategy by considering an induced map (a first return map for a well-chosen subset of phase space). In this paper we show that such a construction can be applied to the two-dimensional border-collision normal form (a continuous piecewise-linear map) if a certain set of conditions are satisfied and develop an algorithm for checking these conditions. The…
We construct self-adjoint Laplacians and symmetric Markov semigroups on partially hyperbolic attractors and on hyperbolic attractors with singularities, endowed with Gibbs u-measures. If the measure has full support, we can also guarantee the existence of an associated symmetric Hunt diffusion process. In the special case of partially hyperbolic diffeomorphisms induced by geodesic flows on manifolds of negative sectional curvature the Laplacians we consider are self-adjoint extensions of well-known classical leafwise Laplacians.
We consider strange attractors of two dimensional generalized map with one nonlinearity such that Lozi, Hénon and Belykh maps are particular cases of it. We describe technique of invariant expanding and contracting cones creation for study of hyperbolic properties. Theorems of singular hyperbolic attractors for new modifications of Lozi, Hénon and Belykh-type maps are presented.
An attractor $Λ$ for a 3-vector field $X$ is singular-hyperbolic if all its singularities are hyperbolic and it is partially hyperbolic with volume expanding central direction. We prove that $C^{1+α}$ singular-hyperbolic attractors, for some $α>0$, always have zero volume, thus extending an analogous result for uniformly hyperbolic attractors. The same result holds for a class of higher dimensional singular attractors. Moreover, we prove that if $Λ$ is a singular-hyperbolic attractor for $X$ then either it has zero volume or $X$ is an Anosov flow. We also present examples of $C^1$ singular-hyp…
A recent problem in dynamics is to determinate whether an attractor $Λ$ of a $C^r$ flow $X$ is $C^r$ robust transitive or not. By {\em attractor} we mean a transitive set to which all positive orbits close to it converge. An attractor is $C^r$ robust transitive (or {\em $C^r$ robust} for short) if it exhibits a neighborhood $U$ such that the set $\cap_{t>0}Y_t(U)$ is transitive for every flow $Y$ $C^r$ close to $X$. We give sufficient conditions for robustness of attractors based on the following definitions. An attractor is {\em singular-hyperbolic} if it has singularities (all hyperbolic) an…
In this paper we consider the realization of DE attractors by self-diffeomorphisms of manifolds. For any expanding self-map $ϕ:M\to M$ of a connected, closed $p$-dimensional manifold $M$, one can always realize a $(p,q)$-type attractor derived from $ϕ$ by a compactly-supported self-diffeomorphsm of $\RR^{p+q}$, as long as $q\geq p+1$. Thus lower codimensional realizations are more interesting, related to the knotting problem below the stable range. We show that for any expanding self-map $ϕ$ of a standard smooth $p$-dimensional torus $T^p$, there is compactly-supported self-diffeomorphism of $…
Given a lamination in a compact space and a laminated vector field $X$ which is hyperbolic when restricted to the leaves of the lamination, we distinguish a class of $X$-invariant probabilities that describe the behaviour of almost every $X$-orbit in every leaf, that we call u-Gibbs states. We apply this to the case of foliations in compact manifolds having leaves with negative curvature, using the foliated hyperbolic vector field on the unit tangent bundle to the foliation generating the leaf geodesics. When the Lyapunov exponents of such an ergodic u-Gibbs states are negative, it is an SRB-m…
In the case of smooth non-invertible maps which are hyperbolic on folded basic sets $Λ$, we give approximations for the Gibbs states (equilibrium measures) of arbitrary Hölder potentials, with the help of weighted sums of atomic measures on preimage sets of high order. Our endomorphism may have also stable directions on $Λ$, thus it is non-expanding in general. Folding of the phase space means that we do not have a foliation structure for the local unstable manifolds (they depend on the whole past and may intersect each other both inside and outside $Λ$), and moreover the number of preimages r…
We give a review of results on superpolynomial decay of correlations, and polynomial decay of correlations for nonuniformly expanding semiflows and nonuniformly hyperbolic flows. A self-contained proof is given for semiflows. Results for flows are stated without proof (the proofs are contained in separate joint work with Balint and Butterley). Applications include intermittent solenoidal flows, suspended Henon attractors, Lorenz attractors, and various Lorentz gas models including the infinite horizon Lorentz gas.
