We consider infinite staircase translation surfaces with varying step sizes. For typical step sizes we show that the translation flow is uniquely ergodic in almost every direction. Our result also hold for typical configurations of the Ehrenfest wind-tree model endowed with the Hausdorff topology. In contrast, we show that the translation flow on a periodic translation surface can not be uniquely ergodic in any direction.
For a Z-cover of a translation surface, which is a lattice surface, and which admits infinite strips, we prove that almost every direction for the straightline flow is ergodic.
Continuing the work in \cite{ergodic-infinite}, we show that within each stratum of translation surfaces, there is a residual set of surfaces for which the geodesic flow in almost every direction is ergodic for almost-every periodic group extension produced using a technique referred to as \emph{cuts}.
This preliminary report contains a sketch of the proof of the following result: a slowly divergent Teichmuller geodesic satisfying a certain logarithmic law is determined by a uniquely ergodic measured foliation.
We consider the geodesic flow defined by periodic Eaton lens patterns in the plane and discover ergodic ones among those. The ergodicity result on Eaton lenses is derived from a result for quadratic differentials on the plane that are pull backs of quadratic differentials on tori. Ergodicity itself is concluded for $\mathbb{Z}^d$-covers of quadratic differentials on compact surfaces with vanishing Lyapunov exponents.
In this paper the authors find examples of translation surfaces that have infinitely generated Veech groups, satisfy the topological dichotomy property that for every direction either the flow in that direction is completely periodic or minimal, and yet have minimal but non uniquely ergodic directions.
We give a short proof of the unique ergodicity of the strong stable foliation of the geodesic flow on the frame bundle of a hyperbolic manifold admitting a finite measure of maximal entropy. Equivalently, let G = S0o(n, 1), $Γ$ \textless{} G be a discrete subgroup of G, and G = N AK the Iwasawa decomposition of G. If the geodesic flow on $Γ$\G admits a finite measure of maximal entropy, we prove that the action of N on $Γ$\G by right multiplication admits a unique invariant measure supported on points whose A-orbit does not diverge.
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 extend the notion of rational ergodicity to $β$-rational ergodicity for $β> 1$. Given $β\in \mathbb R$ such that $β> 1$, we construct an uncountable family of rank-one infinite measure preserving transformations that are weakly rationally ergodic, but are not $β$-rationally ergodic. The established notion of rational ergodicity corresponds to 2-rational ergodicity. Thus, this paper answers an open question by showing that weak rational ergodicity does not imply rational ergodicity.
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 ergodicity and consensus problem for a discrete-time linear dynamic model driven by random stochastic matrices, which is equivalent to studying these concepts for the product of such matrices. Our focus is on the model where the random matrices have independent but time-variant distribution. We introduce a new phenomenon, the infinite flow, and we study its fundamental properties and relations with the ergodicity and consensus. The central result is the infinite flow theorem establishing the equivalence between the infinite flow and the ergodicity for a class of independent ran…
In this paper, we consider generalized wind-tree models and $\Z^d$-covers over compact translation surfaces. Under suitable hypothesis, we prove recurrence of the linear flow in a generic direction and non-ergodicity of Lebesgue measure.
There is only one fully supported ergodic invariant probability measure for the adic transformation on the space of infinite paths in the graph that underlies the Eulerian numbers. This result may partially justify a frequent assumption about the equidistribution of random permutations.
We study equivalence relations that arise from translation actions $Γ\curvearrowright G$ which are associated to dense embeddings $Γ<G$ of countable groups into second countable locally compact groups. Assuming that $G$ is simply connected and the action $Γ\curvearrowright G$ is strongly ergodic, we prove that $Γ\curvearrowright G$ is orbit equivalent to another such translation action $Λ\curvearrowright H$ if and only if there exists an isomorphism $δ:G\rightarrow H$ such that $δ(Γ)=Λ$. If $G$ is moreover a real algebraic group, then we establish analogous rigidity results for the translation…
We study the dynamics of a class of interval translation map on three intervals. We show that in this class the typical ITM is of finite type (reduce to an interval exchange transformation) and that the complement contains a Cantor set. We relate our maps to substitution subshifts. Results on Hausdorff dimension of the attractor and on unique ergodicity are obtained.
