We study mixing properties of generalized $T, T^{-1}$ transformation. We discuss two mixing mechanisms. In the case the fiber dynamics is mixing, it is sufficient that the driving cocycle is small with small probability. In the case the fiber dynamics is only assumed to be ergodic, one needs to use the shearing properties of the cocycle. Applications include the Central Limit Theorem for sufficiently fast mixing systems and the estimates on deviations of ergodic averages.
We prove mixing on a general class of rank-one transformations containing all known examples of rank-one mixing, including staircase transformations and Ornstein's constructions, and a variety of new constructions.
We prove that irreducible, linearly recurrent, type W interval exchange transformations are always mild mixing. For every irreducible permutation the set of linearly recurrent interval exchange transformations has full Hausdorff dimension.
For different classes of measure preserving transformations, we investigate collections of sets that exhibit the property of lightly mixing. Lightly mixing is a stronger property than topological mixing, and requires that a lim inf is positive. In particular, we give a straightforward proof that any mildly mixing transformation T has a dense algebra C such that T is lightly mixing on C. Also, we provide a hierarchy for the properties of lightly mixing, sweeping out and uniform sweeping out for dense collections, and dense algebras of sets. As a result, stronger mixing realizations are given fo…
We consider suspension flows built over interval exchange transformations with the help of roof functions having an asymmetric logarithmic singularity. We prove that such flows are strongly mixing for a full measure set of interval exchange transformations.
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 show that every invertible strong mixing transformation on a Lebesgue space has strictly over-recurrent sets. Also, we give an explicit procedure for constructing strong mixing transformations with no under-recurrent sets. This answers both parts of a question of V. Bergelson. We define $ε$-over-recurrence and show that given $ε> 0$, any ergodic measure preserving invertible transformation (including discrete spectrum) has $ε$-over-recurrent sets of arbitrarily small measure. Discrete spectrum transformations and rotations do not have over-recurrent sets, but we construct a weak mixing rigi…
It is shown that if $E$ is any subset of $\Bbb N$ such that either $1\in E$ or $2\in E$ then there is a mixing transformation whose set of spectral multiplicities is $E$.
We show that a typical interval exchange transformation is either weakly mixing or it is an irrational rotation. We also conclude that a typical translation flow on a surface of genus $g \geq 2$ (with prescribed singularity types) is weakly mixing.
We prove that for any compact connected Lie group G and a typical interval exchange transformation T, not isomorphic to a rotation, the skew product of T with a typical G-valued function, constant on the intervals, is weakly mixing.
We prove that mixing on rank-one transformations is equivalent to the spacer sequence being slice-ergodic. Slice-ergodicity, introduced in this paper, generalizes the notion of ergodic sequence to the uniform convergence of ergodic averages (as in the mean ergodic theorem) over subsequences of partial sums. We show that polynomial staircase transformations satisfy this condition and therefore are mixing.
For any infinite zero-density integer set M, we found a rigid measure-preserving transformation mixing along M by answering Bergelson's question. Gaussian and Poisson suspensions over infinite constructions are suggested as suitable examples. We also discuss rigid rank one transformations of finite measure that mix along a given set, and some analogues for group actions.
A technique is presented for multiplexing two ergodic measure preserving transformations together to derive a third limiting transformation. This technique is used to settle a question regarding rigidity of weak mixing transformations. Namely, given any rigidity sequence for an ergodic measure preserving transformation, there exists a weak mixing transformation which is rigid along the same sequence. This establishes a wide range of rigidity sequences for weakly mixing dynamical systems.
We prove that a rank one transformation satisfying a condition called restricted growth is a mixing transformation if and only if the spacer sequence for the transformation is uniformly ergodic. Uniform ergodicity is a generalization of the notion of ergodicity for sequences in the sense that the mean ergodic theorem holds for what we call dynamical sequences. In particular, Adams' class of staircase transformations and Ornstein's class constructed using ``random spacers'' have both restricted growth and uniformly ergodic spacer sequences.
J.-P. Thouvenot and the author showed via different approaches that the centralizer of a mixing rank-one infinite measure preserving transformation was trivial. In this note the author presents his joining proof. We also consider constructions with algebraic spacers as well as a class of Sidon constructions to produce new examples of mixing rank one transformations. In connection with Gordin's question on the existence of homoclinic ergodic actions for a zero entropy system we also discuss Poisson suspensions of some modifications of Sidon rank one constructions.
