Lukas Weingartner

Peter Hehenberger

Michael Friedl

Andreas Kellner

Stefan Boschert

Roland Rosen

A Lightweight Approach to Manage Engineering Parameters in Mechatronic Design Processes

Keywords: Mechatronic design process, Engineering parameter management, Parameter database, Data exchange process, Model management

In mechatronic design processes the exchange of information between stakeholders from different disciplines is essential to enable simultaneous engineering and a successful integration of domain specific subsystems. Although there is a comprehensive range of methodologies and tools to support collaboration, intentions to implement a central data management platform covering all stakeholders often do not succeed. Reasons are the heterogeneous model landscape, the variety of stand-alone authoring tools, departments', disciplines' or companies' boundaries and a lack of flexibility of established solutions regarding the support of unpredictable and quickly changing design processes. This paper focuses on the management of individual parameters and parameter instances (values) within a multidisciplinary development team. The presented lightweight approach can extend existing methods and data management infrastructure by adding functionalities to provide access to individual parameters using a dedicated database. The procedure is shown by the example of a technical mechanism.

Introduction

During the course of the multi-firm project SyMMDe [START_REF]SyMMDe -System Models for Mechatronic Design, Multi-Firm-Project[END_REF], mechatronic design processes of four different companies were analyzed in detail and presented in the form of model dependency maps [START_REF] Friedl | Model Dependency Maps for transparent concurrent engineering processes[END_REF]. From this point of view, every model needs input parameters and generates results (e.g., output parameters in various forms) for a particular purpose. Heterogeneous models on various hierarchical levels are usually created and implemented within specific authoring systems (e.g., CAx-tools) by different stakeholders and experts (e.g., from the disciplines of mechanics, electronics and software). In most cases, models (e.g., parametric geometry models) are stored using a kind of file-based model storage. There is a wide range of software solutions (e.g., file systems, "PDM -product data management" systems, "ECM -enterprise content management" systems) to manage (mostly proprietary) files within a central platform covering essential functionalities like versioning, access management, configuration management, and much more. For pragmatic reasons like performance advantages and interoperability, individual model storage systems can be settled locally at the expert's computer or close to the authoring systems (e.g., team data management). From our observations, there is usually no overall (e.g., companywide) consistent model backbone for all kinds of models. Although tailor-made configured PLM systems can meet these challenges in principle, in many cases their potential has not yet been exhausted, since only a limited scope (e.g., PDM aspects) of possible functionalities is used, without covering all stakeholders and models.

During mechatronic modeling processes, different types of parameters appear. As discussed in [START_REF] Hehenberger | Meta-model of PLM for Design of Systems of Systems[END_REF], a discipline-specific component can consist of a user-defined part described by design parameters and a standardized part described by configuration parameters. In addition to product descriptive parameters (including, for example, process parameters in process models) there are many other parameters that are relevant for the various discipline-specific models (e.g., regarding rules and knowhow, or solver parameters and settings in simulation processes). The subset of parameters that influence more than one stakeholder or rather multiple models play a significant role across the product development process. It is essential that all stakeholders gain a common understanding of these important parameters. Globally unique and therefore complex names and descriptions of all parameters used are usually inconvenient. Experts from the different disciplines naturally have varied interpretations and views at the system and demand their own naming conventions. If the parameter management concept fails to unify these different views or to regulate access on the parameters, redundancies and consistency problems may follow. To achieve a close integration of domain specific subsystems, domain experts with diversified knowledge and skills often are in typical conflicts of interests. Conflicting objectives and targets concerning subsystems force stakeholders to negotiate about parameters (e.g., available design space). Therefore, parameter and parameter values can change several times due to improvements or coordination between stakeholders. Such engineering changes can have various triggers (e.g., unforeseeable events) and arise throughout the whole design process [START_REF] Abramovici | A Knowledge-based Assistant for Real-time Planning and Execution of PSS Engineering Change Processes[END_REF]. To enable an integration of components and solutions developed from several disciplines, substantial main parameters must be defined and made available to all stakeholders. Lacking access to a central parameter management system often leads to mainly spread, unregulated, and not standardized information exchange (e.g., using telephone, post-it notes, social media, email or additional stand-alone groupware or project management applications). Despite the availability of PLM tools, one commonly used mostly manual and file-based method is to manage parameters in rather simple lists with ordinary spreadsheet tools. Although this is a proper solution for documentation purposes, problems occur if multiple users have frequent change requests, as in the case of usual iterations during design processes. Fig. 1 shows a comparison of the basic mechanisms with a central database (type 3) and without (type 1 and 2).

The aim of this paper is to introduce and implement a methodology that covers the following needs and requirements related to parameter management as observed during typical interdisciplinary development processes at our industrial examples: (i) Access to individual parameters across multiple models (e.g., from proprietary files), without having to use the respective authoring system, e.g., by different stakeholders across multiple departments/locations. (ii) Multi-user access including rights management. (iii) Management of parameters including capabilities to keep an overview by using metadata and attributes or functionalities like filtering and highlighting of major and product defining parameters. (iv) Representation of relationships between parameters and possibilities to analyze impacts in the case of changes. (v) Traceability and history of changes including the possibility to add knowledge regarding decision making (e.g., rules) and documentation about decisions made (e.g., implicit assumptions, findings, reasons for occurring iterations). (vi) Several models (e.g., on different hierarchical levels created in specific authoring tools) should be able to access individual parameters using simple adapters and interfaces without the need of an additional overall authoring tool. (vii) Widespread introduction of the parameter management system should be possible, e.g., by using thin clients or web interfaces. (viii) Ability to extend and customize the platform using standardized tools or common programming languages. (ix) No adverse impacts and limitations on the established development process and flexibility to deal with highly dynamic changes within the development process.

Discussion of established methodologies and tool concepts

Badin et al. [START_REF] Badin | Knowledge Configuration Management for Product Design and Numerical Simulation[END_REF] present a "KCModel" methodology to "Capitalize, Trace, Re-use and ensure the Consistency" of technical data using centralized knowledge management. Parameters are shared between different models and a consistency management (e.g., expert rules) between configurations can detect conflicts in parameter values. Other concepts and tools to support collaborative design based on web technologies were already introduced back in 2002 by Riboulet et al. [START_REF] Riboulet | Tools for Dynamic Sharing of Collaborative Design Information[END_REF]. The presented application "CoDISS" (cooperative data & information sharing system) enables communication (e.g., sharing of knowledge, concepts and related parameters) between various models and different stakeholders [START_REF] Vu-Thi | Integrating Product Model and Whiteboard to Ease Collaborative Work in Global Product Development[END_REF].

Although it's difficult to draw a clear line in between, there are mainly three categories of established tool concepts that are used successfully in the industry to manage product data in a central repository accessible to users and experts from various disciplines. Similar differentiations are discussed in the work of Panchal et. al. [START_REF] Panchal | A Modular Decision-centric Approach for Reusable Design Processes[END_REF][START_REF] Panchal | Facilitating Meta-Design via Separation of Problem, Product, and Process Information[END_REF]. Differentiations between PDM and "SDM -Simulation Data Management" [START_REF] Badin | Knowledge Configuration Management for Product Design and Numerical Simulation[END_REF] as well as weaknesses regarding management of fine grained parameters are also discussed in [START_REF] Penciuc | Towards a PLM Interoperability for a Collaborative Design Support System[END_REF]. Each of the following three options has strengths and weaknesses in different applications.

The first option ("model backbone") are IT systems that focus on file-based product data management to enable collaboration between the stakeholders. There are powerful options to classify and structure the files by using metadata or structural modeling, but their focus is not to manage fine grained information like individual parameters within the files. The most commonly used concepts and tools can be grouped under the headings ECM, "DMS -document management system" and PDM as a part of "PLM -product lifecycle management". Prominent representatives in the field of mechatronic engineering tools are, e.g., Siemens Teamcenter ® [START_REF] Siemens Lms Imagine | Lab™ System Synthesis software, NX™ software and Teamcenter ® software[END_REF] or Dassault Systèmes ENOVIA ® [START_REF]Enovia software[END_REF]. PLM systems (as an extension of PDM systems) even provide possibilities to extract and exchange fine-grained information and parameters from files, e.g., to initiate and execute customized and purpose-built workflows or simulation processes. Some authoring systems can be seamlessly integrated including bidirectional associativity of attributes and parameter values. However, the common fundamental PLM tool concepts are not designed or intended to support highly dynamic engineering processes of different models on a parameter level. Functionalities to analyze changes of parameter values and their impact within the model landscape are the aim of additional tools (e.g., requirements management and systems engineering applications), that can be connected to or installed within the PLM system. Clearly defined processes -even on parameter-level -are wellsupported by tool adapters and workflows customized by PLM experts. To set up quick changing processes (e.g., experimental simulation processes) domain experts need ability and flexibility to implement specific processes by their own, e.g., using an end-user-friendly interface.

This leads to a second option ("model-based workflow management"), that is provided by systems and tools that focus on models and parameter exchange to realize workflows. So-called "PIDO -process integration and design optimization" frameworks or SDM solutions have to deal with a huge amount of individual parameters. Within a framework workflows consisting of interconnected (mainly external) models can be defined and executed. This means that parameters from different sources can be connected and exchanged within the tool environment. In the area of simulation processes, the second option is well suited for a limited number of users. The central simulation process model within the framework (usually a proprietary system platform) allows to run preconfigured simulations and calculations including parameter studies or optimization. The simulations can run at the local machine, without the need to connect to a server, which causes improved performance in some cases. Prominent simulation environments like Siemens LMS Imagine.Lab™ Amesim with Sysdm [START_REF] Siemens Lms Imagine | Lab™ System Synthesis software, NX™ software and Teamcenter ® software[END_REF] or MathWorks ® MATLAB ® Simulink ® [START_REF]MathWorks® software[END_REF] provide extensive possibilities to manage parameters within the associated environment.

A third option ("descriptive system model") is to build a model that contains essential knowledge (parameters, relationships, rules). Such approaches can be implemented using (model based) systems engineering methodologies and modelling languages (e.g., SysML) [START_REF] Sakairi | Model Based Control System Design Using SysML, Simulink, and Computer Algebra System[END_REF]. A major advantage of general-purpose modeling languages like SysML is the standardization, which is a basis for a tool-neutral integration. A well-known problem is that systems engineering tools follow a generic approach and must be adapted to fit concrete applications. Like a survey [START_REF] Bone | The Current State of Model Based Systems Engineering: Survey Results from the OMG™ SysML Request for Information[END_REF] pointed out, the large learning curve to understand SysML is a large inhibitor. In traditional engineering disciplines, only few domain experts have sufficient knowledge in this kind of modeling, mainly inspired by computer science (e.g., UML). From our observations, the role of systems engineers is not widespread introduced in industry at the present time. In the recent past, tool vendors developed modeling applications that enable analysis and execution of diagrams, e.g., SysML parametric diagrams, including expressions in form of constraints [START_REF] Sakairi | Model Based Control System Design Using SysML, Simulink, and Computer Algebra System[END_REF] and management of instances.