A simple proof of the fact that each rank-one infinite measure preserving (i.m.p.) transformation is subsequence weakly rationally ergodic is found. Some classes of funny rank-one i.m.p. actions of Abelian groups are shown to be subsequence weakly rationally ergodic. A constructive definition of finite funny rank for actions of arbitrary infinite countable groups is given. It is shown that the ergodic i.m.p. transformations of balanced finite funny rank are subsequence weakly rationally ergodic. It is shown that the ergodic probability preserving transformations of exact finite rank, the ergod…
We show that generic infinite group extensions of geodesic flows on square tiled translation surfaces are ergodic in almost every direction, subject to certain natural constraints. Recently K. Frcaczek and C. Ulcigrai have shown that certain concrete staircases, covers of square-tiled surfaces, are not ergodic in almost every direction. In contrast we show the almost sure ergodicity of other concrete staircases. An appendix provides a combinatorial approach for the study of square-tiled surfaces.
Consider the non-compact billiard in the first quandrant bounded by the positive $x$-semiaxis, the positive $y$-semiaxis and the graph of $f(x) = (x+1)^{-α}$, $α\in (1,2]$. Although the Schnirelman Theorem holds, the quantum average of the position $x$ is finite on any eigenstate, while classical ergodicity entails that the classical time average of $x$ is unbounded.
We discuss multiple versions of rational ergodicity and rational weak mixing for "nice" transformations, including Markov shifts, certain interval maps and hyperbolic geodesic flows. These properties entail multiple recurrence.
In a previous paper we defined a class of non-compact polygonal billiards, the infinite step billiards: to a given decreasing sequence of non-negative numbers $\{p_{n}$, there corresponds a table $\Bi := \bigcup_{n\in\N} [n,n+1] \times [0,p_{n}]$. In this article, first we generalize the main result of the previous paper to a wider class of examples. That is, a.s. there is a unique escape orbit which belongs to the alpha and omega-limit of every other trajectory. Then, following a recent work of Troubetzkoy, we prove that generically these systems are ergodic for almost all initial velocities,…
We study the limiting behavior of a random dynamic system driven by a stochastic chain. Our main interest is in the chains that are not necessarily ergodic but rather decomposable into ergodic classes. To investigate the conditions under which the ergodic classes of a model can be identified, we introduce and study an l_1-approximation and infinite flow graph of the model. We show that the l_1-approximations of random chains preserve certain limiting behavior. Using the l_1-approximations, we show how the connectivity of the infinite flow graph is related to the structure of the ergodic groups…
Any coded subshift X defined by a set C of code words contains a subshift, which we call L, consisting of limits of single code words. We show that when C satisfies a unique decomposition property, the topological entropy h(X) of X is determined completely by h(L) and the number of code words of each length. More specifically, we show that h(X) = h(L) exactly when a certain infinite series is less than or equal to 1, and when that series is greater than 1, we give a formula for h(X_C). In the latter case, an immediate corollary (using results of Climenhaga and Thompson) is that X has a unique …
In this note we give asymptotic estimates for the volume growth associated to suitable infinite graphs. Our main application is to give an asymptotic estimate for volume growth associated to translation surfaces.
