We propose to study the multifractal behavior of weighted ergodic averages. Our study in this paper is concentrated on the symbolic dynamics. We introduce a thermodynamical formalism which leads to a multifractal spectrum. It is proved that this thermodynamical formalism applies to different kinds of dynamically defined weights, including stationary ergodic random weights, uniquely ergodic weights etc. But the validity of the thermodynamical formalism for very irregular weights, like Möbius function, is an unsolved problem. The paper ends with some other unsolved problems.
This article is devoted to the study of the multifractal analysis of ergodic averages in some nonuniformly hyperbolic systems. In particular, our results hold for the robust classes of multidimensional nonuniformly expanding local diffeomorphisms and Viana maps.
In this paper, we study the multiple ergodic averages of a locally constant real-valued function in linear Cookie-Cutter dynamical systems. The multifractal spectrum of these multiple ergodic averages is completely determined.
In this paper we present a complete solution to the problem of multifractal analysis of multiple ergodic averages in the case of symbolic dynamics for functions of two variables depending on the first coordinate.
We study certain multiple ergodic averages of an iterated functions system generated by two contractions on the unit interval. By using the dynamical coding ${0,1}^{\mathbb{N}}$ of the attractor, we compute the Hausdorff dimension of the set of points with a given frequency of the pattern 11 in positions $k, 2k$.
In this paper we study the multiple ergodic averages $$ \frac{1}{n}\sum_{k=1}^n φ(x_k, x_{kq}, ..., x_{k q^{\ell-1}}), \qquad (x_n) \in Σ_m $$ on the symbolic space $Σ_m ={0, 1, ..., m-1}^{\mathbb{N}^*}$ where $m\ge 2, \ell\ge 2, q\ge 2$ are integers. We give a complete solution to the problem of multifractal analysis of the limit of the above multiple ergodic averages. Actually we develop a non-invariant and non-linear version of thermodynamic formalism that is of its own interest. We study a large class of measures (called telescopic measures) and the special case of telescopic measures defi…
In this paper, we study the topological spectrum of weighted Birkhoff averages over aperiodic and irreducible subshifts of finite type. We show that for a uniformly continuous family of potentials, the spectrum is continuous and concave over its domain. In case of typical weights with respect to some ergodic quasi-Bernoulli measure, we determine the spectrum. Moreover, in case of full shift and under the assumption that the potentials depend only on the first coordinate, we show that our result is applicable for regular weights, like Möbius sequence.
Any ergodic measure of a smooth map on a compact manifold has a multifractal spectrum with one point - the dimension of the measure itself - at the diagonal. We will construct examples where this fails in the most drastic way for invariant measures invariant under linear maps of the circle.
Multifractal analysis refers to the study of the local properties of measures and functions, and consists of two parts: the fine multifractal theory and the coarse multifractal theory. The fine and the coarse theory are linked by a web of conjectures known collectively as the Multifractal Formalism. Very roughly speaking the Multifractal Formalism says that the multifractal spectrum from fine theory equals the Legendre transform of the Renyi dimensions from the coarse theory. Recently {\it fine} multifractal zeta-functions, i.e. multifractal zeta-functions designed to produce detailed informat…
We prove that fully oscillating sequences are orthogonal to multiple ergodic realizations of affine maps of zero entropy on compact abelian groups. It is more than what Sarnak's conjecture requires for these dynamical systems.
We develop a thermodynamic formalism for quasi-multiplicative potentials on a countable symbolic space and apply these results to the dimension theory of infinitely generated self-affine sets. The first application is a generalisation of Falconer's dimension formula to include typical infinitely generated self-affine sets and show the existence of an ergodic invariant measure of full dimension whenever the pressure function has a root. Considering the multifractal analysis of Birkhoff averages of general potentials $Φ$ taking values in $\R^{\N}$, we give a formula for the Hausdorff dimension o…
Let $(X, \mathcal{A},μ)$ be a probability space and let $T$ be a contraction on $L^2(μ)$. We provide suitable conditions over sequences $(w_k)$, $(u_k)$ and $(A_k)$ in such a way that the weighted ergodic limit $\lim\limits_{N\rightarrow\infty}\frac{1}{A_N}\sum_{k=0}^{N-1} w_k T^{u_k}(f)=0$ $μ$-a.e. for any function $f$ in $L^2(μ)$. As a consequence of our main theorems, we also deal with the so-called one-sided weighted ergodic Hilbert transforms.
We consider sequences of Davenport type or Gelfond type and prove that sequences of Davenport exponent larger than $\frac{1}{2}$ are good sequences of weights for the ergodic theorem, and that the ergodic sums weighted by a sequence of strong Gelfond property is well controlled almost everywhere. We prove that for any $q$-multiplicative sequence, the Gelfond property implies the strong Gelfond property and that sequences realized by dynamical systems can be fully oscillating and have the Gelfond property.
We propose to study multiple ergodic averages from multifractal analysis point of view. In some special cases in the symbolic dynamics, Hausdorff dimensions of the level sets of multiple ergodic average limit are determined by using Riesz products.
Universal projection theorems with applications to multifractal analysis and the dimension of every ergodic measure on self-conformal sets simultaneously
In the present paper we prove weighted ergodic theorems and multiparameter weighted ergodic theorems for positive contractions acting on $L_p(\hat{\nabla},\hatμ)$. Our main tool is the use of methods of measurable bundles of Banach-Kantorovich lattices.
We prove weighted and vector-valued variational estimates for ergodic averages on $\mathbb{R}^d$. The weighted square function estimate relating ergodic averages to the dyadic martingale is obtained using an $\ell^r$ version of a reverse Hölder inequality for variation seminorms.