As a conclusion, there are many individual solutions but no universal parameter management approach, which meets all the requirements as stated in chapter 1. The objective was now to develop a powerful but not overloaded ("lightweight") approach, that can be adapted and used productively with reasonable implementation effort, e.g., as a supplement to established design processes.

Generic methodology for lightweight parameter management

Bearing in mind that parameters only get a meaning and a significance in conjunction with models, the following approach is based on a separation of models and parameters and values. There are advantages (e.g., regarding transparency and simplified access to parameters) if models and parameters are considered in separate terms and also get technically implemented separately (e.g., in different databases). As mentioned above, models are nowadays managed successfully using different methodologies and tools (e.g., PLM systems) within a "model backbone". In addition, to manage fine grained parameters, a "parameter backbone" is introduced. In contrast to other approaches, already existing models do not have to be changed and there is only one additional data model necessary, but no centralized knowledge configuration model or dependency model between parameters. Any information (e.g., functional or mathematical relations) how OUT-parameters are generated from the IN-parameters are provided by existing models.

Fig. 2 describes the basic methodology by the example of two models that are created with different (non-integrated) authoring systems. Once stakeholder 1 creates "model 1" using a specific tool, the model is transferred to the "model backbone" (e.g., a common model management system). In addition to this task, the essential parameters are transferred to a separate "parameter backbone" (e.g., by manual input or by using tailored interfaces). These essential parameters can be identified by the fact, that they are generated as (interim) results for a particular purpose and they are used several times as inputs for different models. If parameters appear only once within the development process (e.g., as an input for one downstream model within a seamless tool chain), the central storage may be not appropriate. For every parameter supplementing attributes (e.g., description, context information) need to be specified. Dependent on the type of every attribute, that user task can also be achieved either manually, (semi-)automatically or supported by using functionalities of the parameter backbone (e.g., DBMS "database management system") like (automatic) creation of unique IDs, access and change logs, or versioning. One important attribute that needs to be defined is which model generated the parameter. If stakeholder 2 starts creating "model 2" using another authoring system, the parameter management system provides access to already entered parameters. It is essential, that parameters can be found and identified due to unique attributes. There are supporting functionalities to retrieve the desired parameters, like using filtering by multiple criteria such as parameter name, description, creation date, creator, owner and/or many other context information. To establish the link between "model 2" and every required parameter, attributes are set. Every user of the parameter backbone now has the information, who (which model) generates and which models utilize the parameters. This information is helpful to estimate effects on other models in case of parameter changes. If "model 2" generates further essential parameters, they can be taken into account in the same way.

Fig. 2. Example how to use the introduced methodology in case of two models created by two stakeholders using different authoring tools.

To implement the parameter management approach (e.g., in an enterprise), a welldesigned data model is a prerequisite for a transparent and performant implementation (e.g., within a DBMS). To develop the data model, at first an overview of the parameters and parameter types to be managed must be created. Parameters (and parameter values) are usually spread and redundantly stored at different locations within physical files, that contain a particular selection (view) of parameters. Pragmatic reasons for the occurrence of such parameter sets can be: (i) Software tools need certain machine-readable file formats that can be imported/exported. (ii) Sets contain all parameters that belong to a model. (iii) Sets contain parameters structured by areas of responsibility (e.g., due to expert knowledge or organizational reasons). (iv) Sets contain all relevant parameters that relate to a subsystem (e.g., parameters that are relevant to production). If parameters appear multiple times within different files, all influences have to be considered in the database concept. At the implementation of a central parameter backbone, required views can be provided using adequate attributes to allow grouping or filtering. As an example, there are 10000 essential parameters necessary to describe a certain machine tool. Thereof a set of 50 parameters belong to an electric motor of a drivetrain (e.g., type, manufacturer, power, weight, nominal speed). Various stakeholders from mechanical, electrical and automation departments need different views at these motor parameters. While some of the parameters are created and solely used by one department, other parameters (e.g., like the nominal speed) play an important role for all stakeholders. At this point, the approach brings enormous advantages, since all departments have access to current parameter data (following specified rules) including history and transparency (regarding affected stakeholders and dependent models). To sort or filter parameters, additional attributes like responsible department, project name, assembly can be defined easily. In case of using a relational database for the implementation, there are significant advantages in terms of consistency, transparency and data integrity. Relations between separate tables containing sets of essential parameters (e.g., common motor parameters) avoid data duplication and make changes easier.

Example: Implementation of a parameter management system

The presented methodology is currently under test in industrial case studies in the field of mechatronic product development. The parameter management system is implemented using MySQL™ [16] as a DBMS and Visual Studio [START_REF]Microsoft Visual Studio software[END_REF] VBA programming to build a graphical user interface for user login, manual parameter/attribute input or manipulation and to visualize database queries. In following example of the development of a mechanism, there are many models used by different stakeholders from different departments. A first stakeholder uses a model to perform a two dimensional multi criteria optimization of a mechanism using MATLAB ® [START_REF]MathWorks® software[END_REF]. The output parameters of this model are characteristic dimensions (e.g., L 1 =350mm … length of connection rod) that can be stored within the database using the ODBC API. In addition to every parameter, further corresponding attributes must be defined by using the following relations: "Model" (list and description of models used), "Units", "Projects", "Visualization" (e.g., a sketch to show the context information of a parameter or even a lightweight representation for example based on a JT visualization format [START_REF] Ding | Product representation in lightweight formats for product lifecycle management (PLM)[END_REF]), "Person" (general user management) and "Status" (e.g., to indicate a release status of a parameter). These relations are shown in Fig. 3 with an exemplary data model. Based on the created database information, a second stakeholder is able to identify and reuse essential parameters to parameterize a CAD representation of the mechanism. This three-dimensional model is used to perform a collision check using the advanced capabilities of the Siemens NX™ CAD system. Input-parameters are loaded semi-automatically by the help of NX journaling functionalities to import expressions. Solutions of the collision check are also stored as parameters within the database, since they are important for other stakeholders. This means, every user of the parameter backbone has access to all actual parameter values, even without having access to the discipline-specific models (e.g., 3D-CAD model), and without the need to open the models within specific authoring systems.

Fig. 4 shows the prototype of the graphical user interface to create and manipulate individual parameters like the displayed parameter "L1". In the lower area, a database viewer shows the query results depending on actual filter settings. The button "load model list" gives access to a separate list of models, that can be linked to the parameter. Write access (e.g., to parameter values) can be limited (e.g., to the "owner"/creator of the parameter) by generating access permissions, e.g., using the MySQL Access Privilege System and/or GUI programming.

Summary and Conclusion

The presented approach essentially fulfills the initial requirements regarding parameter management, as described in chapter 1. The exemplary implementation is done using a DBMS. Interfaces between models (authoring tools) and the central database are set up using several APIs. Although, the attempts to keep it lightweight lead to deliberate limitations regarding traceability. The approach shows direct effects to subsequent models in case of parameter value changes, but effects over several models in series are not displayable, since all the information about parameter correlations remain within the models. Nevertheless, first implementations showed that the presented methodology brings transparency into multi-disciplinary development processes. Several users get access to parameter values and receive now additional information about relationships between parameters and models considering responsibilities of stakeholders, after a reasonably small effort in implementation. A difficulty that has to be addressed in future work is to keep the parameter values synchronized, also considering release status and access permissions. It is not always possible to design models in a way that they can handle input parameters or directly link to the parameter database. As a result, parameter values are stored within the models that are probably not up-to-date. Here a notification mechanism could be implemented, to inform the responsible stakeholder to update the model in case of relevant parameter changes.

Fig. 1 .

 1 Fig. 1. This example shows three models (f 1 , f 2 , f 3 ) and three basic types of information exchange (parameter values of a, b, x, y) between different stakeholders and authoring tools: (I) Manual exchange of parameter values. (II) File-based exchange of parameter values. (III) Parameter value exchange using a central repository with an additional graphical user interface.

Fig. 3 .

 3 Fig. 3. Data model as a description of individual parameters and assigned attributes using relations (crow's foot database notation).

Fig. 4 .

 4 Fig. 4. Prototype of a graphical user interface to access the parameter backbone.

Acknowledgments. This work has been supported by the Austrian COMET-K2 programme of the Linz Center of Mechatronics (LCM), and was funded by the Austrian federal government and the federal state of Upper Austria.


Damien Thomine

Keplerian shear in ergodic theory

come

Figure 1: Equirepartition of a cloud of dust in Saturn's rings. On the left: the cloud (thick black line) at initial time. In the middle: the same cloud, after 6 hours. On the right: the same cloud, after 48 hours.

Keplerian shear is a more general feature of many integrable Hamiltonian dynamical systems. Using action-angle coordinates, the phase space is foliated by invariant Lagrangian tori, and the dynamics of a point belonging to the phase space is conjugate to a translation on one of these tori. Provided that the translations on the Lagrangian tori are (in some sense) asynchronous, the dynamics shear the transversals to the invariant tori, so that in large time, densities equidistribute along the tori. In the case of planetary rings, the invariant tori are orbits of given radius, and the asynchronicity comes from the variation of the orbital period: we recover classical Keplerian shear.

Figure 2: Propagation of a wavefront at unit speed in a unit square torus. The wave starts from the corner, and propagates at unit speed. On the left: the wavefront at time 0.5. In the middle: the wavefront at time 10. On the right: the wavefront at time 500.

Other systems with Keplerian shear are the geodesic flow on a flat torus (see Fig 2), or the dynamics of a ball bouncing in a square box.

In this article, we frame Keplerian shear in the more general context of ergodic theory, as a conditional version of the notion of strong mixing. Definition 0.1 (Keplerian shear).

A dynamical system (Ω, µ, (g t ) t∈R ) which preserves a probability measure is said to exhibit Keplerian shear if, for all f ∈ L 2 (Ω, µ),

lim t→+∞ f • g t = E µ (f |I), (0.1)
where I is the invariant σ-algebra and the convergence is for the weak topology on L 2 (Ω, µ).

Recall that a system (Ω, µ, (g t ) t∈R ) is mixing if and only if, for any function f ∈ L 2 (Ω, µ),

lim t→+∞ f • g t = Ω f dµ = E µ (f ),
where the limit is taken in the weak topology on L 2 (Ω, µ), so a system (Ω, µ, (g t ) t∈R ) is mixing if and only if it is ergodic and exhibits Keplerian shear. As such, Keplerian shear is a conditional version of the notion of strong mixing. Informally, if the system restricted to its invariant subsets is mixing, then (Ω, µ, (g t ) t∈R ) has Keplerian shear. The interesting examples occur when these restrictions are ergodic, but not mixing: that is the case, for instance, of translation flows on a torus. In this article, we give a criterion ensuring Keplerian shear for a large class of such systems; for instance, one of our result is: Proposition 0.2 (Corollary of Theorem 2.3 and Proposition 2.4). Let M be a Riemannian manifold, d ≥ 1 and k ∈ [1, ∞]. Let v ∈ C k (M, R d ), and put g t (x, y) := (x, y + tv(x)) for (x, y) ∈ M × T d . If:

Vol M (∇ ξ, v ) = 0 ∀ξ ∈ Z d \ {0},
then (M × T d , Vol M ⊗ Leb T d , (g t )) exhibits Keplerian shear. Moreover, the criterion above is satisfied for a generic v ∈ C k (M, R d ).