Given $\a \in [0,1]$ and $φ: \T \to \R$ measurable, the {\it cylindircal cascade} $S_{\a,φ}$ is the map from $\T \times \R$ to itself given by $S_{\a,φ} (x,y) = (x+\a,y+φ(x))$ that naturally appears in the study of some ordinary differential equations on $\R^3$. In this paper, we prove that for a set of full Lebesgue measure of $\a \in [0,1]$ the cylindrical cascades $S_{\a,φ}$ are ergodic for every smooth function $φ$ with a logarithmic singularity, provided that the average of $φ$ vanishes. Closely related to $S_{\a,φ}$ are the special flows constructed above $R_\a$ and under $φ+c$ where $c …
We study two properties of nonsingular and infinite measure-preserving ergodic systems: weak double ergodicity, and ergodicity with isometric coefficients. We show that there exist infinite measure-preserving transformations that are ergodic with isometric coefficients but are not weakly doubly ergodic. We also give type $\text{III}_λ$ examples of such systems, $0<λ\leq 1$. We prove that under certain hypotheses, systems that are weakly mixing are ergodic with isometric coefficients and along the way we give an example of a uniformly rigid topological dynamical system along the sequence $(n_i)…
We define an infinite measure-preserving transformation to have infinite symmetric ergodic index if all finite Cartesian products of the transformation and its inverse are ergodic, and show that infinite symmetric ergodic index does not imply that all products of powers are conservative, so does not imply power weak mixing. We provide a sufficient condition for $k$-fold and infinite symmetric ergodic index and use it to answer a question on the relationship between product conservativity and product ergodicity. We also show that a class of rank-one transformations that have infinite symmetric …
We give a criterion which allows to prove non-ergodicity for certain infinite periodic billiards and directional flows on Z-periodic translation surfaces. Our criterion applies in particular to a billiard in an infinite band with periodically spaced vertical barriers and to the Ehrenfest wind-tree model, which is a planar billiard with a $Z^2$-periodic array of rectangular obstacles. We prove that, in these two examples, both for a full measure set of parameters of the billiard tables and for tables with rational parameters, for almost every direction the corresponding billiard flow is not erg…
In this paper we provide a necessary and sufficient condition for the amenability of the automorphism group of Fraïssé structures and apply it to prove the non-amenability of the automorphism groups of the directed graph $\mathbf{S}(3)$ and the Boron tree structure $\mathbf{T}$. Also, we provide a negative answer to the Unique Ergodicity-Generic Point problem of Angel-Kechris-Lyons [AKL]. By considering $\mathrm{GL}(\mathbf{V}_\infty)$, where $\mathbf{V}_\infty$ is the countably infinite dimensional vector space over a finite field $F_q$, we show that the unique invariant measure on the univer…
The Gauss-Bonnet formula for classical translation surfaces relates the cone angle of the singularities (geometry) to the genus of the surface (topology). When considering more general translation surfaces, we observe so-called wild singularities for which the notion of cone angle is not applicable any more. We study whether there still exist relations between the geometry and the topology for translation surfaces with wild singularities. By considering short saddle connections, we determine under which conditions the existence of a wild singularity implies infinite genus. We apply this to sho…
In this paper, we compare two definitions of Rauzy classes. The first one was introduced by Rauzy and was in particular used by Veech to prove the ergodicity of the Teichmüller flow. The second one is more recent and uses a "labeling" of the underlying intervals, and was used in the proof of some recent major results about the Teichmüller flow. The Rauzy diagrams obtained from the second definition are coverings of the initial ones. In this paper, we give a formula that gives the degree of this covering. This formula is related to moduli spaces of "framed" translation surfaces, which correspon…
The Ergodicity of Geodesic Flows on Rank One Manifolds of Nonpositive Curvature
College of Mathematics and Systems Science, Shandong University of Science and Technology（山东科技大学数学与系统科学学院）；Department of Mathematics, Southern University of Science and Technology（南方科技大学数学系）
We consider smooth area-preserving flows (also known as locally Hamiltonian flows) on surfaces of genus $g\geq 1$ and study ergodic integrals of smooth observables along the flow trajectories. We show that these integrals display a \emph{power deviation spectrum} and describe the cocycles that lead the pure power behaviour, giving a new proof of results by Forni (Annals 2002) and Bufetov (Annals 2014) and generalizing them to observables which are non-zero at fixed points. This in particular completes the proof of the original formulation of the Kontsevitch-Zorich conjecture. Our proof is base…