We consider one parameter families of vector fields introduced by Rovella, obtained through modifying the eigenvalues of the geometric Lorenz attractor, replacing the expanding condition on the eigenvalues of the singularity by a contracting one. We show that there is no statistical stability within the set of parameters for which there is a physical measure supported on the attractor. This is achieved obtaining a similar conclusion at the level of the corresponding one-dimensional contracting Lorenz maps.
In this paper, we provide a technique result on the existence of Gibbs cu-states for diffeomorphisms with dominated splittings. More precisely, for given $C^2$ diffeomorphim $f$ with dominated splitting $T_ΛM = E\oplus F$ on an attractor $Λ$, by considering some suitable random perturbation of $f$, we show that for any zero noise limit of ergodic stationary measures, if it has positive integrable Lyapunov exponents along invariant sub-bundle $E$, then its ergodic components contain Gibbs cu-states associated to $E$. With this technique, we show the existence of SRB measures and physical measur…
We study the Fourier transforms $\widehatμ(ξ)$ of non-atomic Gibbs measures $μ$ for uniformly expanding maps $T$ of bounded distortions on $[0,1]$ or Cantor sets with strong separation. When $T$ is totally non-linear, then $\widehatμ(ξ) \to 0$ at a polynomial rate as $|ξ| \to \infty$.
We construct self-adjoint Laplacians and symmetric Markov semigroups on hyperbolic attractors, endowed with Gibbs $u$-measures. If the measure has full support, we can also conclude the existence of an associated symmetric diffusion process. In the special case of partially hyperbolic diffeomorphisms induced by geodesic flows on negatively curved manifolds the Laplacians we consider are self-adjoint extensions of well-known classical leafwise Laplacians. We observe a quasi-invariance property of energy densities in the $u$-conformal case and the existence of nonconstant functions of zero energ…
We show the existence and uniqueness of the maximal entropy probability measure for partially hyperbolic diffeomorphisms which are semi-conjugate to nonuniformly expanding maps. Using the theory of projective metric on cones we then prove exponential decay of correlations for Hölder continuous observables and the central limit theorem for the maximal entropy probability measure. Moreover, for systems derived from solenoid we also prove the statistical stability for the maximal entropy probability measure. Finally, we use such techniques to obtain similar results in a context containing partial…
Let $ T:[0,1]\to[0,1] $ be an expanding Markov map with a finite partition. Let $ μ_ϕ$ be the invariant Gibbs measure associated with a Hölder continuous potential $ ϕ$. In this paper, we investigate the size of the uniform approximation set \[\mathcal U^κ(x):=\{y\in[0,1]:\forall N\gg1,~\exists n\le N, \text{ such that }|T^nx-y|<N^{-κ}\},\] where $ κ>0 $ and $ x\in[0,1] $. The critical value of $ κ$ such that $ \textrm{dim}_{\textrm H}\mathcal U^κ(x)=1 $ for $ μ_ϕ$-a.e.$ \, x $ is proven to be $ 1/α_{\max} $, where $ α_{\max}=-\int ϕ\,dμ_{\max}/\int\log|T'|\,dμ_{\max} $ and $ μ_{\max} $ is the…
In this paper, we explore the three-dimensional chaotic set near a homoclinic cycle to a hyperbolic bifocus at which the vector field has negative divergence. If the invariant manifolds of the bifocus satisfy a non-degeneracy condition, a sequence of hyperbolic suspended horseshoes arises near the cycle, with one expanding and two contracting directions. We extend previous results on the field and we show that the first return map to a given cross section may be approximated by a map exhibiting heteroclinic tangencies associated to two periodic orbits. When the cycle is broken, under an additi…