We also study the rate of decay of conditional covariance for the geodesic flow on T 1 T d , and give non-trivial examples of non-Hamiltonian systems with Keplerian shear.

Keplerian shear for the geodesic flow on the flat torus is related to two famous problems. The first is Landau's damping for plasma dynamics on a torus (see Landau's article [START_REF] Landau | On the vibrations of the electronic plasma[END_REF], and [START_REF] Mouhot | On Landau damping[END_REF]Theorem 3.1] for a version which follows closely our formalism), where the effect is qualitatively similar, although the underlying mechanism is different. The second is Gauss's circle problem, which consists in counting integral points in a large disc; we shall discuss it in Sub-subsection 2.4.2. The methods used to tackle these problems are either through Fourier transform (e.g. for Landau damping), or with a big arc/small arc decomposition (typical for Gauss's circle problem). While both work in our setting, we shall only use the Fourier transform.

In the context of ergodic theory, a notion closely related with Keplerian shear was used independently by F. Maucourant [START_REF] Maucourant | Unique ergodicity of asynchronous rotations, and application[END_REF] to prove that the some hyperbolic actions on (R d SL d (R)) /(Z d SL d (Z)) are ergodic for a large class of measures. The presentation in [START_REF] Maucourant | Unique ergodicity of asynchronous rotations, and application[END_REF] is however very different, as the phenomenon -named asynchronicity -is described as a version of unique ergodicity for measures with prescribed marginals.

Organization of the article

Section 1 gives general results on the notion of Keplerian shear (including equivalences between distinct definitions), and gives us some tools to use for the remainder of the article.

Section 2 deals with a first family of systems which may exhibit Keplerian shear: fibrations by tori, where the flow acts by translation on each torus. using action-angle coordinates, this family includes integrable Hamiltonian flows. We give an explicit criterion ensuring Keplerian shear, check that it is C r -generic (r ≥ 1) and satisfied for some explicit systems, then give rates of convergence for the geodesic flow on T 1 T n . We also detail the link between Keplerian shear and the unique ergodicity as investigated in [START_REF] Maucourant | Unique ergodicity of asynchronous rotations, and application[END_REF].

Section 3 deals with another family of dynamical systems (roughly, "fibrations by suspension flows"), which includes many non-Hamiltonian examples, and uses a different mechanism to ensure Keplerian shear.

The shorter Section 4 gives examples of systems without Keplerian shear.

A note on the terminology

Given that Keplerian shear is a conditional version of the notion mixing, one could want to use a terminology such as conditional (strong) mixing. We prefer to eschew this option, and to keep the name of Keplerian shear; indeed, we think that otherwise the name of conditional (strong) mixing would be overloaded.

Indeed, in probability theory, there are already multiple notions of conditional mixing; compare for instance [START_REF] Rao | Conditional independence, conditional mixing and conditional association[END_REF] (where it refers to conditional α-mixing) and [START_REF] Kacem | Some mixing properties of conditionally independent processes[END_REF], among others.

More worryingly, in ergodic theory, the notion of conditionally weakly mixing systems is wellestablished (see e.g. [START_REF] Tao | Lecture 14: Weakly mixing extensions[END_REF]), but if one where to conceive a notion of conditional strong mixing along this line, the resulting notion would be stronger than Keplerian shear, essentially requiring that almost every subsystem in its ergodic decomposition be mixing.

Open problems

We sum up here some further leads which seem worth pursuing. The setting of Section 2 covers integrable Hamiltonian systems. However, it requires some regularity, and in particular it does not cover singular systems. A conjecture by Boshernitzan asserts that given a compact translation surface S, the geodesic flow on (T 1 S, Liouv) exhibits Keplerian shear. This question, mentioned as illumination by circles, also appears in [START_REF] Monteil | Illumination dans les billards polygonaux et dynamique symbolique[END_REF], and admits a partial answer by J. Chaika and P. Hubert [START_REF] Chaika | Circle averages and disjointness in typical flat surfaces on every Teichmüller disc[END_REF], where the convergence of Cov(f, g • g t |I) to zero is shown along a density 1 subsequence for all continuous observables f and g1 .

In Subsection 2.5, we investigate the speed of Keplerian shear for the geodesic flow on T 1 T n . The problem is simplified by the particularities of the geometry of the sphere, more precisely the fact that its principal curvatures do not vanish. What would the speed of convergence be if the curvature vanishes (e.g. in a topologically or measure-theoretically generic setting)?

Finally, while the settings of Sections 2 and 3 are distinct, it could be that they are a special case of a more general structure. A natural candidate would be spaces fibrated by suspension tori, but we need new tools to prove Keplerian shear (or even to get a description of the invariant σ-algebra I).

General properties of Keplerian shear

The following lemma from basic functional analysis is quite useful to prove the ergodicity and mixing of any given dynamical system, and will be instrumental in the remainder of our article.

Lemma 1.1.

Let B be a Banach space. Let (T t ) t≥0 be a family of operators on B, such that sup t∈R + T t B→B < +∞. Let T be an operator on B.

Let E and E * be subsets of B and B * respectively, whose span is dense in their respective space. Assume that, for all f ∈ E and g ∈ E * , lim t→+∞ g, T t f = g, T f .

(1.1)

Then (T t f ) t≥0 converges weakly to T f for all f ∈ B.

Proof.

By bilinearity, Equation (1.1) holds for all f ∈ span(E) and g ∈ span(E * ).

Since sup t∈R + T t B→B < +∞, the family of functions T t : B * × B → C is locally equicontinuous, and by the remark above, it converges to T on a dense subset. Hence, the convergence of Equation (1.1) holds for all f ∈ B and g ∈ B * .

When we use Lemma 1.1, the operator T t shall correspond to the composition by the flow g t at time t, and the operator T to the projection f → E(f |I). Since the flow is assumed to preserve the measure, for all t ≥ 0 and all p ∈ [1, +∞], the operator T t acting on L p (Ω, µ) is unitary. Lemma 1.1 implies that to prove the Keplerian shear in one of those Banach space B (potentially different from L 2 ), it is enough to restrict ourselves to subsets E of B and E * of B * whose linear span is dense. As a first consequence, in the definition of Keplerian shear, one may replace L 2 by L p for any p ∈ [1, +∞):

Proposition 1.2.
Let (Ω, µ, (g t ) t∈R ) be a flow which preserves a probability measure. Let I be the invariant σ-algebra of the system. Then there is equivalence between:

• There exists p ∈ [1, +∞) such that, for all f ∈ L p (Ω, µ), we have f • g t → E(f |I) weakly in L p .

• The system exhibits Keplerian shear.

• For all p ∈ [1, +∞), for all f ∈ L p (Ω, µ), we have f • g t → E(f |I) weakly in L p .

Proof. We only prove the non-trivial implication. Let p ∈ [1, +∞). Assume such that, for all f ∈ L p (Ω, µ), we have f

• g t → E(f |I) weakly in L p . Then, since L ∞ ⊂ L p ∩ (L p ) * , for all f 1 and f 2 in L ∞ , lim t→+∞ f 1 , f 2 • g t = f 1 , E(f 2 |I) .
Let q ∈ [1, +∞). Since L ∞ is dense in both L q and (L q ) * , by Lemma 1.1, the convergence above occurs for all f 1 and f 2 in L q and (L q ) * respectively.

A second consequence is that Keplerian shear is not uniquely a property of the invariant measure µ, but of the class of µ.

Proposition 1.3.
Let (Ω, µ, (g t ) t∈R ) be a flow which preserves a probability measure and exhibits Keplerian shear. Let ν µ be a probability measure which is also (g t )-invariant. Then (Ω, ν, (g t ) t∈R ) also exhibits Keplerian shear.

Proof. Let (Ω, µ, (g t ) t∈R ) and ν be as in assumptions of the proposition. Let h

:= dν/dµ. Let f 1 be in L ∞ (Ω, µ) be such that f 1 h ∈ L 2 (Ω, µ), and let f 2 ∈ L ∞ (Ω, µ). Since h is I-measurable, ν-almost surely, E ν (f 2 |I) = E µ (f 2 |I). Let t ≥ 0. Then: E ν (f 1 • f 2 • g t ) = E µ ((f 1 h) • f 2 • g t ).
Since the initial system is assumed to have Keplerian shear, f ∈ L ∞ (Ω, µ) and gh ∈ L ∞ (Ω, µ), we get:

lim t→+∞ E ν (f 1 • f 2 • g t ) = E µ (f 1 hE µ (f 2 |I)) = E ν (f 1 E ν (f 2 |I)). The canonical projection L ∞ (Ω, µ) → L ∞ (Ω, ν) is surjective, so its image is dense in L 2 (Ω, ν). The image of the set of functions f 1 ∈ L ∞ (Ω, µ) such that f 1 h ∈ L 2 (Ω, µ
) by this projection is also dense in L 2 (Ω, ν). We use Lemma 1.1 to conclude.

The last lemma asserts that, in the definition of Keplerian shear, the limit object E µ (f |I) cannot be meaningfully modified.

Proposition 1.4.

Let (Ω, µ, (g t ) t∈R ) be a flow which preserves a probability measure. Let f, h ∈ L 2 (Ω, µ).

If (f • g t ) t∈R converges weakly to h, then h = E µ (f |I).
Proof. Let g ∈ L 2 (Ω, µ). Our hypotheses imply that lim t→+∞ E µ (g • f • g t ) = E µ (gh). In addition, the function t → E µ (g • f • g t ) is measurable and bounded. By taking the Cesàro average, we get:

lim t→+∞ E µ g t t 0 f • g s ds = lim t→+∞ 1 t t 0 E µ (g • f • g s ) ds = E µ (gh).
On the other hand, by von Neumann's ergodic theorem,

lim t→+∞ E µ g t t 0 f • g s ds = E µ (gE µ (f |I)).
Since this holds for all g ∈ L 2 , we have h = E µ (f |I).