In the present paper we prove Besocovich weighted ergodic theorem for positive contractions acting on Orlich-Kantorovich space. Our main tool is the use of methods of measurable bundles of Banach-Kantorovich lattices.
We present a simple way to produce good weights for several types of ergodic theorem including the Wiener-Wintner type multiple return time theorem and the multiple polynomial ergodic theorem. These weights are deterministic and come from orbits of certain bounded linear operators on Banach spaces. This extends the known results for nilsequences and return time sequences of the form (g(S^ny)) for a measure preserving system (Y,S) and g\in L^\infty(Y), avoiding in the latter case the problem of finding the full measure set of appropriate points y.
In this article, we consider the weighted ergodic optimization problem Axiom A attractors of a $C^2$ flow on a compact smooth manifold. The main result obtained in this paper is that for a generic observable from function space $\mc C^{0,\a}$ ($\a\in(0,1]$) or $\mc C^1$ the minimizing measure is unique and is supported on a periodic orbit.
This paper is devoted to study ergodic optimisation problems for almost-additive sequences of functions (rather than a fixed potential) defined over countable Markov shifts (that is a non-compact space). Under certain assumptions we prove that any accumulation point of a family of Gibbs equilibrium measures is a maximising measure. Applications are given in the study of the joint spectral radius and to multifractal analysis of Lyapunov exponent of non-conformal maps.
Let $μ$ be the geometric realization on $[0,1]$ of a Gibbs measure on $Σ=\{0,1\}^{\mathbb{N}}$ associated with a Hölder potential. The thermodynamic and multifractal properties of $μ$ are well known to be linked via the multifractal formalism. In this article, the impact of a random sampling procedure on this structure is studied. More precisely, let $\{I_w\}_{w\in Σ^*}$ stand for the collection of dyadic subintervals of $[0,1]$ naturally indexed by the set of finite dyadic words $Σ^*$. Fix $η\in(0,1)$, and a sequence $(p_w)_{w\in Σ^*}$ of independent Bernoulli variables of parameters $2^{-|w|…
We conduct the multifractal analysis of the level sets of the asymptotic behavior of almost-additive continuous potentials $(ϕ_n)_{n=1}^\infty$ on a topologically mixing subshift of finite type $X$ endowed itself with a metric associated with such a potential. We work without bounded distorsion property assumption. We express the whole Hausdorff spectrum in terms of a conditional variational principle, as well as a new large deviations principle. Our approach provides a new description of the structure of the spectrum in terms of {\it weak} concavity. Another new point is that we consider sets…
We study the problem on the weak-star decomposability of a topological $\mathbb{N}_{0}$-dynamical system $(Ω,φ)$, where $φ$ is an endomorphism of a metric compact set $Ω$, into ergodic components in terms of the associated enveloping semigroups. In the tame case (where the Ellis semigroup $E(Ω,φ)$ consists of $B_{1}$-transformations $Ω\rightarrow Ω$), we show that (i) the desired decomposition exists for an appropriate choice of the generalized sequential averaging method; (ii) every sequence of weighted ergodic means for the shift operator $x\rightarrow x\circφ$, $x\in C(Ω)$, contains a point…
We study mean convergence results for weighted multiple ergodic averages defined by commuting transformations with iterates given by integer polynomials in several variables. Roughly speaking, we prove that a bounded sequence is a good universal weight for mean convergence of such averages if and only if the averages of this sequence times any nilsequence converge. Key role in the proof play two decomposition results of independent interest. The first states that every bounded sequence in several variables satisfying some regularity conditions is a sum of a nilsequence and a sequence that has …
We prove that if $μ_n$ are probability measures on $Z$ such that $\hat μ_n$ converges to 0 uniformly on every compact subset of $(0,1)$, then there exists a subsequence $\{n_k\}$ such that the weighted ergodic averages corresponding to $μ_{n_k}$ satisfy a pointwise ergodic theorem in $L^1$. We further discuss the relationship between Fourier decay and pointwise ergodic theorems for subsequences, considering in particular the averages along $n^2+ \lfloor ρ(n)\rfloor$ for a slowly growing function $ρ$. Under some monotonicity assumptions, the rate of growth of $ρ'(x)$ determines the existence of…
Let $(X,T)$ and $(Y,S)$ be two subshifts so that $Y$ is a factor of $X$. For any asymptotically sub-additive potential $Φ$ on $X$ and $\ba=(a,b)\in\R^2$ with $a>0$, $b\geq 0$, we introduce the notions of $\ba$-weighted topological pressure and $\ba$-weighted equilibrium state of $Φ$. We setup the weighted variational principle. In the case that $X, Y$ are full shifts with one-block factor map, we prove the uniqueness and Gibbs property of $\ba$-weighted equilibrium states for almost additive potentials having the bounded distortion properties. Extensions are given to the higher dimensional wei…
We study correlation estimates of automatic sequences (that is, sequences computable by finite automata) with polynomial phases. As a consequence, we provide a new class of good weights for classical and polynomial ergodic theorems, not coming themselves from dynamical systems. We show that automatic sequences are good weights in $L^2$ for polynomial averages and totally ergodic systems. For totally balanced automatic sequences (i.e., sequences converging to zero in mean along arithmetic progressions) the pointwise weighted ergodic theorem in $L^1$ holds. Moreover, invertible automatic sequenc…
Motivated by recent investigations of ergodic optimisation for matrix cocycles, we study the measures of maximum top Lyapunov exponent for pairs of bounded weighted shift operators on a separable Hilbert space. We prove that for generic pairs of weighted shift operators the Lyapunov-maximising measure is unique, and show that there exist pairs of operators whose unique Lyapunov-maximising measure takes any prescribed value less than $\log 2$ for its metric entropy. We also show that in contrast to the matrix case, the Lyapunov-maximising measures of pairs of bounded operators are in general no…
