Centralizer and liftable centralizer of special flows over rotations

The liftable centralizer for special flows over irrational rotations is studied. It is shown that there are such flows under piecewise constant roof functions which are rigid and whose liftable centralizer is trivial

Limit law for some modified ergodic sums

An example due to Erdos and Fortet shows that, for a lacunary sequence of integers (q_n) and a trigonometric polynomial f, the asymptotic distribution of normalized sums of f(q_k x) can be a mixture of gaussian laws. Here we give a generalization of their example interpreted as the limiting behavior of some modified ergodic sums in the framework of dynamical systems

Remarks on step cocycles over rotations, centralizers and coboundaries

By using a cocycle generated by the step function $\varphi_{\beta, \gamma} = 1_{[0, \beta]} - 1_{[0, \beta]} (. + \gamma)$ over an irrational rotation $x \to x + \alpha \mod 1$, we present examples which illustrate different aspects of the general theory of cylinder maps. In particular, we construct non ergodic cocycles with ergodic compact quotients, cocycles generating an extension $T_{\alpha, \varphi}$ with a small centralizer. The constructions are related to diophantine properties of $\alpha, \beta, \gamma$

On multiple ergodicity of affine cocycles over irrational rotations

Let T_\alpha denote the rotation T_{\alpha}x=x+\alpha (mod 1) by an irrational number \alpha on the additive circle T=[0,1). Let \beta_1,..., \beta_d be d\geqslant 1 parameters in [0, 1). One of the goals of this paper is to describe the ergodic properties of the cocycle (taking values in R^(d+1)) generated over T_\alpha by the vectorial function \Psi_{d+1}(x):=(\phi(x), \phi(x+\beta_1),..., \phi(x+\beta_d)), with \phi(x)={x}-1/2. It was already proved in \cite{LeMeNa03} that \Psi_{2} is regular for \alpha with bounded partial quotients. In the present paper we show that \Psi_{2} is regular for any irrational \alpha. For higher dimensions, we give sufficient conditions for regularity. While the case d=2 remains unsolved, for d=3 we provide examples of non-regular cocycles \Psi_{4} for certain values of the parameters \beta_1,\beta_2,\beta_3. We also show that the problem of regularity for the cocycle \Psi_{d+1} reduces to the regularity of the cocycles of the form \Phi_{d} =(1_{[0, \beta_j]} - \beta_j)_{j= 1, ..., d} (taking values in R^d). Therefore, a large part of the paper is devoted to the classification problems of step functions with values in R^{d}.Comment: 34 pages; revisions in response to referees' comment

Griechische Kohlenbecken

Caption title.Offprint: Jahrbuch des Kaiserlich Deutschen Archäologischen Instituts, Band V, zweites Heft.Mode of access: Internet

The Invariant Measures of some Infinite Interval Exchange Maps

We classify the locally finite ergodic invariant measures of certain infinite interval exchange transformations (IETs). These transformations naturally arise from return maps of the straight-line flow on certain translation surfaces, and the study of the invariant measures for these IETs is equivalent to the study of invariant measures for the straight-line flow in some direction on these translation surfaces. For the surfaces and directions for which our methods apply, we can characterize the locally finite ergodic invariant measures of the straight-line flow in a set of directions of Hausdorff dimension larger than 1/2. We promote this characterization to a classification in some cases. For instance, when the surfaces admit a cocompact action by a nilpotent group, we prove each ergodic invariant measure for the straight-line flow is a Maharam measure, and we describe precisely which Maharam measures arise. When the surfaces under consideration are finite area, the straight-line flows in the directions we understand are uniquely ergodic. Our methods apply to translation surfaces admitting multi-twists in a pair of cylinder decompositions in non-parallel directions.Comment: 107 pages, 11 figures. Minor improvement

The different effect of consumer learning on incentives to differentiate in Cournot and Bertrand competition

We combine two extensions of the differentiated duopoly model of Dixit (1979), namely Caminal and Vives (1996) and Brander and Spencer (2015a,b), to analyze the effect of consumer learning on firms' incentives to differentiate their products in models of Cournot and Bertrand competition. Products are of different quality, consumers buy sequentially and are imperfectly informed about the quality of the goods. Before simultaneously competing in quantities, firms simultaneously choose their investment into differentiation. Late consumers can observe earlier consumers' decisions and extract information about the quality of the goods. This influences the firms' incentives to differentiate. If firms compete in quantities, they are more likely to invest in differentiation with consumer learning than without. This is in line with implications of the recommendation effect introduced in Conze and Kramm (2016) in a model of spatial differentiation. We also examine the case in which firms compete in prices. Here, the effect of consumer learning is reversed, so that differentiation is less likely with consumer learning. Thus, we find an information-based difference between Cournot and Bertrand competition: in the Bertrand setting consumer learning increases the competition, i.e. products are more likely to be substitutes, and it weakens it in the Cournot model.Wir kombinieren zwei Erweiterungen des Duopol Models mit differenzierten Gütern von Dixit (1979), genauer die von Caminal und Vives (1996) und die von Brander und Spence (2015a,b), um den Effekt von (Bayesianisch) lernenden Konsumenten auf Anreize der Firmen zur Produktdifferenzierung im Cournot und im Bertrand Wettbewerb zu analysieren. Die Produkte haben eine unterschiedliche Qualität, Konsumenten kaufen sequentiell und sind unvollständig über die Produktqualität informiert. Bevor die Firmen simultan über ihre produzierten Mengen entscheiden, wählen sie simultan ihre Investitionen in Produktdifferenzierung. "Spätere" Konsumenten können die Konsumentscheidungen "früherer" Konsumenten beobachten und so Information über die Produktqualität erhalten, also Bayesianisch lernen. Im Cournot Wettbewerb mit lernenden Konsumenten ist es wahrscheinlicher, dass Firmen in Produktdifferenzierung investieren, als ohne. Dieses Ergebnis stimmt überein mit dem Wirken des "Recommendation Effects", welcher in Conze und Kramm (2016a) gezeigt und analysiert wird. Wir untersuchen auch den Bertrand Wettbewerb. Hier ist der Effekt von Konsumentenlernen umgekehrt, so dass Differenzierung mit Konsumentenlernen weniger wahrscheinlich ist also ohne. Wir finden demnach einen informationsbasierten Unterschied zwischen Cournot und Bertrand Wettbewerb: im Bertrand Modell erhöht Konsumentenlernen den Wettbewerb, das heißt Produkte sind mit lernenden Konsumenten eher Substitute als ohne, und im Cournot Modell schwächt es ihn ab

Ergodicity for Infinite Periodic Translation Surfaces

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