2 Affine tori bundles

Setting and main theorem

We generalize our introductory examples to a class of flows on fibre bundles by tori which leave the basis invariant. More specifically, the spaces on which we work are the following:

Definition 2.1. An affine tori bundle is a C 1 manifold Ω which is a fiber bundle by d-dimensional tori, with group structure T d GL d (Z). In other words, there exist:

• two integers n, d ≥ 1; • a n-dimensional C 1 real manifold M ; • a C 1 projection π : Ω → M ;
• a maximal atlas A on M , such that, for all U ∈ A, we have a diffeomorphism

ψ U : π -1 (U ) → U × T d such that π 1 • ψ U = π,
and the change of charts are given by:

ψ V • ψ -1 U : (U ∩ V ) × T d → (U ∩ V ) × T d (x, y) → (x, α U,V (x) + A U,V (y)) , where α U,V is C 1 and A U,V ∈ GL d (Z).
The notions of "subset of zero Lebesgue measure" or "subset of full Lebesgue measure" are welldefined on C 1 manifolds (as they are invariant by diffeomorphisms), and thus so is the notion of "probability measure absolutely continuous with respect to the Lebesgue measure". We will abuse notations and write Leb(A) = 0 for a measurable subset of zero Lebesgue measure A, and µ Leb for an absolutely continuous measure. Definition 2.2. Let Ω be an affine tori bundle. A flow (g t ) t∈R on Ω is said to be compatible on a chart ψ U :

π -1 (U ) → U × T d if there exists v ψ U ∈ C 1 (U, R d ) such that, for all t ∈ R, ψ U • g t • ψ -1 U (x, y) = (x, y + tv ψ (x)). A σ-finite measure µ on Ω is said to be compatible on a chart ψ U : π -1 (U ) → U × T d if ψ U, * µ |π -1 (U ) = (π * µ) |U ⊗ Leb T d .
A flow or a measure is said to be compatible if it is compatible on all charts.

A compatible measure is always invariant under a compatible flow. In addition, this notion behaves well with respect to the affine structure on the manifolds we work with. If a flow or a measure is compatible on some chart

ψ U : U ∩ V → π(U ∩ V ) × T d and if ψ U,V is a change of charts, then the flow or the measure is compatible on the chart ψ V |U ∩V : U ∩ V → π(U ∩ V ) × T d .
In what follows, we are working mostly with absolutely continuous measures. In this case, what happens on a subset of zero Lebesgue measure does not matter: the assumption that M be a manifold can be weakened to account for singularities or boundaries.

In light of the previous paragraph, the introduction of the structure group T d GL d (Z) might look gratuitous: one can always cut out the manifold M along a set of zero Lebesgue measure to get a disjoint union of simply connected domains, on which there is no holonomy. However, this structure appears naturally in many examples. For instance, for all n ≥ 1, we can work with the geodesic flow on T S n : if we ignore the set of null tangent vectors, which is negligible, we get a fibre bundle over R * + × Gr(2, n + 1) with fibre S 1 . With the same adaptation, our setting also includes billiards in ellipsoids or the geodesic flow on ellipsoids (see C. Jacobi [START_REF] Jacobi | Vorlesungen über Dynamik[END_REF] for the geodesic flow on ellipsoids, J. Moser [START_REF] Moser | Various aspects of integrable Hamiltonian systems[END_REF] for similar examples, and S. Tabachnikov [START_REF] Tabachnikov | Ellipsoids, complete integrability and hyperbolic geometry[END_REF] for the relation between the geodesic flow and the billiard). Let us also mention the study of the geodesic flow on (R d SL d (R)) /Z d SL d (Z) done by F. Maucourant [START_REF] Maucourant | Unique ergodicity of asynchronous rotations, and application[END_REF], in which the same structure appears.

Another important remark is that, when we change charts from chart U to chart V , we have

v ψ V |U ∩V = A U,V v ψ U |U ∩V .
So, while there is in general no well-defined function v : M → R d which gives the direction of the flow, the set of functions {x → ξ, v(x) } ξ∈Z d -{0} is well-defined.

We are now ready to state our main theorem.

Theorem 2.3.

Let π : Ω → M be an affine d-dimensional tori bundle over a manifold M . Let (g t ) t∈R be a compatible flow, and µ be an absolutely continuous compatible probability measure.

If Leb( ξ∈Z d -{0} {d ξ, v = 0}) = 0 on M , then the dynamical system (Ω, µ, (g t )) exhibits Keplerian shear.

Proof.

Assume that Leb(

ξ∈Z d -{0} {d ξ, v = 0}) = 0. Then Leb( ξ∈Z d -{0} { ξ, v = 0}) = 0, so (g t (x, y)) t∈R equidistributes in {x} × T d for Lebesgue-almost every x.
Hence, up to completion by the measure µ, the invariant σ-algebra of the flow is

I := π * B M , where B M is the Borel σ-algebra of M .
Our goal is to find a family of observables which is large enough to generate a dense subset of L2 (Ω, µ), and specific enough to make our computations manageable. Roughly, we choose a specific frequency in the direction of the torus T d . Under the hypothesis of the theorem, we can rectify the differential form ξ, v so that it has a very simple expression. Then we choose observables which split into an observable a in the direction of ξ, v , and another observable b in the direction of the kernel. The later observable b does not see the shearing at all, so the shearing only affects a.

Let (U i , ϕ i ) i∈I be a countable cover of M by disjoint open charts 2 , up to a Lebesgue-negligible set, with ϕ i :

U i → W i ⊂ R n . Let ψ i : π -1 (U i ) → U i × T d be
a family of triviliazing charts for Ω, and let

v i := v ψ i . For ξ ∈ Z d \ {0}, let V ξ i := V i ∩ {d ξ, v i = 0}.
Using the local normal form of submersions, we can find a finite or countable family (V ξ ij ) j∈J(i,ξ) of open sets which are pairwise disjoint, cover V ξ i up to a Lebesgue-negligible set, and with charts ϕ ξ ij :

V ξ ij → W ξ ij ⊂ R n such that ξ, v i • ϕ ξ,-1 ij (x) = x 1 .
For ξ = 0, we choose J(i, ξ) to be a singleton and take V 0 ij := V i . Given a point p ∈ R n , we write p x its first coordinate in R n , and p y for its remaining n -1 coordinates in R n . Given a point p ∈ M × T d , we write p z for its coordinate in T d . We apply Lemma 1.1, with the Banach space B = B * = L 2 (Ω, µ), and:

E = E * = i∈I ξ∈Z d j∈J(i,ξ) a((ϕ ξ ij • π) x )b((ϕ ξ ij • π) y )e 2πi ξ,ψ i,z : a, b ∈ L ∞ , ab ∈ L ∞ (W ξ ij , Leb) .
Let f j = a j b j e ξ j ,• , with j ∈ {1, 2}, be in E. If the corresponding indices i ∈ I are different, then

f 1 and f 2 • g t have disjoint support for all t, so E µ (f 1 • f 2 • g t ) = 0 = E µ (f 1 • E µ (f 2 |I)) for all t ∈ R.
We can thus assume without loss of generality that they are supported by the same open set π -1 (V i ).

If the corresponding frequencies ξ j ∈ 2πZ d are different, then the integral of f 1 • f 2 • g t on each torus T d vanishes, and a least one of E µ (f 2 |I) or E µ (f 2 |I) vanishes, so for all t ∈ R:

E µ (f 1 • f 2 • g t ) = 0 = E µ (E µ (f 1 |I)E µ (f 2 |I)) = E µ (f 1 • E µ (f 2 |I)).
We can thus assume without loss of generality that their frequencies ξ j are the same; let us denote it by ξ. If ξ = 0, then f 1 and f 2 are invariant under the flow, so there is nothing more to prove. We further assume that ξ = 0.

If the corresponding indices j ∈ J(i, ξ) are different, then the supports of f 1 and f 2 •g t are disjoint for all t, so then again there is nothing more to prove. We thus fruther assume that these indices are the same.

Write

h ξ ij := d(ϕ ξ ij, * π * µ)/d Leb ∈ L 1 (W ξ ij , Leb).
Then, for all t ∈ R:

E µ (f 1 • f 2 • g t ) = W ξ ij a 1 (x)b 1 (y)a 2 (x)b 2 (y) T d e 2πi ξ,z+tv i •ϕ -1 ij (x,y)-z dzh ξ ij (x, y) dx dy = W ξ ij ∩{0}×R n-1 (b 1 b 2 )(y)e 2πi ξ,tv i •ϕ -1 ij (0,y) W ξ ij ∩(y+R×{0}) (a 1 a 2 )(x)e ixt h ξ ij (x, y) dx dy. The function x → (a 1 a 2 )(x)h ξ ij (x, y
) is integrable for almost every y. By the Riemann-Lebesgue lemma, the inner integral decay to 0 as t → ±∞. The inner integral is bounded by:

a 1 a 2 L ∞ W ξ ij ∩(y+R×{0}) h ξ ij (x, y) dx,
which is integrable as a function of y. Hence, by the dominated convergence theorem,

lim t→±∞ E µ (f 1 • f 2 • g t ) = 0 = E µ (f 1 • E µ (f 2 |I)).

Genericity

We check in this subsection that the sufficient condition in Theorem 2.3 is C r -generic for all r ∈ [1, +∞]. Given a C r affine tori bundle Ω, we begin by endowing the space of C r compatible flows with a topology. Let r ∈ [1, +∞], and π : Ω → M be a C r affine d-dimensional tori bundle over a manifold M . Let (U i ) i∈I be a locally finite open cover of M with trivializing charts ϕ i :

U i → W i ⊂ R n . Let (K i ) i∈I be a cover of M by compact sets subordinated to (U i ) i∈I . Denote by F r (M, R d ) the set of C r compatible flows on Ω. For each v ∈ F r (M, R d ),
there is a unique family of function (v i ) i∈I which generates the flow, where each

v i belongs to C r (U i , R d ). A sequence (v n ) of elements of F r (M, R d ) converges to v ∈ F r (M, R d ) if,
for all i ∈ I, all the derivatives of (v n,i ) n≥0 (up to order r) converge to those of v uniformly on each K i . This topology does not depends on the choice of the charts (U i ) i∈I nor on that of the compacts (K i ) i∈I , and makes F r (M, R d ) a Baire space. Proposition 2.4.

Let r ∈ [1, +∞]. Let π : Ω → M be a C r affine d-dimensional tori bundle over a manifold M . For a Baire generic subset of compatibles flows in F r (M, R d ), the dynamical system (Ω, µ, (g t )) exhibits Keplerian shear for all absolutely continuous compatible measures µ.

Proof.

