We prove the following dichotomy for vector fields in a C1-residual subset of volume-preserving flows: for Lebesgue almost every point all Lyapunov exponents equal to zero or its orbit has a dominated splitting. As a consequence if we have a vector field in this residual that cannot be C1-approximated by a vector field having elliptic periodic orbits, then, there exists a full measure set such that every orbit of this set admits a dominated splitting for the linear Poincare flow. Moreover, we prove that a volume-preserving and C1-stably ergodic flow can be C1-approximated by another volume-preserving flow which is non-uniformly hyperbolic.
The conservative sequence of a set $A$ under a transformation $T$ is the set of all $n \in \mathbb{Z}$ such that $T^n A \cap A \not = \varnothing$. By studying these sequences, we prove that given any countable collection of nonsingular transformations with no finite invariant measure $\{T_i\}$, there exists a rank-one transformation $S$ such that $T_i \times S$ is not ergodic for all $i$. Moreover, $S$ can be chosen to be rigid or have infinite ergodic index. We establish similar results for $\mathbb{Z}^d$ actions and flows. Then, we find sufficient conditions on rank-one transformations $T$ …
We construct a class of rank-one infinite measure-preserving transformations such that for each transformation $T$ in the class, the cartesian product $T\times T$ of the transformation with itself is ergodic, but the product $T\times T^{-1}$ of the transformation with its inverse is not ergodic. We also prove that the product of any rank-one transformation with its inverse is conservative, while there are infinite measure-preserving conservative ergodic Markov shifts whose product with their inverse is not conservative.
We show stable ergodicity of a class of conservative diffeomorphisms which do not have any hyperbolic invariant subbundle. Moreover the uniqueness of SRB measures for non-conservative $C^1$ perturbations of such diffeomorphisms. This class contains strictly non-partially hyperbolic robustly transitive diffeomorphisms by Bonatti-Viana and so we answer their question about the stable ergodicity of such systems.
We show that for a $C^1$-open and $C^{r}$-dense subset of the set of ergodic iterated function systems of conservative diffeomorphisms of a finite-volume manifold of dimension $d\geq 2$, the extremal Lyapunov exponents do not vanish. In particular, the set of non-uniform hyperbolic systems contains a $C^1$-open and $C^r$-dense subset of ergodic random products of i.i.d. conservative surface diffeomorphisms.
We study the rich behavior of ergodicity and conservativity of Cartesian products of infinite measure preserving transformations. A class of transformations is constructed such that for any subset $R\subset \mathbb Q\cap (0,1)$ there exists $T$ in this class such that $T^p\times T^q$ is ergodic if and only if $\frac{p}{q} \in R$. This contrasts with the finite measure preserving case where $T^p\times T^q$ is ergodic for all nonzero $p$ and $q$ if and only if $T\times T$ is ergodic. We also show that our class is rich in the behavior of conservative products. For each positive integer $k$, a fa…
Given $k>0$ and an Abelian countable discrete group $G$ with elements of infinite order, we construct $(i)$ rigid funny rank-one infinite measure preserving (i.m.p.) $G$-actions of ergodic index $k$, $(ii)$ 0-type funny rank-one i.m.p. $G$-actions of ergodic index $k$, $(iii)$ funny rank-one i.m.p. $G$-actions $T$ of ergodic index 2 such that the product $T\times T^{-1}$ is not ergodic. It is shown that $T\times T^{-1}$ is conservative for each funny rank-one $G$-action $T$.