We study the notions of weak rational ergodicity and rational weak mixing as defined by Jon Aaronson. We prove that various families of infinite measure-preserving rank-one transformations possess (or do not posses) these properties, and consider their relation to other notions of mixing in infinite measure.
We examine some of the properties of uniformly rigid transformations, and analyze the compatibility of uniform rigidity and (measurable) weak mixing along with some of their asymptotic convergence properties. We show that on Cantor space, there does not exist a finite measure-preserving, totally ergodic, uniformly rigid transformation. We briefly discuss general group actions and show that (measurable) weak mixing and uniform rigidity can coexist in a more general setting.
We construct a rank one infinite measure preserving transformation $T$ such that for all sequences of nonzero integers $\{k_{1},..., k_{r}\}$, $T^{k_{1}}\times...\times T^{k_{r}}$ is ergodic.
We show that a residual set of non-degenerate IETs on more than 3 letters is topologically mixing. This shows that there exists a uniquely ergodic topologically mixing IET. This is then applied to show that some billiard flows in a fixed direction in an L-shaped polygon are topologically mixing.
In paper [1] S. V. Tikhonov introduced a complete metric on the space of mixing transformations. It generates a topology called leash-topology. In [2] Tikhonov states the following problem: for what mixing transformation T its conjugacy class is dense in the space of mixing transformations with leash-topology? We show that it is true for every mixing T. As a corollary we get that generic mixing is rank-1.
We introduce high staircase infinite measure preserving transformations and prove that they are mixing under a restricted growth condition. This is used to (i) realize each subset $E\subset\Bbb N\cup\{\infty\}$ as the set of essential values of the multiplicity function for the Koopman operator of a mixing ergodic infinite measure preserving transformation, (ii) construct mixing power weakly mixing infinite measure preserving transformations, (iii) construct mixing Poissonian automorphisms with a simple spectrum, etc.
In the class of Ornstein transformations the mixing property satisfies a 0-1 law. Here we consider Ornstein's construction with bounded cutting parameter. In fact, these latter transformations are not mixing, however it is proved that the weak mixing property occurs with probability one. Our situation is similar to the case of interval exchange transformations whose link with the cutting and stacking construction relies in a dynamical process called Rauzy induction.
Let $d >1$. In this paper we show that for an irreducible permutation $π$ which is not a rotation, the set of $[λ]\in \mathbb{P}_+^{d-1}$ such that the interval exchange transformation $f([λ],π)$ is not weakly mixing does not have full Hausdorff dimension. We also obtain an analogous statement for translation flows. In particular, it strengthens the result of almost sure weak mixing proved by G. Forni and the first author. We adapt here the probabilistic argument developed in their paper in order to get some large deviation results. We then show how the latter can be converted into estimates o…
We prove that minimal area-preserving flows locally given by a smooth Hamiltonian on a closed surface of any genus are typically (in the measure-theoretical sense) not mixing. The result is obtained by considering special flows over interval exchange transformations under roof functions with symmetric logarithmic singularities and proving absence of mixing for a full measure set of interval exchange transformations.
Let $(X, T)$ be a weakly mixing minimal system, $p_1, \cdots, p_d$ be integer-valued generalized polynomials and $(p_1,p_2,\cdots,p_d)$ be non-degenerate. Then there exists a residual subset $X_0$ of $X$ such that for all $x\in X_0$ $$\{ (T^{p_1(n)}x, \cdots, T^{p_d(n)}x): n\in \mathbb{Z}\}$$ is dense in $X^d$.
A classical result of Halmos asserts that among measure preserving transformations the weak mixing property is generic. We extend Halmos' result to the collection of ergodic extensions of a fixed, but arbitrary, ergodic transformation $T_0$. We then use a result of Connes, Feldman and Weiss to extend this relative theorem to the general (countable) amenable group.
We derive sufficient conditions for the mixing of all orders of interacting transformations of a spatial Poisson point process, under a zero-type condition in probability and a generalized adaptedness condition. This extends a classical result in the case of deterministic transformations of Poisson measures. The approach relies on moment and covariance identities for Poisson stochastic integrals with random integrands.