We use the criterion of Theorem 2.3. It is enough to prove that, for all ξ ∈ Z d \ {0} and all i ∈ I:

A ξ,i := {v ∈ F r (M, R d ) : Leb({d ξ, v i = 0} ∩ K i ) = 0} is Baire generic. But A c ξ,i = n≥1 m≥1 B ξ,i,n,m , with: B ξ,i,n,m = {v ∈ F r (M, R d ) : Leb({ d ξ, v i • ϕ -1 i ≤ 1/m} ∩ ϕ i (K i )) ≥ 1/n}.
All is left is to prove that m≥1 B ξ,i,n,m is meager. Note that:

B c ξ,i,n,m = v ∈ F r (M, R d ) : Leb({ d ξ, v i • ϕ -1 i > 1/m} ∩ ϕ i (K i )) > Leb(ϕ i (K i )) -1/n . Let v ∈ B c ξ,i,n,m
. By inner regularity of the Lebesgue measure on ϕ i (K i ), there exists

K ⊂ K i compact such that d ξ, v i • ϕ -1 i > 1/m on ϕ i (K ) and Leb(ϕ i (K )) > Leb(ϕ i (K i )) -1/n. By compactness, for all v close enough to v, we have d ξ, v i • ϕ -1 i > 1/m on ϕ i (K ), and thus v ∈ B c
ξ,i,n,m . Hence, each B ξ,i,n,m is closed. We only need to show that the sets m≥1 B ξ,i,n,m have empty interior.

Fix

ξ ∈ Z d \ {0}, i ∈ I and n ≥ 1. Let χ i ∈ C r (V i , [0, 1]
), with Supp(χ i ) ⊂ V i compact and χ i ≡ 1 on ϕ i (K i ). For t ∈ R, let v(t) be defined by:

v i (t) • ϕ -1 i (x) := v i • ϕ -1 i (x) + tx 1 χ i (x)ξ on (U i , ϕ i ), v j (t) • ϕ -1 j (x) := v j • ϕ -1 j (x) + tx 1 χ i • ϕ i • ϕ -1 j (x)1 U i ∩U j (x)A U i ,U j (ξ) on (U j , ϕ j ), j = i. Then lim t→0 v(t) = v in F r (M, R d ). On ϕ i (K i ), we have χ i ≡ 1, therefore: d ξ, v i (t) • ϕ -1 i = d ξ, v i (0) • ϕ -1 i + t ξ 2 e *
1 , with e * 1 = (1, 0, . . . , 0). By the pigeonhole principle, for all m ≥ 1, at least one of the functions v(2k/( ξ 2 m)), with 0 ≤ k ≤ n Leb(ϕ i (K i )) , belongs to B c ξ,i,n,m . Thus there exists a sequence (t m ) m≥1 such that v(t m ) ∈ B c ξ,i,n,m and lim m→+∞ t m = 0. This finishes the proof. Remark 2.5.

If Ω = M × T d and r ≥ 2, we can conclude using the (well known, but more difficult to prove) fact that a generic function in C r (M, R) is Morse.

Examples

The simplest non-trivial example of Keplerian shear is given by the map

T = 1 1 0 1 ,
acting on T 2 = {(x, y) : x, y ∈ T}. This transformation preserves the Lebesgue measure, as well as all the circles T × {y}. Keplerian shear is rather easy to prove3 , as there is no need to play with charts; one can use directly the Fourier basis on L 2 (T 2 , Leb), which behaves well under T . A slightly more sophisticated version of this argument is used in Sub-subsection 2.5.1 to compute the speed of decay of correlations. All systems are not that simple. Besides genericity, Theorem 2.3 provides a useful criterion to prove that a given dynamical system exhibits Keplerian shear. We now use it to prove Keplerian shear for two dynamical systems: the billiard in the unit ball B n ⊂ R n , and the unit speed geodesic flow on T n (with the flat metric).

Billiard in a ball

Let B n be the unit ball in R n , with n ≥ 2. Consider a particle moving with unit speed in B n , which reflects specularly on the boundary S n-1 . The phase state is an orbifold T 1 B n , and the flow (g t ) t∈R preserves the Liouville measure µ n (which here is essentially the Lebesgue measure on B n × S n-1 ).

Proposition 2.6.

The dynamical system (T 1 B n , µ n , (g t ) t∈R ) exhibits Keplerian shear.

Proof.

If we exclude trajectories which go through the origin, then any given trajectory lie in the unique plane generated by the position and the speed at any given time. Restricted to any such plane, the billiard is isomorphic to the billiard in B 2 . Since a disjoint union of systems with Keplerian shear still has Keplerian shear, it is enough to prove that (T 1 B 2 , µ 2 , (g t ) t∈R ) has Keplerian shear.

The space T 1 B 2 is 3-dimensional. The angle θ ∈ (-π/2, π/2) with which the trajectories hit the boundary is an invariant of the flow. Hence, (T 1 B 2 , µ 2 , (g t ) t∈R ) is isomorphic to (Ω, μ, (g t ) t∈R ), where:

• Ω = (-π/, π/2) × T 2 ; • μ = 2 -1 cos(θ)dθ ⊗ Leb T 2 ; • gt (θ, x) = (θ, x + tv(θ)), and v(θ) = 2 cos(θ)(1, 1/2 -θ/π). In particular, v (θ) = -2 sin(θ) 1 1 2 -θ π + 1 π cot(θ)
.

For all ξ ∈ Z 2 \ {0}, the function ξ, v is analytic and non-zero, and thus its zero set is discrete. By Theorem 2.3, the system (T 1 B 2 , µ 2 , (g t ) t∈R ) has Keplerian shear.

A similar proof applies to the billiard in an ellipsoid, or the geodesic flow on an ellipsoid.

Geodesic flow on the torus

The second example we discuss is the unit speed geodesic flow on the torus T n , with n ≥ 1. This flow, again, preserves the Liouville measure.

Proposition 2.7.

The dynamical system (T 1 T n , Liouv, (g t ) t∈R ) exhibits Keplerian shear.

Proof.

The manifold T 1 T n is trivializable, and thus isomorphic to T n × S n-1 . The geodesic flow (g t ) t∈R acts on T 1 T n by:

g t (x, v) = (x + tv, v).
Let ξ ∈ Z n \ {0}. Then d ξ, v vanishes at only two points, which are ±ξ/ ξ . By Theorem 2.3, the system (T 1 T n , Liouv, (g t ) t∈R ) has Keplerian shear.

Unique ergodicity

In this subsection, we describe the relation between Keplerian shear and the unique ergodicity of a transformation acting on spaces of probability measures, as introduced by F. Maucourant [START_REF] Maucourant | Unique ergodicity of asynchronous rotations, and application[END_REF]. We drop the assumption that the function v generating the flow be C 1 : here, continuity is enough.

Definition and relation with Keplerian shear

Let π : Ω → M be a compact affine tori bundle, (g t ) a compatible flow on Ω, and ν ∈ P(M ). Denote by P ν ⊂ P(Ω) the subspace of probability measures μ such that π * μ = ν, and by ν ⊗ Leb the unique compatible measure on Ω such that π * (ν ⊗ Leb) = ν. Let G t := g t, * act continuously on P(Ω), which is compact when endowed with the weak convergence. Since the flow is compatible, (G t ) preserves P ν , which is also compact. Note that ν ⊗ Leb is a fixed point of (G t ), so δ ν⊗Leb is (G t )-invariant.

Theorem 2.8.

Let π : Ω → M be a compact affine tori bundle. Let (g t ) be a compatible flow on Ω. Let ν ∈ P(M ).

The system (Ω, ν ⊗ Leb, (g t )) exhibits Keplerian shear if and only if G t (µ) → ν ⊗ Leb for all µ ∈ P ν . Then (P ν , (G t )) is uniquely ergodic.

Proof.

Let π : Ω → , (g t ) and ν be as in the hypotheses of the theorem. First, we assume that (Ω, ν ⊗ Leb, (g t )) exhibits Keplerian shear. We can find a countable cover of M by disjoint open charts (U i ) i∈I , up to a ν-negligible subset. Then all (U i × T d , ν |U i ⊗ Leb, (g t )) exhibit Keplerian shear.

Let µ be in M(U i × T d ) with π * µ = ν |U i . Endow U i with any bounded Riemannian metric, and T d with a flat metric. This yields a Riemannian metric on U i × T d (e.g. the product metric), from which we get a Wasserstein distance d W , which metrizes the weak convergence.

We denote by * the fiberwise convolution on each torus. Fix ε > 0, and let ρ ε be an absolutely continuous measure supported on B T d (0, ε). Then d W (ρ ε , δ 0 ) ≤ ε, whence, for all t:

d W (G t (µ * ρ ε ), G t (µ)) = d W (µ * G t (ρ ε ), µ * G t (δ 0 )) ≤ ε.
On the other hand, µ * ρ ε ν |U i ⊗ Leb and π * (µ * ρ ε ) = ν. As we see by integrating against test functions, Keplerian shear implies that G t (µ * ρ ε ) → ν |U i ⊗ Leb weakly. In particular, d W (G t (µ * ρ ε ), ν |U i ⊗ Leb) ≤ ε for all large enough t, whence d W (G t (µ), ν U i ⊗ Leb) ≤ 2ε. As this is true for all ε > 0, we get G t (µ) → ν |U i ⊗ Leb. Since this is true for all i, G t (µ) → ν ⊗ Leb for all µ ∈ P ν . Hence, (P ν , (G t )) is uniquely ergodic.

Assume now that G t (µ) → ν ⊗ Leb for all µ ∈ P ν . By [7, Theorem 1], g 1 is asynchronuous, so the set of points x of M such that (g t ) acts on {x} × T d by an irrational translation has full ν-measure. Hence, the invariant σ-algebra is π * B M .

Let (U i ) i∈I be an open cover of M by charts. Let f ∈ C(Ω, C). Let i ∈ I and ρ(x, y) = a(x)b(y)

on U i × T d , for a ∈ C c (U i , R * + ) and b ∈ C(T d , R * + ) such that T d b d Leb = 1. Take ρ ≡ 0 on π -1 (U c i ) and a ≡ 0 on U c i . Let µ be the probability measure on Ω defined by µ |π -1 (U i ) := ν U i ⊗ (b d Leb) and µ |π -1 (U i ) c := ν U c i ⊗ Leb.
Then µ ∈ P ν , and, for all t:

Ω f • g t • ρ dν ⊗ Leb = U i f a • g t, * (ν |U i ⊗ b d Leb) = Ω f a • G t (µ).
By assumption, G t (µ) converges weakly to ν ⊗ Leb, so the quantity above converges to:

Ω f a dν ⊗ Leb = E ν⊗Leb (E ν⊗Leb (f |I)E ν⊗Leb (ρ|I)).
By Lemma 1.1, (Ω, ν ⊗ Leb, (g t )) exhibits Keplerian shear.

Remark 2.9 (Keplerian shear is stronger than unique ergodicity). F. Maucourant gives an example [START_REF] Maucourant | Unique ergodicity of asynchronous rotations, and application[END_REF] of a compatible flow and a measure ν such that (P ν , (G t )) is uniquely ergodic, but the fixed point ν ⊗ Leb behaves like an indifferent fixed point: there are exceptional sequences of times (t i ) for which G t i (ν ⊗ δ 0 ) is far from ν ⊗ Leb. As a corollary, the unique ergodicity of (P ν , (G t )) does not imply that (Ω, ν ⊗ Leb, (g t )) has Keplerian shear.