We present a new approach to the proof of ergodic theorems for actions of free groups based on geometric covering and asymptotic invariance arguments. Our approach can be viewed as a direct generalization of the classical geometric covering and asymptotic invariance arguments used in the ergodic theory of amenable groups. We use this approach to generalize the existing maximal and pointwise ergodic theorems for free group actions to a large class of geometric averages which were not accessible by previous techniques. Some applications of our approach to other groups and other problems in ergod…
We prove a substantial part of a conjecture of Kontsevich and Zorich on the Lyapunov exponents of the Teichmuller geodesic flow on the deviation of ergodic averages for generic conservative flows on higher genus surfaces. The result on the Teichmuller flow is formulated in terms of a (symplectic) cocycle on the real cohomology bundle over the moduli space of holomorphic differentials introduced by Kontsevich and Zorich. We prove that such a cocycle is non-uniformly hyperbolic, that is, all of its Lyapunov exponents are different from zero. In particular, the number of strictly positive exponen…
In this paper free harmonic analysis tools are used to study parabolic iteration in the complex upper half-plane. The main result here is a complete characterization for the norming constants in the monotonic central limit theorem. This allows us to construct a new class of conservative and ergodic measure-preserving transformations on the real line with Lebesgue measure. Among all, we mention that the generalized Boole transformation with infinitely many poles is shown to be conservative, as long as its residues are summable.
We show that a certain type of quasi finite, conservative, ergodic, measure preserving transformation always has a maximal zero entropy factor, generated by predictable sets. We also construct a conservative, ergodic, measure preserving transformation which is not quasi finite; and consider distribution asymptotics of information showing that e.g. for Boole's transformation, information is asymptotically mod-normal with square root normalization. Lastly we see that certain ergodic, probability preserving transformations with zero entropy have analogous properties and consequently entropy dimen…
The main goal in this paper is to describe the geometric structure of invariant graphs of a certain class of skew products. Our focus is on attracting multi-graphs. An invariant multi-graph is an invariant compact set which is a finite union of invariant graphs, and thus consists of a finite number of points on each fiber. We introduce invariant bony multi-graphs and construct an open set of skew products over an invertible base map (solenoid map) having attracting invariant multi-graphs and bony multi-graphs which support finitely many ergodic SRB measures. In this study some thermodynamic pr…
Let $G$ be an amenable discrete countable infinite group, $A$ a finite set, and $(μ_g)_{g\in G}$ a family of probability measures on $A$ such that $\inf_{g\in G}\min_{a\in A}μ_g(a)>0$. It is shown (among other results) that if the Bernoulli shiftwise action of $G$ on the infinite product space $\bigotimes_{g\in G}(A,μ_g)$ is nonsingular and conservative then it is weakly mixing. This answers in positive a question by Z.~Kosloff who proved recently that the conservative Bernoulli $\Bbb Z^d$-actions are ergodic. As a byproduct, we prove a weak version of the pointwise ratio ergodic theorem for n…
We consider the system of $N$ ($\ge2$) elastically colliding hard balls of masses $m_1,...,m_N$ and radius $r$ on the flat unit torus $\Bbb T^ν$, $ν\ge2$. We prove the so called Boltzmann-Sinai Ergodic Hypothesis, i. e. the full hyperbolicity and ergodicity of such systems for every selection $(m_1,...,m_N;r)$ of the external geometric parameters, provided that almost every singular orbit is geometrically hyperbolic (sufficient), i. e. the so called Chernov-Sinai Ansatz is true. The present proof does not use the formerly developed, rather involved algebraic techniques, instead it employs excl…
We consider the system of $N$ ($\ge2$) elastically colliding hard balls of masses $m_1,...,m_N$ and radius $r$ on the flat unit torus $\Bbb T^ν$, $ν\ge2$. We prove the so called Boltzmann-Sinai Ergodic Hypothesis, i. e. the full hyperbolicity and ergodicity of such systems for every selection $(m_1,...,m_N;r)$ of the external geometric parameters. The present proof does not use the formerly developed, rather involved algebraic techniques, instead it employs exclusively dynamical methods and tools from geometric analysis.