We establish that there are non-mixing maps that are mixing on appropriate sequences including sequences $(s_i)$ which satisfy the Rajchman dissociated property. Our examples are based on the staircase rank one construction, $M$-towers constructions and the Gaussian transformations. As a consequence, we obtain there are non-mixing maps which are mixing along the squares. We further prove that a sequence $M=(m_n)$ is a mixing sequence for some weak mixing ${1}/{2}$-rigid transformation $T$ if and only if the complement of $M$ is a thick set. This result is generalized to ${r}/{(r+1)}$-rigid tra…
We establish a dichotomy for the rate of the decay of the Cesàro averages of correlations of sufficiently regular functions for typical interval exchange transformations (IET) which are not rigid rotations (for which weak mixing had been previously established in the works of Katok-Stepin, Veech, and Avila-Forni). We show that the rate of decay is either logarithmic or polynomial, according to whether the IET is of rotation class (i.e., it can be obtained as the induced map of a rigid rotation) or not. In the latter case, we also establish that the spectral measures of Lipschitz functions have…
We introduce generalized $(α,β)$-transformations, which include all $(α,β)$ and generalized $β$-transformations, and prove that all transitive generalized $(α,β)$-transformations satisfy the level-2 large deviation principle with a unique measure of maximal entropy. A crucial step in our proof is to establish density of periodic measures in the set of ergodic measures.
We introduce a class of rank-one transformations, which we call extremely elevated staircase transformations. We prove that they are measure-theoretically mixing and, for any $f : \mathbb{N} \to \mathbb{N}$ with $f(n)/n$ increasing and $\sum 1/f(n) < \infty$, that there exists an extremely elevated staircase with word complexity $p(n) = o(f(n))$. This improves the previously lowest known complexity for mixing subshifts, resolving a conjecture of Ferenczi.
Consider a class of skew product transformations consisting of an ergodic or a periodic transformation on a probability space (M, B, m) in the base and a semigroup of transformations on another probability space (W,F,P) in the fibre. Under suitable mixing conditions for the fibre transformation, we show that the properties ergodicity, weakly mixing, and strongly mixing are passed on from the base transformation to the skew product (with respect to the product measure). We derive ergodic theorems with respect to the skew product on the product space. The main aim of this paper is to establish u…
We show that for generalized Baker's transformations there is a parameter domain where we have an absolutely continuous ergodic measure and in direct neighborhood there is a parameter domain where not even the variational principle for Hausdorff dimension holds.
It is proved that each Gaussian cocycle over a mildly mixing Gaussian transformation is either a Gaussian coboundary or sharply weak mixing. The class of nonsingular infinite direct products $T$ of transformations $T_n$, $n\in\Bbb N$, of finite type (IDPFT) is studied. It is shown that if $T_n$ is mildly mixing, $n\in\Bbb N$, the sequence of the Radon-Nikodym derivatives of $T_n$ is asymptotically translation quasi-invariant and $T$ is conservative then the Maharam extension of $T$ is sharply weak mixing. This techniques provides a new approach to the nonsingular Gaussian transformations studi…
Via (C,F)-construction we produce a 2-fold simple mixing transformation which has uncountably many non-trivial proper factors and all of them are prime.
Isospectral transformations (IT) of matrices and networks allow for compression of either object while keeping all the information about their eigenvalues and eigenvectors.We analyze here what happens to generalized eigenvectors under isospectral transformations and to what extent the initial network can be reconstructed from its compressed image under IT. We also generalize and essentially simplify the proof that eigenvectors are invariant under isospectral transformations and generalize and clarify the notion of spectral equivalence of networks.