An application : Gauss' circle problem

The alternative characterization of Keplerian shear given by Theorem 2.8 is also useful in settings which use non-absolutely continuous measures. Let us give an elementary application to a variation on Gauss' circle problem. Let S(x, r) be the sphere of center x and radius r in R n , with n ≥ 2. Let ε ∈ (0, 1/2). What is the number of integer points in an ε-neighborhood of S(x, r)?

Let σ x,r be the uniform measure on S(x, r), and the canonical projection from R n to T n . Take Ω := S n-1 × T n , with g t (v, y) = (v, y + tv) and ν the uniform measure on S n-1 . Let f (y) := 1 |y|≤ε on T n . Then:

σ x,r ({y ∈ R n : d(y, Z n ) ≤ ε}) = ( * σ x,r )({y ∈ T n : d(y, 0) ≤ ε}) = G t (ν ⊗ δ (x) )(f ).
The system (Ω, ν ⊗ Leb, (g t )) has Keplerian shear by Proposition 2.7, so that:

lim r→+∞ σ x,r ({y ∈ R n : d(y, Z n ) ≤ ε}) = Leb(B R n (0, ε)) = ε n Leb(B R n (0, 1).
In addition, S(x, r) ∩ B(Z n , ε) consists of finitely many caps, which get flatter and flatter as r increases; the number of integer points ε-close to S(x, r) is the number of such caps. Let us direct there caps by the outward normal at their center. Since the measure supported by the projection on S n-1 × T n of these caps equidistributes in S n-1 × B(0, ε), we get that the average area (for * σ x,r ) of each cap converges to:

Average cross-section of B R n (0, ε) Leb n-1 (S(0, r)) = ε n-1 Leb(B R n (0, 1)) 2r n-1 Leb n-1 (S n-1 ) .
Hence, the number of integer points in an ε-neighborhood of S(x, r) converges, as r goes to infinity, to:

ε n Leb(B R n (0, 1)) • 2r n-1 Leb n-1 (S n-1 ) ε n-1 Leb(B R n (0, 1)) = 2εr n-1 Leb n-1 (S n-1 ).
This stays true if the sphere is replaced by any compact manifold, under non-resonancy conditions which ensure Keplerian shear for the relevant dynamical system. Note also that for the sphere, by integrating over r, one recovers the more elementary fact that the number of integral points at distance r from the origin is equivalent to r n Leb(B R n (0, 1)).

This result is not optimal. For instance, the best known bounds for Gauss' circle problem [START_REF] Huxley | Exponential sums and lattice points. III[END_REF] imply that:

Card[Z 2 ∩ (S(0, r) + B(0, ε))] ∼ 2εr n-1 Leb n-1 (S n-1 ) + O(r 131 208 ln(r) 18627 8320 ),
and this error bound holds if the circle is replaced by a closed C 3 curve with non-vanishing curvature. The proof of this result, however, requires more technology4 .

Speed of mixing

Keplerian shear is a qualitative property of a measure-preserving dynamical system, which asserts the convergence to zero on average of the conditional correlations:

E(Cov t (f 1 , f 2 |I)) = E(f 1 • f 2 • g t ) -E(E(f 1 |I)E(f 2 |I)).
As with the notion of mixing, one cannot expect a rate of convergence for all observables f 1 , f 2 ∈ L 2 . However, we may get a rate of convergence if f 1 and f 2 are regular enough. We may also need assumptions of the measure µ and the critical points of the functions ξ, v .

In the examples we discuss below, f 1 and f 2 shall belong to anisotropic Sobolev spaces (or, more precisely, weighted anisotropic Sobolev spaces). The regularity of such observables depends on the direction. We refer the reader to the monography by H. Triebel for additional information [15, Chapters 5-6] 5 .

In our setting, we need relatively little regularity in the direction of the invariant tori: what matters most is the regularity transversaly to the invariant tori. This is not surprising in view of Theorem 2.8, which asserts roughly that E(Cov t (f 1 , f 2 |I)) vanishes, where f 1 is Lipschitz and f 2 is e.g. Leb ⊗δ 0 on M × T d . In this case, f 2 is a distribution which is more regular transversaly to the invariant tori than in the direction of the invariant tori.

Instead of working out a general statement, we discuss two simple systems: the parabolic automorphism of T 2 at the beginning of Subsection 2.3, and the unit speed geodesic flow on T n .

Transvection on T 2

Consider the map

T = 1 0 1 1 ,
acting on T 2 , endowed with the Lebesgue measure. Let us define suitable anisotropic Sobolev spaces. For ξ ∈ R 2 , let:

h(ξ) :=    1 + ξ 2 1 ξ 2 2 1 2 if ξ 2 = 0 1 if ξ 2 = 0 .
For any real number s ≥ 0, let:

H s,0 (T 2 ) :=    f ∈ L 2 (T 2 ) : f 2 H s,0 (T 2 ) := ξ∈2πZ 2 h(ξ) 2s | f | 2 (ξ) < +∞    .
The following proposition gives decay bounds on the correlation coefficients for Sobolev or analytic observables.

Proposition 2.10.

Let f 1 , f 2 be in H s,0 (T 2 , R). Then:

|E(Cov n (f 1 , f 2 |I))| ≤ 4 s n 2s f 1 H s,0 (T 2 ) f 2 H s,0 (T 2 ) .
If f 1 and f 2 are analytic, then there exist constants c, C > 0 (depending on f 1 and f 2 ) such that, for all n ∈ Z,

|E(Cov n (f 1 , f 2 |I))| ≤ Ce -c|n| .
5 A small difference is that our spaces H s,0 and H s, n-1 2 below do not fit exactly in the framework of Triebel, because the weights do not satisfy the assumptions at the beginning of [START_REF] Triebel | Theory of function spaces[END_REF]Chapters 6]. However, one can write for instance H s,0 (T 2 ) = L 2 (T 1 ) ⊕ H s,0 (T 2 ), where L 2 (T 1 ) has no effect on the correlations and H s,0 (T 2 ) fits into Triebel's framework.

Proof.

Let f 1 , f 2 be in H s,0 (T 2 ). By Plancherel's theorem,

E(f 1 • f 2 • T n ) = ξ∈2πZ 2 f 1 (ξ) f2 (T * n ξ),
so that:

|E(Cov n (f 1 , f 2 |I))| = ξ∈2πZ 2 ξ 2 =0 f 1 (ξ) f2 (T * n ξ) ≤ ξ∈2πZ 2 ξ 2 =0 | f 1 h s | • | f2 h s | • T * n • h -s • h -s • T * n (ξ) ≤ f 1 H s,0 (T 2 ) f 2 H s,0 (T 2 ) sup ξ∈2πZ 2 ξ 2 =0 {h -s (ξ)h -s (T * n ξ)}. Let ξ 2 ∈ 2πZ \ {0}. The function ξ 1 → h -s (ξ 1 , ξ 2 )h -s (ξ 1 + nξ 2 , ξ 2 ) is maximal for ξ 1 = -nξ 2 /2
, where its value is (1 + n 2 /4) -s , so that:

|E(Cov n (f 1 , f 2 |I))| ≤ 4 s n 2s f 1 H s,0 (T 2 ) f 2 H s,0 (T 2 ) .
The proof for analytic functions is essentially the same. The only remark needed is that, if f is analytic on the torus, then there exist constants c ,

C > 0 such that | f |(ξ) ≤ C e -c |ξ| .
The map T is especially well-behaved: not only does it acts nicely on Fourier series, but its shearing (the derivative of v) does not vanish. The estimates of Proposition 2.10 are thus a best case behaviour, that we do not expect to hold for more general systems.

Speed for the geodesic flow on the torus

The geodesic flow is harder to analyse than the previous example: not only does it lack its algebraic structure, but the functions ξ, v have vanishing gradient at two points for any non-zero ξ. Hence, we cannot expect the same rate of convergence. We use the stationary phase method to compute the speed of convergence. This yields a polynomial rate of decay for a large space of observables belonging again to some anisotropic Sobolev spaces (Proposition 2.11).

The definition of these anisotropic Sobolev spaces is however slightly more delicate. Let n ≥ 2 and s > (n -1)/2.

For (k, ξ) ∈ R n-1 × 2πZ n , let: h(k, ξ) := 1 + (1+ k 2 ) s 2 ξ n-1 4 if ξ = 0 1 if ξ = 0 . We see T 1 T n as S n-1 × T n .
Fix a finite open cover by charts (U i , ϕ i ) of S n-1 , and a smooth partition of the unit (χ i ) subordinated to (U i ). Then define:

H s, n-1 2 (S n-1 × T n ) :=    f ∈ L 2 : i ξ∈2πZ 2 R n-1 h 2 | [(f χ i ) • (ϕ -1 i , id)]| 2 (x, ξ) dx < +∞    ,
and denote by

• 2 H s, n-1 2
the norm appearing in this definition. In the same way, we define the Sobolev space H s (S n-1 ). These spaces do not depend on the choice of the family of charts and of the partition of the unit.

The following proposition gives decay bounds on the correlation coefficients for observables in H s, n-1 2 .

Proposition 2.11.

Let n ≥ 2 and s > (n -1)/2. There exists a constant C such that, for all

f 1 , f 2 ∈ H s, n-1 2 (S n-1 × T n ), |E(Cov t (f 1 , f 2 |I))| ≤ C t n-1 2 f 1 H s, n-1 2 f 2 H s, n-1 2 . ( 2.1)
Proof.

In this proof, the letter C shall denote a constant which may change from line to line, but which depends only on the dimension n and on the parameter s.

