In this paper we investigate maps of the two-torus $\mathbb{T}^2$ of the form $T(x,y)=(x+ω,g(x)+f(y))$ for Diophantine $ω\in\mathbb{T}$ and for a class of maps $f,g:\mathbb{T}\to\mathbb{T}$, where each $g$ is strictly monotone and of degree 2, and each $f$ is an orientation preserving circle homeomorphism. For our class of $f$ and $g$ we show that $T$ is minimal and has exactly two invariant and ergodic Borel probability measures. Moreover, these measures are supported on two $T$-invariant graphs. One of the graphs is a Strange Nonchaotic Attractor whose basin of attraction consists of (Lebesgue) almost all points in $\mathbb{T}^2$. Only a low regularity assumption (Lipschitz) is needed on the maps $f$ and $g$, and the results are robust with respect to Lipschitz-small perturbations of $f$ and $g$.
It is known that locally constant toral extensions of hyperbolic systems can never mix at an exponential rate. In this note we exhibit some examples of non-abelian locally constant compact extensions of the shift map which are exponentially mixing for Holder-$L^2$ observables. The proof rests on a result of Bourgain-Gamburd and a decoupling argument.
The statistical and Milnor attractors of step skew products over the Bernoulli shift are studied. For the case of the fiber a circle we prove that for a topologically generic step skew product the statistical and the Milnor attractor coincide and are Lyapunov stable. For this end we study some properties of the projection of the attractor onto the fiber, which might be of independent interest. For the case of the fiber being a segment we give a description of the Milnor attractor as the closure of the union of graphs of finitely many almost everywhere defined functions from the base of the ske…
Consider the skew product $F:\mathbb{T}^2 \to \mathbb{T}^2$, $F(x,y)= (f(x),y+τ(x))$, where $f:\mathbb{T}^1\to \mathbb{T}^1$ is a piecewise $\mathscr{C}^{1+α}$ expanding map on a countable partition and $τ:\mathbb{T}^1 \to \mathbb{R}$ is piecewise $\mathscr{C}^1$. It is shown that if $τ$ is not Lipschitz-cohomologous to a piecewise constant function on the joint partition of $τ$ and $f$, then $F$ is mixing at a stretched-exponential rate.
We consider the skew product $F: (x,u) \mapsto (f(x), u + τ(x))$, where the base map $f : \mathbb{T}^{1} \to \mathbb{T}^{1}$ is piecewise $\mathcal{C}^{2}$, covering and uniformly expanding, and the fibre map $τ: \mathbb{T}^{1} \to \mathbb{R}$ is piecewise $\mathcal{C}^{2}$. We show the dichotomy that either this system mixes exponentially or $τ$ is cohomologous (via a Lipschitz function) to a piecewise constant.
We prove there is a class of maps $γ:\mathbb{T}^{2n}\rightarrow\mathbb{S}^1$ such that a conservative dynamically coherent partially hyperbolic skew-product on $\mathbb{T}^{2n}\times\mathbb{S}^1$ with fixed hyperbolic dynamics on the base and rotation by angle $γ$ acting on the fibers have integrable hyperbolic structure which also implies in particular that they are not contact diffeomorphisms. In dimension $3$, we prove the same result using a standard technique in Contact Geometry, namely, that of \emph{characteristic foliations}, which gives a simple proof of the result but with more tight…
We consider a family of skew-products of the form $(Tx, g_x(t)) : X \times \mathbb{R} \to X \times \mathbb{R}$ where $T$ is a continuous expanding Markov map and $g_x : \mathbb{R} \to \mathbb{R}$ is a family of homeomorphisms of $\mathbb{R}$. A function $u: X \to \mathbb{R}$ is said to be an invariant graph if $\mathrm{graph}(u) = \{(x,u(x)) \mid x\in X\}$ is an invariant set for the skew-product; equivalently if $u(T(x)) = g_x(u(x))$. A well-studied problem is to consider the existence, regularity and dimension-theoretic properties of such functions, usually under strong contraction or expans…
Let $f : (\mathbb{C}^2, 0) \to (\mathbb{C}^2, 0)$ be a germ of holomorphic skew product with a superattracting fixed point at the origin. If it has a suitable weight, then we can construct a Böttcher coordinate which conjugates $f$ to the associated monomial map. This Böttcher coordinate is defined on an open set whose interior or boundary contains the origin.