We prove a perturbation (pasting) lemma for conservative (and symplectic) systems. This allows us to prove that $C^{\infty}$ volume preserving vector fields are $C^1$-dense in $C^{1}$ volume preserving vector fields (After the conclusion of this work, Ali Tahzibi pointed out to us that this result was proved in 1979 by Carlos Zuppa, although his proof is different from ours.). Moreover, we obtain that $C^1$ robustly transitive conservative flows in three-dimensional manifolds are Anosov and we conclude that there are no geometrical Lorenz-like sets for conservative flows. Also, by-product of t…
The ultimate goal of our book is to present a unified approach to the dynamics, ergodic theory, and geometry of elliptic functions from $\C$ to $\oc$. We consider elliptic functions as a most regular class of transcendental meromorphic functions. Poles form an essential feature of such functions but the set of critical values is finite and an elliptic function is "the same" on its of its fundamental regions. In a sense this is the class of transcendental meromorphic functions which resembles rational functions most. On the other hand, the differences are huge. We will touch on them in the cour…
We consider the system of $N$ ($\ge2$) elastically colliding hard balls of masses $m_1,...,m_N$ and radius $r$ on the flat unit torus $\Bbb T^ν$, $ν\ge2$. We prove the so called Boltzmann-Sinai Ergodic Hypothesis, i. e. the full hyperbolicity and ergodicity of such systems for every selection $(m_1,...,m_N;r)$ of the external geometric parameters, without exceptional values. The present proof does not use at all the formerly developed, rather involved algebraic techniques, instead it employs exclusively dynamical methods and tools of geometric analysis.
We study limit laws for return time processes defined on infinite conservative ergodic measure preserving dynamical systems. Especially for the critical cases with purely atomic limiting distribution we derive distorted processes posessing non-degenerated limits. For these processes also large deviation asymptotics are stated. The Farey map is used as an illustrating example giving new insides into the metric number theory of continued fractions.
We prove that a class of infinite measure preserving transformations, satisfying a "strong" weak mixing condition, generates all rigidity sequences of all conservative ergodic invertible measure preserving transformations defined on a Lebesgue $σ$-finite measure space.
We define generalized currents associated with immersions of abstract oriented solenoids with a transversal measure. We realize geometrically the full real homology of a compact manifold with these generalized currents, and more precisely with immersions of minimal uniquely ergodic solenoids. This makes precise and geometric De Rham's realization of the real homology by only using a restricted geometric subclass of currents.
We continue our study of the dynamics of mappings with small topological degree on (projective) complex surfaces. Previously, under mild hypotheses, we have constructed an ergodic ``equilibrium'' measure for each such mapping. Here we study the dynamical properties of this measure in detail: we give optimal bounds for its Lyapunov exponents, prove that it has maximal entropy, and show that it has product structure in the natural extension. Under a natural further assumption, we show that saddle points are equidistributed towards this measure. This generalize results that were known in the inve…
We construct an example of a uniquely ergodic measured foliation on a surface such that the associated translation flow on the orientation double cover is minimal but not uniquely ergodic. We then prove a geometric criterion for the horizontal foliation of a quadratic differential to be uniquely ergodic. The second theorem generalizes a result of Treviño for the horizontal flow on a translation surface, as well as Masur's criterion for unique ergodicity of the horizontal foliation.
We consider the system of $N$ ($\ge2$) hard balls with masses $m_1,...,m_N$ and radius $r$ in the flat torus $\Bbb T_L^ν=\Bbb R^ν/L\cdot\Bbb Z^ν$ of size $L$, $ν\ge3$. We prove the ergodicity (actually, the Bernoulli mixing property) of such systems for almost every selection $(m_1,...,m_N; L)$ of the outer geometric parameters. This theorem complements my earlier result that proved the same, almost sure ergodicity for the case $ν=2$. The method of that proof was primarily dynamical-geometric, whereas the present approach is inherently algebraic.
For a geometrically finite group Gamma of G=SO(n,1), we survey recent developments on counting and equidistribution problems for orbits of Gamma in a homogeneous space H\G where H is trivial, symmetric or horospherical. Main applications are found in an affine sieve on orbits of thin groups as well as in sphere counting problems for sphere packings invariant under a geometrically finite group. In our sphere counting problems, spheres can be ordered with respect to a general conformal metric.