Let

s > (n -1)/2. Let f 1 , f 2 be in C ∞ (S n-1 × T n , C).
Denote by f ξ i (x) the Fourier transform of f i (x, •) evaluated in ξ ∈ 2πZ n . By Plancherel's and Fubini-Lebesgue theorems, the conditional covariance is equal to:

E(Cov n (f 1 , f 2 |I)) = ξ∈2πZ n ξ =0 S n-1 f ξ 1 (x)f ξ 2 (x)e it ξ,x dx. Let χ ∈ C ∞ (S n-1 , [0, 1]) be such that χ ≡ 1 near N := (1, 0, . . . , 0) and χ(-x) = 1 -χ(x). Let ϕ + : S n-1 \ {S} → R n-1 (resp. ϕ -: S n-1 \ {N } → R n-1
) be the stereographic projection from the North (resp. South) pole. Let ξ ∈ 2πZ n , and R ξ a rotation which send ξ/ ξ to N . Finally, let ψ ξ,± := (ϕ ± • R ξ ) -1 . Then:

S n-1 f ξ 1 (x)f ξ 2 (x)e it ξ,x dx = R n-1 f ξ 1 f ξ 2 • ψ ξ,+ (x)e it ξ,ψ ξ,+ (x) χ • ϕ -1 + (x) Jac(ϕ -1 + )(x) dx + R n-1 f ξ 1 f ξ 2 • ψ ξ,-(x)e it ξ,ψ ξ,-(x) (1 -χ) • ϕ -1 -(x) Jac(ϕ -1 -)(x) dx.
The function

1/ Jac(ϕ -1 ± ) is in C ∞ b (R n-1
) and the function x → ξ, ψ ξ,± (x) has a unique critical point in 0 which is non-degenerate. By the stationary phase method [2, Chapter 7.7], there exists a constant C such that:

S n-1 f ξ 1 (x)f ξ 2 (x)e it ξ,x dx ≤ C (t ξ ) n-1 2 R n-1 f ξ 1 f ξ 2 • ψ ξ,+ • χ • ϕ -1 + (k) dk + R n-1 f ξ 1 f ξ 2 • ψ ξ,-• (1 -χ) • ϕ -1 -(k) dk ≤ C (t ξ ) n-1 2 f ξ 1 H s (S n-1 ) f ξ 2 H s (S n-1 )
,

where we used the fact that f g

L 1 (R n-1 ) ≤ C f H s (R n-1 ) g H s (R n-1
) whenever s > (n-1)/2. Hence:

E(Cov n (f 1 , f 2 |I)) ≤ C t n-1 2 ξ∈2πZ n ξ =0 f ξ 1 H s (S n-1 ) f ξ 2 H s (S n-1 ) ξ n-1 2 ≤ C t n-1 2 ξ∈2πZ n ξ =0 f ξ 1 2 H s (S n-1 ) ξ n-1 2 ξ∈2πZ n ξ =0 f ξ 2 2 H s (S n-1 ) ξ n-1 2
.

Finally, using our local charts (U i , ϕ i ) on S n-1 :

ξ∈2πZ n ξ =0 f ξ 1 2 H s (S n-1 ) ξ n-1 2 ≤ C i ξ∈2πZ n ξ =0 R n-1 (1 + k 2 ) s ξ n-1 2 | [(f 1 χ i ) • (ϕ -1 i , id)]| 2 (x, ξ) dx ≤ C f 1 2 H s, n-1 2 (S n-1 ×T n )
.

That finishes the proof for smooth observables f 1 and f 2 . But, for fixed t, the correlation function

E(Cov t (•, •|I)) is bilinear and continuous from L 2 to C. Since the H s, n-1 2 norm is stronger than the L 2 norm, E(Cov t (|I)) is also continuous from H s, n-1 2 to C. But C ∞ is dense in H s, n-1
2 , so the bound (2.1) actually holds for any two observables in H s, n-1 2 .

Assuming that the observables f 1 and f 2 have higher regularity, standard formulations of the stationary phase method yield a higher order development of E(Cov t (f 1 , f 2 |I)) as t goes to infinity.

Assume now that we change the flow on S n-1 × T n , for instance by making the velocity depend on the direction. Then the rates we got in Proposition 2.11 may not be generic. We shall sketch the difficulties encountered with more general systems. Let n ≥ 3 and M be a compact connected (n -1)-dimensional smooth manifold, and let v : M → R n be smooth. Consider the flow g t (x, y) = (x, y + tv(x)) on M × T n . If Dv is never degenerate (which is a C 1 -open condition on v), then v is an immersion. If in addition the extrinsic curvature of the immersed manifold is never degenerate, then we get rates of convergence as in Proposition 2.11. However, if the extrinsic curvature is never degenerate, then the Gauss map M → S n-1 is a local diffeomorphism, so a diffeomorphism (since n ≥ 3), and thus M is a sphere.

In other words, if M is not a sphere, then we have to deal with degenerescences of the extrinsic curvature of v(M ). If such a degenerescence happens in a rational direction of R n , then we would get a speed of convergence in O(t -n-1-r

2

), where r is the corank of the Hessian in the given direction. If this degenerescence happens in a direction u which is not rational, then this bound could be improved, although any improvement would depend on the Diophantine properties of u (the bound getting better if u is badly approximable by rationals). In particular, one cannot hope to get a significantly better bound than O(t -n-1-r 2 ) in a Baire generic setting, as Baire generic directions are Liouville.

For n ≥ 2, the same kind of obstruction may happen for v : S n-1 → R n . For a C 3 -open set of such functions v, the map v has non-degenerate inflexion points. Without further argument about the directions these inflexion points occur, this would for instance yield a rate of decay of only O(t -1

3 ) if n = 2.

Stretched Birkhoff sums

We present in this sub-section another class of systems which may exhibit Keplerian shear. The examples of Subsection 2.1 are based on translations on the torus, which are a family of non-mixing dynamical systems. In this section, the elementary brick will be given by suspension flows with constant roof function. The family of examples we get includes many non-Hamiltonian systems.

Let (A, ν, T ) be a measure-preserving dynamical system. For v > 0, the suspension flow with constant roof 1 and speed v is the measure-preserving semi-flow ( A, ν, (g v t ) t≥0 ) defined by:

• A := (A × [0, 1]) (x,1)∼(T (x),0) ; • g v t [(x, s)] = [(x, s + vt)];
• ν := ν ⊗ Leb on the fundamental domain A × [0, 1).

Such a suspension flow is ergodic, but cannot be mixing, as it has the rotation on the circle as a factor. Now, we give ourselves:

• a n-dimensional C 1 manifold M , with n ≥ 1;
• a measure-preserving ergodic dynamical system (A, ν, T );

• a measurable function v : M → R * + .
With this data we construct a new semi-flow (Ω, (g t ) t≥0 ) with Ω := A×M and g t (x, y) := (g v(y) t (x), y). A measure µ ∈ P(Ω) is said to be compatible if it is equal to ν ⊗ μ for some μ ∈ P(M ). Compatible measures are preserved by (g t ).

If (A, ν, T ) is invertible, the suspension semi-flow can be extended to a flow, in which case v may take negative values. The following theorem also holds in this alternative setting. Theorem 3.1.

Let (Ω, (g t ) t≥0 ) be a system defined as above, with v ∈ C 1 (M, R * + ). Let µ be an absolutely continuous compatible measure. If Leb(dv = 0) = 0, then (Ω, µ, (g t ) t≥0 ) exhibits Keplerian shear.

Proof.

Let I A be the invariant σ-algebra of (A, T ), and B M the Borel σ-algebra of M . As a measured space, we can see Ω as A × S 1 × M . Up to completion with respect to µ, the invariant σ-algebra of (Ω, (g t ) t≥0 ) is I :

= I A ⊗ {∅, S 1 } ⊗ B M .
Let U := {dv = 0} ⊂ M . Let (U i , ψ i ) i∈I be a countable cover of U by charts, with ϕ i : U i → W i ⊂ R n and W i bounded. Using the local normal form of submersions, we assume that v•ϕ -1 i (z) = z 1 > 0. We write z = (z 2 , . . . , z n ). Let (V i ) i∈I be a partition of U by open sets, up to a Lebesgue negligible subset of U , such that V i ⊂ U i for all i. We write W i := ϕ i (V i ).

We apply Lemma 1.1, with the Banach space B = B * = L 2 (Ω, µ), and:

E = E * = i∈I f (x, y, z) = a(x)e iξy b(ϕ i (z) 1 )c(ϕ i (z) ) : a ∈ L 2 (A, ν), ξ ∈ 2πZ, b, c ∈ C 1 c , bc ∈ C 1 c (W i ) . Let us write d(z) := b(ϕ i (z) 1 )c(ϕ i (z) ) for z ∈ U i .
Let (p i ) i∈I be a sequence of positive numbers such that i∈I p i Leb(W i ) = 1. By Proposition 1.3, without loss of generality, we replace μ by μ :

= i∈I p i ϕ * i Leb |W i . Let f j = a j e iξ j • d j , with j ∈ {1, 2}, be in E. If the d j have disjoint support, then E(f 1 • f 2 • g t ) = 0 = E(f 1 E(f 2 |I
)) for all t, and there is nothing more to prove. We assume without loss of generality that the h j are supported by the same open set

V i . Let h(z 1 ) := b 1 (z 1 )b 2 (z 1 ) ∈ C 1 c (R * + ).
Then, for all t ≥ 0:

Ω f 1 • f 2 • g t dµ = M d 1 (z)d 2 (z) A a 1 (x)a 2 (T v(x)t x) 1 0 e -iξ 1 y e iξ 2 (y+v(z)t) dy dν(x) dμ(z) = δ ξ 1 ξ 2 M d 1 (z)d 2 (z)e iξ 1 v(z)t A a 1 (x)a 2 (T v(z)t x) dν(x) dμ(z) = δ ξ 1 ξ 2 p i W i d 1 (ϕ -1 i (z))d 2 (ϕ -1 i (z))e iξ 1 v(ϕ -1 i (z))t A a 1 (x)a 2 (T v(ϕ -1 i (z))t x) dν(x) dz = δ ξ 1 ξ 2 p i R n-1 c 1 (z )c 2 (z ) dz • +∞ 0 h(z 1 )e iξ 1 z 1 t A a 1 (x)a 2 (T z 1 t x) dν(x) dz 1 .
If ξ 1 = ξ 2 , there is nothing more to prove. Assume that ξ 1 = ξ 2 =: ξ. Then:

+∞ 0 h(z 1 )e iξz 1 t A a 1 (x)a 2 (T z 1 t x) dν(x) dz 1 = A a 1 (x) +∞ 0 e iξz 1 t a 2 (T z 1 t x)h(z 1 ) dz 1 dν(x) = A a 1 (x) +∞ k=0 a 2 (T k x) 1 t 0 e iξts h k t + s ds dν(x).
We now distinguish between two cases, depending on whether ξ = 0 or not.

Case 1: ξ = 0.

In the spirit of Riemann-Lebesgue's lemma, we use an integration by parts to show that the oscillations make the integral decay.

+∞ 0 h(z 1 )e iξz 1 t A a 1 (x)a 2 (T z 1 t x) dν(x) dz 1 = 1 |ξ|t A a 1 (x) +∞ k=0 a 2 (T k x) 1 t 0 (1 -e iξts )h k t + s ds dν(x) ≤ 2 |ξ|t A |a 1 |(x) +∞ k=0 |a 2 |(T k x) 1 t 0 |h | k t + s ds dν(x) ≤ 2 a 1 L 2 a 2 L 2 h BV |ξ|t .
By integrating over z , we get:

E µ (f 1 • f 2 • g t ) ≤ 2 a 1 L 2 a 2 L 2 c 1 L 2 c 1 L 2 h BV |ξ|t → t→+∞ 0. But E µ (f 2 |I) = 0, so the integral converges to E µ (f 1 E µ (f 2 |I)).
Case 2: ξ = 0.