In this paper, we study quasi-periodic CMV matrices with Verblunsky coefficients given by the skew-shift. We prove the positivity of Lyapunov exponents and Anderson localization for most frequencies, which establish the analogous results of one-dimensional Schrödinger operators proved by Bourgain, Goldstein and Schlag.
We first study the discrete Schrödinger equations with analytic potentials given by a class of transformations. It is shown that if the coupling number is large, then its logarithm equals approximately to the Lyapunov exponents. When the transformation becomes the skew-shift, we prove that the Lyapunov exponent is week Hölder continuous, and the spectrum satisfies Anderson Localization and contains large intervals. Moreover, all of these conclusions are non-perturbative.
The repetition property of a sequence in a metric space, a notion introduced by us in an earlier paper, is of importance in the spectral analysis of ergodic Schrödinger operators. It may be used to exclude eigenvalues for such operators. In this paper we study the question of when a sequence on a torus that is generated by a polynomial or a skew-shift has the repetition property. This provides classes of ergodic Schrödinger operators with potentials generated by skew-shifts on tori that have, contrary to earlier belief, no eigenvalues.
In this paper we address the question of proving Anderson localization (AL) for the operator [H(x,ω)ψ](n) := - \vp(n+1) - \vp(n-1) + V\bigl(T^n_ωx\bigr)ψ(n), n\in\mathbb Z where T:\tor^2\to\tor^2 is either the shift or the skew-shift and V is only C^α(\tor^2) for some α>0. We show that under the assumption of positive Lyapunov exponents, (AL) takes place for a.e. frequency, phase, and energy.
We consider continuous cocycles arising from CMV and Jacobi matrices. Assuming the Verblunsky and Jacobi coefficients arise from generalized skew-shifts, we prove that uniform hyperbolicity of the associated cocycles is $C^0$-dense. This implies that the associated CMV and Jacobi matrices have Cantor spectrum for a generic continuous sampling map.
In this paper, for a discontinuous skew-product transformation with the integrable observation function, we obtain uniform ergodic theorem and semi-uniform ergodic theorem. The main assumptions are that discontinuity sets of transformation and observation function are neglected in some measure-theoretical sense. The theorems extend the classical results which have been established for continuous dynamical systems or continuous observation functions. Meanwhile, on the torus $\mathbb{T}^{d}$ with special rotation, we prove the pointwise convergence of multiple ergodic average $\disp \f 1 N \sum_…
We establish a prime number theorem for all uniquely ergodic, analytic skew products on the $2$-torus $\mathbb{T}^2$. More precisely, for every irrational $α$ and every $1$-periodic real analytic $g:\mathbb{R}\to\mathbb{R}$ of zero mean, let $T_{α,g} : \mathbb{T}^2 \rightarrow \mathbb{T}^2$ be defined by $(x,y) \mapsto (x+α,y+g(x))$. We prove that if $T_{α, g}$ is uniquely ergodic then, for every $(x,y) \in \mathbb{T}^2$, the sequence $\{T_{α, g}^p(x,y)\}$ is equidistributed on $\mathbb{T}^2$ as $p$ traverses prime numbers. This is the first example of a class of natural, non-algebraic and smo…
Let $\mathbb{T}$ be the unit circle and $Γ\backslash G$ the $3$-dimensional Heisenberg nilmanifold. We prove that a class of skew products on $\mathbb{T} \times Γ\backslash G$ are distal, and that the Möbius function is linearly disjoint from these skew products. This verifies the Möbius Disjointness Conjecture of Sarnak.
We show that for $\varepsilon > 0$, every $C^{1 + \varepsilon}$ skew product on $\mathbb{T}^2$ over a rotation of $\mathbb{T}^1$ satisfies Sarnak's conjecture. This is an improvement of earlier results of Kulaga-Przymus-Lemańczyk, Huang-Wang-Ye, and Kanigowski-Lemańczyk-Radziwill.
Let (Ω,μ) be a shift of finite type with a Markov probability, and (Y,ν) a non-atomic standard measure space. For each symbol i of the symbolic space, let Φ_i be a measure-preserving automorphism of (Y,ν). We study skew products of the form (ω,y) --> (σω,Φ_{ω_0}(y)), where σ=shift map on (Ω,μ). We prove that, when the skew product is conservative, it is ergodic if and only if the Φ_i's have no common non-trivial invariant set. In the second part we study the skew product when Ω={0,1}^Z, μ=Bernoulli measure, and Φ_0,Φ_1 are R-extensions of a same uniquely ergodic probability-preserving automorp…
We examine iteration of certain skew-products on the bidisk whose components are rational inner functions, with emphasis on simple maps of the form $Φ(z_1,z_2) = (ϕ(z_1,z_2), z_2)$. If $ϕ$ has degree $1$ in the first variable, the dynamics on each horizontal fiber can be described in terms of Möbius transformations but the global dynamics on the $2$-torus exhibit some complexity, encoded in terms of certain $\mathbb{T}^2$-symmetric polynomials. We describe the dynamical behavior of such mappings $Φ$ and give criteria for different configurations of fixed point curves and rotation belts in term…
We study the Bernoulli property for a class of partially hyperbolic systems arising from skew products. More precisely, we consider a hyperbolic map $(T,M,μ)$, where $μ$ is a Gibbs measure, an aperiodic Hölder continuous cocycle $ϕ:M\to \mathbb R$ with zero mean and a zero-entropy flow $(K_t,N,ν)$. We then study the skew product $$ T_ϕ(x,y)=(Tx,K_{ϕ(x)}y), $$ acting on $(M\times N,μ\times ν)$. We show that if $(K_t)$ is of slow growth and has good equidistribution properties, then $T_ϕ$ remains Bernoulli. In particular, our main result applies to $(K_t)$ being a typical translation flow on a s…
We consider skew-products on $M\times \mathbb{T}^2$, where $M$ is the two-sphere or the two-torus, which are partially hyperbolic and semi-conjugate to an Axiom A diffeomorphism. This class of dynamics includes the open sets of $Ω$-non-stable systems introduced by Abraham, Smale and Shub. We present sufficient conditions, both on the skew-products and the potentials, for the existence and uniqueness of equilibrium states, and discuss their statistical stability.
We consider large Hermitian matrices whose entries are defined by evaluating the exponential function along orbits of the skew-shift $\binom{j}{2} ω+jy+x \mod 1$ for irrational $ω$. We prove that the eigenvalue distribution of these matrices converges to the corresponding distribution from random matrix theory on the global scale, namely, the Wigner semicircle law for square matrices and the Marchenko-Pastur law for rectangular matrices. The results evidence the quasi-random nature of the skew-shift dynamics which was observed in other contexts by Bourgain-Goldstein-Schlag and Rudnick-Sarnak-Z…
We study skew products where the base is a hyperbolic automorphism of $\mathbb{T}^2$, the fiber is a smooth area preserving flow on $\mathbb{T}^2$ with one fixed point (of high degeneracy) and the skewing function is a smooth non coboundary with non-zero integral. The fiber dynamics can be represented as a special flow over an irrational rotation and a roof function with one power singularity. We show that for a full measure set of rotations the corresponding skew product is $K$ and not Bernoulli. As a consequence we get a natural class of volume-preserving diffeomorphisms of $\mathbb{T}^4$ wh…
In this paper we study the existence of positive Lyapunov exponents for three different types of skew products, whose fibers are compact Riemannian surfaces and the action on the fibers are by volume preserving diffeomorphisms. These three types include skew products with a volume preserving Anosov diffeomorphism on the basis; or with a subshift of finite type on the basis preserving a measure with product structure; or locally constant skew products with Bernoulli shifts on the basis. We prove the $C^1$-density and $C^r$-openess of the existence of positive Lyapunov exponents on a set of posi…
It is known that a one-dimensional quantum particle is localized when subjected to an arbitrarily weak random potential. It is conjectured that localization also occurs for an arbitrarily weak potential generated from the nonlinear skew-shift dynamics: $v_n=2\cos\left(\binom{n}{2}ω+ny+x\right)$ with $ω$ an irrational number. Recently, Han, Schlag, and the second author derived a finite-size criterion in the case when $ω$ is the golden mean, which allows to derive the positivity of the infinite-volume Lyapunov exponent from three conditions imposed at a fixed, finite scale. Here we numerically …
Starting from a discrete $C^*$-dynamical system $(\mathfrak{A}, θ, ω_o)$, we define and study most of the main ergodic properties of the crossed product $C^*$-dynamical system $(\mathfrak{A}\rtimes_α\mathbb{Z}, Φ_{θ, u},\om_o\circ E)$, $E:\mathfrak{A}\rtimes_α\mathbb{Z}\rightarrow\ga$ being the canonical conditional expectation of $\mathfrak{A}\rtimes_α\mathbb{Z}$ onto $\mathfrak{A}$, provided $\a\in\aut(\ga)$ commute with the $*$-automorphism $þ$ up tu a unitary $u\in\ga$. Here, $Φ_{θ, u}\in\aut(\mathfrak{A}\rtimes_α\mathbb{Z})$ can be considered as the fully noncommutative generalisation of …
We prove a conditional theorem on the positivity of the Lyapunov exponent for a Schrödinger cocycle over a skew shift base with a cosine potential and the golden ratio as frequency. For coupling below 1, which is the threshold for Herman's subharmonicity trick, we formulate three conditions on the Lyapunov exponent in a finite but large volume and on the associated large deviation estimates at that scale. Our main results demonstrate that these finite-size conditions imply the positivity of the infinite volume Lyapunov exponent. This paper shows that it is possible to make the techniques devel…
For $τ>2$, let $T$ be a $C^τ$ skew product map of the form $(x+α,y+h(x))$ on $\mathbb T^2$ over a rotation of the circle. We show that if $T$ preserves a measurable section, then it is disjoint to the Möbius sequence. This in particular implies that any non-uniquely ergodic $C^τ$ skew product map on $\mathbb T^2$ has a finite index factor that is disjoint to the Möbius sequence.
We initiate a parametric study of holomorphic families of polynomial skew products, i.e., polynomial endomorphisms of $\mathbb{C}^2$ of the form $F(z,w)= (p(z), q(z,w))$ that extend to holomorphic endomorphisms of $\mathbb{P}^2(\mathbb{C})$. We prove that dynamical stability in the sense of arXiv:1403.7603 preserves hyperbolicity within such families, and give a complete classification of the hyperbolic components that are the analogue, in this setting, of the complement of the Mandelbrot set for the family $z^2 +c$. We also precisely describe the geometry of the bifurcation locus and current …
Let $F$ be a non-discrete non-Archimedean locally compact field such that the characteristic $\mathrm{ch}(F)\ne 2$ and let $\mathcal{O}_F$ be the ring of integers in $F$. The main results of this paper are Theorem 1.2 that classifies ergodic probability measures on the space $\mathrm{Skew}(\mathbb{N}, F)$ of infinite skew-symmetric matrices with respect to the natural action of the group $\mathrm{GL}(\infty,\mathcal{O}_F)$ and Theorem 1.4, that gives an unexpected natural correspondence between the set of $\mathrm{GL}(\infty,\mathcal{O}_F)$-invariant Borel probability measures on $\mathrm{Sym}…
We consider continuous $SL(2,\mathbb{R})$-cocycles over a strictly ergodic homeomorphism which fibers over an almost periodic dynamical system (generalized skew-shifts). We prove that any cocycle which is not uniformly hyperbolic can be approximated by one which is conjugate to an $SO(2,\mathbb{R})$-cocycle. Using this, we show that if a cocycle's homotopy class does not display a certain obstruction to uniform hyperbolicity, then it can be $C^0$-perturbed to become uniformly hyperbolic. For cocycles arising from Schrödinger operators, the obstruction vanishes and we conclude that uniform hype…
Let $X$ be a variety defined over an algebraically closed field $k$ of characteristic $0$, let $N\in\mathbb{N}$, let $g:X\dashrightarrow X$ be a dominant rational self-map, and let $A:\mathbb{A}^N\to \mathbb{A}^N$ be a linear transformation defined over $k(X)$, i.e., for a Zariski open dense subset $U\subset X$, we have that for $x\in U(k)$, the specialization $A(x)$ is an $N$-by-$N$ matrix with entries in $k$. We let $f:X\times\mathbb{A}^N\dashrightarrow X\times \mathbb{A}^N$ be the rational endomorphism given by $(x,y)\mapsto (g(x), A(x)y)$. We prove that if the determinant of $A$ is nonzero…
We consider the spectral and dynamical properties of one-dimensional quantum walks placed into homogenous electric fields according to a discrete version of the minimal coupling principle. We show that for all irrational fields the absolutely continuous spectrum of these systems is empty, and prove Anderson localization for almost all (irrational) fields. This result closes a gap which was left open in the original study of electric quantum walks: a spectral and dynamical characterization of these systems for typical fields. Additionally, we derive an analytic and explicit expression for the L…
We introduce and study skew product Smale endomorphisms over finitely irreducible topological Markov shifts with countable alphabets. We prove that almost all conditional measures of equilibrium states of summable and locally Holder continuous potentials are dimensionally exact, and that their dimension is equal to the ratio of the (global) entropy and the Lyapunov exponent. We also prove for them a formula of Bowen type for the Hausdorff dimension of all fibers. We develop a version of thermodynamic formalism for finitely irreducible two-sided topological Markov shifts with countable alphabe…
We study the one-dimensional discrete Schrödinger operator with the skew-shift potential $2λ\cos\left(2π\left(\binom{j}{2} ω+jy+x\right)\right)$. This potential is long conjectured to behave like a random one, i.e., it is expected to produce Anderson localization for arbitrarily small coupling constants $λ>0$. In this paper, we introduce a novel perturbative approach for studying the zero-energy Lyapunov exponent $L(λ)$ at small $λ$. Our main results establish that, to second order in perturbation theory, a natural upper bound on $L(λ)$ is fully consistent with $L(λ)$ being positive and satisf…
For $p \in \mathbb{Q}_+ \smallsetminus \{ 1 \}$ a positive rational number different from one, we say that the Puisseux series $f \in \mathbb{C}((t))^\text{alg}$ is $p$-Mahler of non-exceptional polynomial type if there is a polynomial $P \in \mathbb{C}(t)^\text{alg}[X]$ of degree at least two which is not conjugate to either a monomial or to plus or minus a Chebyshev polynomial for which the equation $f(t^p) = P(f(t))$ holds. We show that if $p$ and $q$ are multiplicatively independent and $f$ and $g$ are $p$-Mahler and $q$-Mahler, respectively, of non-exceptional polynomial type, then $f$ an…
IRIF, Université Paris Cité（巴黎西大 IRIF）；Département de mathématiques et de statistique, Université de Moncton（蒙克顿大学数学与统计系）；School of Mathematics, Korea Institute for Advanced Study（韩国高等研究院数学学院）；Laboratoire d’Informatique Gaspard Monge, Université Paris Est（巴黎东大加斯帕尔·蒙日计算机科学实验室）；Mathematics department, University of North Carolina（北卡罗来纳大学数学系）
Let $f:\mathbb{C}^2\to \mathbb{C}^2$ be a polynomial skew product which leaves invariant an attracting vertical line $ L $. Assume moreover $f$ restricted to $L$ is non-uniformly hyperbolic, in the sense that $f$ restricted to $L$ satisfies one of the following conditions: 1. $f|_L$ satisfies Topological Collet-Eckmann and Weak Regularity conditions. 2. The Lyapunov exponent at every critical value point lying in the Julia set of $f|_L$ exist and is positive, and there is no parabolic cycle. Under one of the above conditions we show that the Fatou set in the basin of $L$ coincides with the uni…
We show a partial generalization of Sullivan's non-wandering domain theorem in complex dimension two. More precisely, we show the non-existence of wandering Fatou components for polynomial skew products of $ \mathbb{C}^2$ with an invariant attracting fiber, under the assumption that the multiplier $ λ$ is small. We actually show a stronger result, namely that every forward orbit of any vertical Fatou disk intersects a bulging Fatou component.