This text is addressed to students. It is a short story about some problems in ergodic theory, both related and independent. We discuss the factorization of transformations into the product of three involutions; Furstenberg's theorem on weak multiple mixing in the mean along progressions; ergodic Furstenberg-Roth's theorem. In connection with Ledrappier's $Z^2$-action, we consider harmonic $Z_2$-sequences on $Z^2$ and experimentally observe the dominant infinite threads arising in them. The property of multiple mixing for systems with the Gordin ergodic group is proven. We discuss the absence …
The following generalizations of the Chacon map are proposed: instead of classical constant spacer sequence $(0,1,0)$ let a sequence $(0,s_j,0)$ be one with unbounded $s_j$. (We mention also an analogue of the historical Chacon map with spacer sequences in the form $(0,s_j)$.) This narrow class of rank-one transformations may be abundant source of open questions. All such constructions have partial rigidity, but some other properties could be different. For root sequence, $ s_j= [\sqrt{j}]$, (or $ s_j= [\ln{j}]$) the corresponding action is rigid, moreover it possesses all polynomials in its w…
We define a notion of entropy for an infinite family $\mathcal{C}$ of measurable sets in a probability space. We show that the mean ergodic theorem holds uniformly for $\mathcal{C}$ under every ergodic transformation if and only if $\mathcal{C}$ has zero entropy. When the entropy of $\mathcal{C}$ is positive, we establish a strong converse showing that the uniform mean ergodic theorem fails generically in every isomorphism class, including the isomorphism classes of Bernoulli transformations. As a corollary of these results, we establish that every strong mixing transformation is uniformly str…
Let $f\colon X\to X$, $X=[0,1)$, be an ergodic IET (interval exchange transformation) relative to the Lebesgue measure on $X$. Denote by $f_t\colon X_t\to X_t$ the IET obtained by inducing $f$ to the subinterval $X=[0,t)$, $0<t<1$. We show that \[ \{0<t<1\mid f_{t} \text{is weakly mixing}\} \] is a residual subset of $X$ of full Lebesgue measure. The result is proved by establishing a generic Diophantine sufficient condition on $t$ for $f_{t}$ to be weakly mixing.
In the following text we prove that for finite discrete $X$ with at least two elements and infinite $Γ$, the generalized shift transformation semigroup $({\mathcal S},X^Γ)$ is equicontinuous (resp. has at least an equicontinuous point, is not sensitive) if and only if for all $w\inΓ$, $\{φ(w):σ_φ\in{\mathcal S}\}$ is finite. We continue our study regarding distality and expansivity of $({\mathcal S},X^Γ)$.
We present a computational study of finite-time mixing of a line segment by cutting and shuffling. A family of one-dimensional interval exchange transformations is constructed as a model system in which to study these types of mixing processes. Illustrative examples of the mixing behaviors, including pathological cases that violate the assumptions of the known governing theorems and lead to poor mixing, are shown. Since the mathematical theory applies as the number of iterations of the map goes to infinity, we introduce practical measures of mixing (the percent unmixed and the number of interm…
We exhibit rationally ergodic, weakly mixing measure preserving transformations which are not subsequence rationally weakly mixing and give a condition for smoothness of renewal sequences.
Invariant ergodic measures for generalized Boole type transformations are studied using an invariant quasi-measure generating function approach based on special solutions to the Frobenius--Perron operator. New two-dimensional Boole type transformations are introduced, and their invariant measures and ergodicity properties are analyzed.
We present a set of phase-space portraits illustrating the extraordinary oscillatory possibilities of the dynamical systems through the so-called generalized Landau scenario. In its simplest form the scenario develops in N dimensions around a saddle-node pair of fixed points experiencing successive Hopf bifurcations up to exhausting their stable manifolds and generating N-1 different limit cycles. The oscillation modes associated with these cycles extend over a wide phase-space region by mixing ones within the others and by affecting both the transient trajectories and the periodic orbits them…
We study generalized Hopf-Cole transformations motivated by the Schrödinger bridge problem, which can be seen as a boundary value Hamiltonian system on the Wasserstein space. We prove that generalized Hopf-Cole transformations are symplectic submersions in the Wasserstein symplectic geometry. Many examples, including a Hopf-Cole transformation for the shallow water equations, are given. Based on this transformation, energy splitting inequalities are provided.
Consider an invertible measure-preserving transformation of a probability space. A finite partition of the space is called weakly independent if there are infinitely many images of this partition under powers of the transformation that are jointly independent. Krengel proved that a transformation is weakly mixing if and only if weakly independent partitions of the underlying space are dense among all finite partitions. Using the tools developed in the later papers of del Junco-Reinhold-Weiss and del Junco-Begun we obtain Krengel- type results for weakly mixing random dynamical systems (or equi…
The problem of characterizing all new-time transformations preserving the Poisson structure of a finitedimensional Poisson system is completely solved in a constructive way. As a corollary, this leads to a broad generalization of previously known results. Examples are given.