In this case,

+∞ 0 h(z 1 )e iξz 1 t A a 1 (x)a 2 (T z 1 t x) dν(x) dz 1 = A a 1 (x) +∞ k=0 a 2 (T k x) 1 t h k t dν(x) + O(t -1 ), (3.1)
where:

|O(t -1 )| ≤ a 1 L 2 a 2 L 2 h C 1 t .
By von Neumann's ergodic theorem,

lim t→+∞ +∞ k=0 a 2 (T k x) 1 t h k t = +∞ 0 h(z 1 ) dz 1 • lim n→+∞ 1 N N -1 k=0 a 2 • T k = +∞ 0 h(z 1 ) dz 1 • E ν (a 2 |I A ),
where the convergence is in L 2 norm. Hence,

lim t→+∞ +∞ 0 h(z 1 )e iξz 1 t A a 1 (x)a 2 (T z 1 t x) dν(x) dz 1 = +∞ 0 h(z 1 ) dz 1 • A a 1 E ν (a 2 |I A ) dν,
so that:

lim t→+∞ E µ (f 1 • f 2 • g t ) = p i M d 1 d 2 dϕ * i Leb • A a 1 E ν (a 2 |I A ) dν = p i M d 1 E μ(d 2 |B M ) dϕ * i Leb • A a 1 E ν (a 2 |I A ) dν = E µ (f 1 E µ (f 2 |I)).
Since the sufficient criterion in Theorem 3.1 is the same as in Theorem 2.3, genericity follows (as for Proposition 2.4): Corollary 3.2.

Let (A, ν, T ) be a system preserving a probability measure, M a n-dimensional manifold (with n ≥ 1). Let r ∈ [1, +∞]. For v ∈ C r (M, R * + ), let (Ω, (g v t ) t≥0 ) be defined as above. For C r generic roof functions v, the system (Ω, µ, (g v t ) t≥0 ) exhibits Keplerian shear for any absolutely continuous compatible measure µ.

We shall not discuss the speed of decay of correlations for such systems: not only do the critical points of v matter, so do the decay of correlations on (A, ν, T ).

Systems without Keplerian shear

While systems with Keplerian shear are abundant in the classes we discussed -since the conditions in Theorems 2.3 and 3.1 are generic -, we shall finish with a couple of examples of non-ergodic systems without Keplerian shear. The first is the geodesic flow on the sphere, which falls in the setting of Section 2 but lacks asynchronicity; the second is given by a large class of p-adic translations.

Geodesic flow on a sphere

Let n ≥ 2. The manifold T 1 S n is a fibre bundle over the oriented Grassmannian Gr(2, n + 1) with fibre S 1 . This comes from the fact that the orbits of the geodesic flow on this manifold are oriented grand circles, and the space of oriented grand circle is isomorphic to the space of oriented 2-planes in R n+1 . The geodesic flow acts by translations on the grand circles. Hence the dynamical system (T 1 S n , Liouv, (g t )) belongs to the class of examples discussed in Section 2. The invariant σ-algebra I is isomorphic to B Gr(2,n+1) , and thus non trivial.

However, all grand circles are of the same length, so g t+2π = g t . In particular, given any integrable function h which is not I-measurable, the sequence of functions (h • g t ) t cannot converge to a (g t )invariant function.

Finally, the geodesic flow on T 1 S 1 is isomorphic to the disjoint union of two rotations on S 1 , which are ergodic but not mixing. Hence, the system (T 1 S n , Liouv, (g t )) does not have Keplerian shear for any n ≥ 1.

p-adic translations

Until now, we have seen classes of dynamical systems for which Keplerian shear is generic, with the geodesic flow on T 1 S n being an exception rather than the rule. As we shall see now, the situation is completely different for p-adic translations. Recall that, for p a prime number, the ring Z p is the completion of Z for the p-adic norm. It is compact, and thus supports an invariant probability, which we shall denote Leb.

We shall see that, when one replaces translations on a torus by translations on Z p , the system they get typically does not exhibit Keplerian shear. The reason is that, on Z p , errors do not accumulate: if we change a translation on Z p by a small quantity, the iterates of the two translations still stay close one to another at all times. Let p be a prime number, d ≥ 1. Let (M, ν) be a standard probability space. Let v : M → (Z p ) d be measurable. Let:

T : M × (Z p ) d → M × (Z p ) d (x, y) → (x, y + v(x)) .
Then (M × Z p , ν ⊗ Leb, T ) exhibits Keplerian shear if and only if v ≡ 0 almost everywhere.

Proof. If v ≡ 0 almost everywhere, then T is essentially the identity, which has Keplerian shear. Assume that this is not the case. Then one can find A ⊂ M , N ≥ 0, i ∈ {1, . . . , d} and k ∈ {1, . . . , p -1} such that ν(A) > 0 and v i (x) = kp N + (x)p N +1 for all x ∈ A.

Let χ be a non-trivial character on Z /pZ . Let The function f is non-zero on a set of positive measure, and since χ(k) is a non-trivial pth root of the unit, we get that (f • T n ) n≥0 is exactly p-periodic. Hence, the system (M × Z p , ν ⊗ Leb, T ) does not exhibit Keplerian shear.

  ≥0 y ,i p 1≤i≤d → χ(y N,i ) , Then, for (x, y) ∈ A × (Z p ) d , f • T n (x, y) = χ(y N,i + nk) = χ(y N,i )χ(k) n .

Technically, J. Chaika and P. Hubert show the convergence only for observables which do not depend on the direction, but a straightworward generalization and a diagonal argument yield the general case.

The goal of this first decomposition is only to get well-defined speed functions v ψi , and can be bypassed if the fibre bundle is trivial.

This example has been used with some success by the author in a graduate-level exercise course in ergodic theory.

Typically, it uses a decomposition of the circle into "big arcs" and "small arcs", which can also be used to prove Keplerian shear directly without using the Fourier transform.

Acknowledgements

I would like to thank Sébastien Gouëzel, Bassam Fayad and Fraçois Maucourant for their useful comments and some of the references, as well as Jérôme Buzzi for his feedback on the presentation.


Namig J Guliyev

THE REGULARIZED TRACE FORMULA FOR THE STURM-LIOUVILLE EQUATION WITH SPECTRAL PARAMETER IN THE BOUNDARY CONDITIONS

The regularized trace formula of first order for the Sturm-Liouville equation with spectral parameter in the boundary conditions is obtained.

Consider the boundary value problem -y ′′ (x) + q(x)y(x) = λy(x),

x ∈ [0, π],

λ(y ′ (0) -hy(0)) = h 1 y ′ (0) -h 2 y(0),

λ(y ′ (π) + Hy(π)) = H 1 y ′ (π) + H 2 y(π),

where q(x)

∈ W 1 2 [0, π] is a real-valued function, h, h 1 , h 2 , H, H 1 , H 2 ∈ R and δ := hh 1 -h 2 > 0, ρ := HH 1 -H 2 > 0.
Let ϕ(x, λ) be the solution of (1) satisfying the initial conditions

ϕ(0, λ) = h 1 -λ, ϕ ′ (0, λ) = h 2 -λh.
We denote s := √ λ. Since ϕ(x, λ) satisfies the integral equation

ϕ(x, λ) = -s 2 cos sx -   h + 1 2 x 0 q(t)dt   s sin sx + O e | Im s|x , (4)
ϕ ′ (x, λ) = s 3 sin sx -   h + 1 2 x 0 q(t)dt   s 2 cos sx- -   h1 + q(x) + q(0) 4 + h 2 x 0 q(t)dt + 1 8   x 0 q(t)dt   2    s sin sx- - s 4 x 0 q ′ (t) sin[s(x -2t)]dt + O e | Im s|x . (5) 1
Denote by χ(λ) the characteristic function of ( 1)-(3):

χ(λ) := (λ -H 1 )ϕ ′ (π, λ) + (λH -H 2 )ϕ(π, λ).
From (4) and (5) we have

χ(λ) := s 5 sin sπ -ωs 4 cos sπ -ω 1 s 3 sin sπ - s 3 4 π 0 q ′ (t) sin[s(π -2t)]dt+ + O |s| 2 e | Im s|π , (6)
where

ω := h + H + 1 2 π 0 q(t)dt, ω 1 := h 1 + H 1 + hH + q(π) + q(0) 4 + h + H 2 π 0 q(t)dt + 1 8   π 0 q(t)dt   2 .
The eigenvalues of the problem (1)-( 3) are real and coincide with the zeros of the function χ(λ). From (6) we obtain

s n = n -2 + ω π(n -2) + ζ n n 2 , {ζ n } ∈ l 2 .
Then

λ n = s 2 n = (n -2) 2 + 2ω π + ξ n n , {ξ n } ∈ l 2 . (7)
Therefore

s λ := λ 0 + λ 1 + ∞ n=2 λ n -(n -2) 2 - 2ω π < ∞. ( 8
)
The series (8) is called the regularized trace of first order (or the first regularized trace) for the problem (1)-( 3). Analogously, one can define the second, the third etc. regularized traces. Note that the second regularized trace formula for the Sturm-Liouville problem with spectral parameter in a boundary condition has been found in [START_REF] Albayrak | The second regularized trace formula for the Sturm-Liouville problem with spectral parameter in a boundary condition[END_REF]. Since χ(λ) is an entire function of order 1/2, from Hadamard's theorem (see [2, Section 4.2]), using (6) we obtain

χ(λ) = π(λ -λ 0 )(λ -λ 1 )(λ -λ 2 ) ∞ n=3 λ n -λ (n -2) 2 (9)
Let λ = -µ 2 . We calculate the sum s λ of the series (8) by comparing the asymptotic expressions obtained from formulas (6) and (9) as µ → ∞. From the formula (9) we have

χ(-µ 2 ) = -π(µ 2 + λ 0 )(µ 2 + λ 1 )(µ 2 + λ 2 ) ∞ n=3 λ n + µ 2 (n -2) 2 = = - (µ 2 + λ 0 )(µ 2 + λ 1 )(µ 2 + λ 2 ) sh πµ µ Φ(µ), (10)
and from ( 10)

χ(-µ 2 ) = e πµ 2 -µ 5 -ωµ 4 + µ 3 s λ + ω π + ω 2 2 + o µ 3 . (14)
The estimate (6) implies χ(-µ 2 ) = e πµ 2 -µ 5 -ωµ 4 + ω 1 µ 3 + o µ 3 .

(15)

Comparing ( 14) and ( 15) we obtain the following trace formula:

λ 0 + λ 1 + ∞ n=2 λ n -(n -2) 2 - 2ω π = = h 1 + H 1 + q(π) + q(0) 4 - h + H π - h 2 + H 2 2 - 1 2π π 0 q(t)dt.

where

We need the formulas (see [START_REF] Levitan | Calculation of the regularized trace for the Sturm-Liouville operator[END_REF])

and

Using ( 7) and (12) we have

where

From (11), ( 12) and (13) we calculate: