Slow chaos in surface flows

Abstract

Flows on surfaces describe many systems of physical origin and are one of the most fundamental examples of dynamical systems, studied since Poincará. In the last decade, there have been a lot of advances in our understanding of the chaotic properties of smooth area-preserving flows (a class which include locally Hamiltonian flows), thanks to the connection to Teichmueller dynamics and, very recently, to the influence of the work of Marina Ratner in homogeneous dynamics. We motivate and survey some of the recent breakthroughs on their mixing and spectral properties and the mechanisms, such as shearing, on which they are based, which exploit analytic, arithmetic and geometric techniques.

1 Slowly chaotic dynamical systems

1.1 Deterministic chaos and the butterfly effect

Dynamical systems provide mathematical models of systems which evolve in time. Many systems phenomena in our world, from the evolution of the weather to the motion of an electron in a metal, can be described by a dynamical system. While in a model one can include a random component, or external noise, we will restrict ourselves to fully deterministic systems, whose evolution is completely described by a system of pre-determined rules or equations. We will furthermore consider continuous time dynamical systems, namely systems for which the time variable is a real parameter $t\in \mathbb {R}$, described by a flow on a space X, namely a 1-parameter group¹$\varphi _{\mathbb {R}}=(\varphi _t)_{t\in \mathbb {R}}$ of maps $\varphi _\mathbb {R}:X\rightarrow X$ (diffeomorphisms if X is a smooth manifold).

Deterministic dynamical systems often display chaotic features (see Sect. 2 for examples), which make their behaviour as time grows hard to predict. This is a phenomenon known as deterministic chaos. One of the best known features of chaotic behaviour is sensitive dependence on initial conditions (SDIC for short), a property which was popularized as the butterfly effect. In a system which displays SDIC, a small variation of the initial condition can lead to a macroscopically very different evolution after a long time. In particular, given a flow $\varphi _{\mathbb {R}}: X\rightarrow X$ on a metric space (X, d) with SFIC and a point $x\in X$ (the initial condition) one can find arbitrarily close initial conditions $y\in X$ such that the (forward) trajectories of x and y, namely the orbits $(\varphi _t(x))_{t\ge 0}$ and $(\varphi _t(y))_{t\ge 0}$ drift apart.²

1.2 Fast chaos versus slow chaos

Dynamical systems can roughly be divided in three categories (hyperbolic, elliptic and parabolic) according to the speed of divergence (if any) of close orbits. In a hyperbolic flow, the orbits $(\varphi _t(x))_{t\ge 0}$, $(\varphi _t(y))_{t\ge 0}$ of most³ pairs x, y of initial conditions diverge exponentially in time (i.e. the distance $d(\varphi _t(x),\varphi _t(y))$, for small values⁴ of time, is described by an exponential function of time. In a parabolic dynamical system, there is also divergence of (most) nearby orbits, but this divergence happens at subexponential (usually polynomial) speed. Finally while the flow is called elliptic if there is no divergence (or perhaps it is slower than polynomial). Thus, both hyperbolic system and parabolic systems display SDIC, but the butterfly effect happens at different speeds (respectively exponentially or (sub)polynomially). We colloquially call these systems respectively fast chaotic (when the butterfly effect is fast, i.e. exponential) and slowly chaotic (when the butterfly effect is slow, i.e. polynomial or slower than polynomial).

While there is a classical and well-developed theory of hyperbolic systems (starting with the study of uniformely hyperbolic dynamical systems, which was already developed in the 1970s by mathematicians such as Anosov and Sinai, Abel Prize in 2014, among others) and also a systematic study of elliptic ones (starting with the theory of circle diffeomorphisms, whose study is intertwined with Hamiltonian dynamics and KAM theory), there is no general theory which describes the dynamics of parabolic flows and only classical and isolated examples are well-understood.

1.3 Examples of parabolic systems

Slowly chaotic (or parabolic) systems include many dynamical systems of interest in physics, such as the Novikov model of electrons in a metal (which will be discussed below), or the Ehrenfest model (also called windtree model) proposed by Paul and Tatjiana Ehrenfest in 1912 to understand thermodynamics laws.

Among examples arising in mathematics, perhaps the most studied (and better understoood) example of a parabolic flow is given in the context of hyperbolic geometry by the horocycle flow on (the unit tangent bundle of) a compact negatively curved surface:⁵ while the geodesic flow (whose trajectories move along geodesics for the hyperbolic metric) is a classical example of fast chaos and hyperbolic dynamics, when moving along horocycles (which, in the upper half plane $\mathbb {H}$ are circles tangent to the real axis) one can show that divergence of nearbly trajectories is only a quadratic function of time, thus giving slowly chaotic dynamics.

Another fundamental class of homogeneous flows is given by nilflows, or flows on (compact) quotients of nilpotent Lie groups (nilmanifolds); The prototype example in this class are Heisenberg nilflows, given by the action (by left multiplication) of a 1 parameter subgroup of transformations of a compact quotient of the Heisenberg group.⁶

In this survey we will focus on an another fundamental class of parabolic flows, in the context of area-preserving flows on (higher genus) surfaces, given by locally Hamiltonian flows, which are smooth flows which preserve a smooth area-form. The definition is given later on in Sect. 3.2.

Finally, starting from the classical examples of parabolic flows mentioned above, one can build new parabolic flows by considering perturbations: the simplest perturbations are time-changes (or time-reparametrizations) of a given flow, i.e. flows that move points along the same orbits, but with different speed. More precisely, a flow $\tilde{\varphi }_{\mathbb R}$ is called a (smooth) time-change of a flow $\varphi _{\mathbb R}$ on X if for all $x \in X$ and $t \in \mathbb {R}$ we have $\varphi _t(x) = \varphi _{\tau (x,t)}(x)$ for some measurable (smooth) function $\tau : X \times \mathbb {R} \rightarrow \mathbb {R}$. Notice that it follows from this definition that the time-change $\tilde{\varphi }_{\mathbb R}$ has exactly the same trajectories than $\varphi _{\mathbb R}$ (but the motion along the trajectory has different speed). Some time-changes, known as (smoothly) trivial, give rise to flows that are (smoothly) conjugated (i.e. isomorphic as dynamical systems) to the original one and therefore have the same chaotic properties. A feature of parabolic dynamical systems, though, is that among smooth time-changes, smoothly trivial time-changes are rare, i.e. they often form a finite or countable codimension subspace.⁷ Therefore, the study of non trivial time-changes allows to systematically produce new classes of parabolic flows.

2 Chaotic properties

A natural and fundamental question in parabolic dynamics (and dynamics in general) is which chaotic properties -in particular which fine ergodic and spectral properties- are generic among classes of (smooth) slowly chaotic flows. Let us give some examples in this section of which type of chaotic properties one might be interested in. We will comment on possible notions of generic in Sect. 3.5.

Different type of chaotic properties are the focus of different branches of dynamics: properties of topological nature (such as existence of dense trajectories) are studied in topological dynamics, measure-theoretical features (such as equidistribution of a trajectory with respect to an equilibrium measure) in ergodic theory, and properties of spectral nature in spectral theory of dynamical systems.

2.1 Topological dynamics

Focus on the most basic questions about the behaviour of trajectories, such as existence and abundance of periodic trajectories (i.e. trajectories of a point x such that there exists a period $t_0$ for which $\varphi _{t+t_0}(x)=\varphi _t(x)$ for all $t\in {\mathbb R}$), or existence and abundance of trajectories which are dense in X. A (continuous) flow $\varphi _{\mathbb R}:X\rightarrow X$ (where X is a topological- or metric- space) is called minimal if every orbit is dense. In presence of fixed points (which is the case for surface flows in higher genus, which always have singularities for $g\ge 2$), the definition of minimality is slightly different: we only require all regular orbits, namely all orbits $\varphi _{\mathbb R}(x)$ which are neither a fixed point nor a saddle separatrix, to be dense.

2.2 Ergodic theory

Studies flows which preserve a measure: we assume hence that $(X,\mu )$ is a measure space and that $\varphi _{\mathbb R}:X\rightarrow X$ preserves the measure $\mu$, namely for any A measurable set, $\mu (A)=\mu (\varphi _t(A)$ for all $t\in \mathbb {R}$. If a trajectory is dense, one can further ask whether it is equidistributed with respect to the invariant measure $\mu$, namely if the time spent in a measurable set A is proportional (asymptotically) to its measure $\mu (A)$, or, in formulas, whether

$$\begin{aligned} \lim _{T\rightarrow \infty }\frac{1}{T}\int _{0}^T \chi _A \left( \varphi _t(x)\right) \text {d}t = \mu (A). \end{aligned}$$

(1)

Systems for which this is true for almost every initial condition x (with respect to $\mu$) are ergodic.⁸ A stronger conclusion, namely that equidistribution holds for every point $x\in X$ with an infinite trajectoriy, holds for smooth flows which are uniquely ergodic.⁹

A stronger property, mixing, guarantees equidistributions not only of individual orbits, but of sets pushed under the flow $\varphi _{\mathbb R}$: in a mixing system, every measurable set $A\subset X$ equidistributes (with respect to $\mu$) under the flow, i.e.

$$\begin{aligned} \lim _{t\rightarrow \infty }\mu (\varphi _t(A)\cap B) = \mu (A)\mu (B) \end{aligned}$$

(2)

for every measurable set B. This property is equivalent to decay of correlations, i.e. for every two smooth observables $f, g: X\rightarrow {\mathbb R}$,

$$\begin{aligned} \lim _{t\rightarrow \infty }\int _X \left( f\circ \varphi _t \right) g\, \mathrm {d}\mu - \int _X f \mathrm {d}\mu \int _X g\, \mathrm {d}\mu = 0. \end{aligned}$$

(3)

This property is also known as strong mixing, to distinguish it from another (weaker) property known as weak mixing (where the convergence in (2) is only required to happen along a subset of $t\in {\mathbb R}$ of density one). Mixing and weak mixing can also be interpreted as spectral properties, (see footnote 11). Other type of mixing properties in addition to weak mixing and strong mixing include mild mixing and mixing of all orders. The latter generalizes mixing (which is defined using two sets A, B) to more sets: a measure preserving flow $\varphi _{\mathbb R}$ on $(X,\mu )$ is mixing of order N if, for any N-tuple $A_0$,$\dots$, $A_{N-1}$ of measurables sets,

$$\begin{aligned}&\mu \left( A_0 \cap \varphi _{{t_1}}({A_1})\cap \varphi _{{t_1+t_2}}({A_2}) \cap \cdots \cap \varphi _{{t_1+\cdots +t_{N-1}}}({A_{N-1}}) \right) \xrightarrow {t_1,t_2\ldots , t_{N-1} \rightarrow \infty }\nonumber \\&\mu ( { A_0}) \cdots \mu ({A_{N-1}} ) \end{aligned}$$

(4)

and it is mixing of all orders if it is N-mixing for any $N\ge 2$. Equivalently, as for the definition of mixing, this can be reinterpreted as a statement about decay of multi-correlations. It is a famous open conjecture, known as Rohlin’s conjecture and still open, whether mixing implies mixing of all orders.

2.3 Spectral theory of dynamical systems

Study the nature of the spectrum (and spectral measures) associated to the Koopman operator a (family of) operator(s) on $L^2(X,\mathscr {A},\mu )$ associated to measure preserving flow $\varphi _{\mathbb R}$ (which acts by pre-composition $f\mapsto f\circ \varphi _t$ with the dynamics). One of the fundamental questions in spectral theory (see for example the surveys [49] or [58] on spectral theory of dynamical systems) is what is the nature of the spectrum of the Koopman operator. To every $f\in L^2(X,\mu )$ one can associate a spectral measure denoted by $\sigma _f$, i.e. the unique finite Borel measure on ${\mathbb R}$ whose Fourier coefficients are described by correlations, i.e. such that

$$\begin{aligned} \int _X f\circ \varphi _{t} \overline{f} \,d\mu =\int _{\mathbb R}e^{its}\,d\sigma _g(s)\quad \text {for every}\quad t\in {\mathbb R}. \end{aligned}$$

(5)

Spectral measures are useful to describe components of the the unitary representation given by the Koopman operator.¹⁰ We say that the spectrum of $\varphi _{\mathbb R}$ is (absolutely) continuous, or respectively (purely) singular iff for every $f\in L^2(X,\mu )$ the spectral measure $\sigma _f$ is (absolutely) continous, or respectively singular with respect to the Lebesgue measure on ${\mathbb R}$.

Ergodicity, weak mixing and mixing can be expressed in terms of the spectrum of the Koopman operator.¹¹ Spectral results thus provide finer and stronger dynamical information. For example, since mixing (and weak mixing), when they hold, provide, as spectral implication, the information that the spectrum is continuous (see footnote 11), proving that the spectrum is absolutely continuous is e. g.  a strengthening of mixing, while singularity of the spectrum shows that the system studied is far (more formally, spectrally disjoint, a stronger concept of disjointess than that introduced in Sect. 3.13) from strongly, fast chaotic systems.

2.4 Generic chaotic properties in slowly chaotic systems

There is a large and quite extensive literature on topological, ergodic and spectral properties of some of the classical parabolic examples mentioned in Sect. 1. For example, the fine ergodic and spectral properties of the horocycle flow have been studied in great detail¹² and mostly were already well understood in the 1970s (see for example [12, 22, 29, 36, 57, 66, 81] and more in general [80] or [2], and the reference therein, for unipotent flows).

It is well known that the typical¹³ nilflow is minimal and uniquely ergodic [2], however, nilflows are never (weak) mixing, due to an intrinsic obstruction, namely the presence of a toral factor.¹⁴ Results on the speed of equidistribution of Heisenberg nilflows for smooth functions were proved by Flaminio and Forni [23].

A series of recent works [4, 5, 74] indicates that, even though classical nilflows are never mixing (see footnote 14), a typical time-change (in a dense class of smooth time-changes) of a mimimal nilflow on any nilmanifold (different from a torus) is mixing.

In the rest of this survey we will focus on generic chaotic properties of smooth area preserving flows. While the understanding of minimality and ergodicity follows from results from the 1970s and 1980s respectively, the study of mixing properties has been the object of active research in the last decade or so, while the first breakthroughs on spectral properties are only very recent.

These results do not show an entirely coherent picture and make the identification and description of characteristic parabolic features difficult. For example, while the horocycle flow (as well as all its smooth time-changes) are mixing (and actually mixing of all orders, see [61, 62]), as we recalled above (see in particular footnote 14) nilflows are never mixing. This difference in behavior can be attributed to the lack of parabolicity in certain directions (those that live in the toral factor, see foonote 13), which would suggest calling nilflows partially parabolic systems. Nevertheless, this obstruction can be broken by a perturbation (as we show in [5], see also [4] for the special case of Heisenberg nilflows), so that in a dense set of smooth-time changes all flows which are not trivially conjugate to the nilflow itself are indeed mixing. Similarly, recent results seem to indicate that certain disjointess properties (see Sect. 3.13), which do not hold for a classical example such as the horocycle flow, are nevertheless generic among time-changes.

This shows that to better understand slowly chaotic systems it is therefore crucial both to understand what are the finer chaotic properties of the known parabolic examples as well as, at the same time, to study new classes of parabolic examples (such as those produced by time-changes and other parabolic perturbations.¹⁵

3 Chaotic properties of smooth surface flows

We will focus now on the class of slowly chaotic systems given by smooth area-preserving (or locally Hamiltonian) flows on surfaces, surveying the recent advances in our understanding of their mixing, spectral and disjointness properties as well as the mechanisms which explain them.

3.1 Flows on surfaces

Flows on surfaces are one of the most basic and most fundamental examples of dynamical systems, whose study goes back to Poincaré [67] at the end of the Ninetineth century, and coincides with the birth of dynamical systems as a research field. Many models of systems of physical origin are described by surface flows: Poincaré motivation to study surface flows was for example related to celestial mechanics and the two physical systems already mentioned before, the Ehrenfest model in statistical mechanics and the Novikov model in solid state physics, can be described by flows on surfaces (respectively linear flow on an translation surface and to a locally Hamiltonian flows).

In addition to providing a fundamental classes of parabolic dynamical systems, smooth, area-preserving flows on surfaces, are fundamental in dynamics because they are among the lowest possible dimensional smooth dynamical system (on compact manifolds of lower dimension, the other fundamental class of smooth dynamical systems are circle diffeomorphisms, whose rich theory is a cornerstone of dynamics). Despite having zero entropy, as shown in [90], they nevertheless display a rich variety of chaotic properties and, despite their basic nature, there are still many open questions on the mathematical characterization of chaos (in particular on dynamical, spectral and rigidity question) in various natural classes of surface flows.

In this survey we will only be concerned with flows which preserve a (probability) measure (see Sect. 2 for the definition), for example an area-form, since this is the natural setup for ergodic theory (see Sect. 2).

3.2 Locally Hamiltonian flows

Let S be a compact, connected, orientable (smooth) surface and let g denote its genus. We will assume throughout that $g\ge 1$. Perhaps the most natural class of measure-preserving flows on S are smooth flows preserving a smooth measure (with smooth absolutely continuous density). Let $\omega$ be a fixed smooth area form (locally given in coordinates (x, y) by $f(x,y) dx\wedge dy$ where f is a smooth function). Thus, equivalently, the pair $(S, \omega )$ is a two-dimensional symplectic manifold. We will consider a smooth flow $\varphi _{\mathbb R}=(\varphi _t)_{t\in \mathbb {R}}$ on S which preserves a measure $\mu$ given integrating a smooth density with respect to $\omega$. We will assume that the area is normalized so that $\mu (S) =1$. It turns out that such smooth area preserving flows on S are in one-to-one correspondence with smooth closed real-valued differential 1-forms as follows. Given a smooth, closed, real-valued differential 1-form $\eta$, let X be the vector field determined by $\eta = i_X \omega$ where $i_X$ denotes the contraction operator, i.e. $i_X \omega =\omega ( \eta , \cdot )$ and consider the flow $\varphi _{\mathbb R}$ on S given by X. Since $\eta$ is closed, the transformations $\varphi _t$, $t \in \mathbb {R}$, are area-preserving. Conversely, every smooth area-preserving flow can be obtained in this way.

Fig. 1

Trajectories of locally Hamiltonian flows on a surfaces

The flow $\varphi _{\mathbb R}$ is known as the multi-valued Hamiltonian flow associated to $\eta$. Indeed, the flow $\varphi _{\mathbb R}$ is locally Hamiltonian, i.e. locally one can find coordinates (x, y) on S in which $\varphi _{\mathbb R}$ is given by the solution to the equations

$${\left\{ \begin{array}{ll}\dot{x}&{}={\partial H}/{\partial y},\\ \dot{y}&{} =-{\partial H}/{\partial x}\end{array}\right. }$$

for some smooth real-valued Hamiltonian function H. A global Hamiltonian H cannot be in general be defined (see [64, Section 1.3.4]), but one can think of $\varphi _{\mathbb R}$ as globally given by a multi-valued Hamiltonian function.

Fig. 2

Type of singularities (i.e. fixed points) of a locally Hamiltonian flow

Let us remark that locally Hamiltonian flows necessarily have fixed points, or singularities, if $g \ge 2$. Singularities, as shown in Fig. 2, can be either centers (Fig. 2a), simple saddles (Fig. 2b) or multi-saddles (i.e. saddles with 2k pronges, $k\ge 2$, see Fig. 2c for $k=3$), . Examples of flow trajectories are shown in Fig. 1. For $g=1$, i.e. on a torus, if there is a singularity than there has to be another one. The simplest examples of locally Hamiltonian flows with singularities on a torus, i.e. flows with one center and one simple saddle (see Fig. 1b), were studied by Arnold in [1] and are nowadays often called Arnold flows.¹⁶

A lot of interest in the study of multi-valued Hamiltonians and the associated flows–in particular, in their ergodic and mixing properties—was sparked by Novikov [65] in connection with problems arising in solid-state physics as well as in pseudo-periodic topology (see e.g. the survey [91] by Zorich). Indeed, Novikov [65] and his school in the 1990s advocated the study of locally Hamiltonian flows as model to describe the motion of an electron in a metal under a magnetic field in the semi-classical approximation (the surface appears here as Fermi energy level surface). Novikov made some conjectures (known as Novikov problem) on the asymptotic behaviour of trajectories of electrons. At the same time, Arnold [1] made a conjecture on mixing for the flows we call today Arnold flows (see footnote 16). This conjecture has been the motivation for a lot of the work on the mixing properties of locally Hamiltonian flows, see the overview given in Sect. 3.8.

In order to survey the current knowledge of chaotic properties of locally Hamiltonian flows, it is useful to first point out their relation with another well studied class of area-preserving flows on surfaces, namely linear flows (which, we stress for the reader, are not smooth surface flows).

3.3 Linear flows and time-changes of locally Hamiltonian flows

The basic example of a linear flow is the flow given on the torus $\mathbb {R}^2/\mathbb {Z}^2$ by solutions of

$$\begin{aligned} \left( x'(t) , y'(t)\right) =\left( \cos \theta , \sin \theta \right) , \end{aligned}$$

which move along at unit speed along (the image in $\mathbb {R}^2/\mathbb {Z}^2$ of) Euclidean lines. Linear flows (also called translation flows) can be defined more in general on translation surfaces, namely surfaces which are locally Euclidean outside a finite number of conical singularities (with cone angles $2\pi k, k\in \mathbb {N}$, which produce saddles of the flows with 2k prongs, see Fig. 2c for $k=3$). Notice that these flows preserve a Euclidean area (but are discontinuous flows, since singularities are reached in finite time).

It turns out that every minimal locally Hamiltonian flow on S (as well as the restriction of a locally Hamiltonian flow to one of its minimal components, see Sect. 3.6), in suitably chosen coordinates, are time-changes (or time-reparametrizations, we refer to Sect. 1 for the definition) of a linear flow on S (or a subsurface of S in the case of a minimal component). Thus, minimal locally Hamiltonian flows have the same trajectories (up to time-reparametrization) than linear flows on translation surfaces (see for example [91]). In particular, certain properties, such as minimality and ergodicity (as well as homological aspects such as the asymptotic behaviour in the Novikov problem), which depend only on trajectories as sets and not on their time-parametrization, can be deduced for locally Hamiltonian flows by studying them in linear flows. This was in part one of the original motivations (in addition to unfolding of rational billiards in the West) that sparked the interest of mathematicians such as Zorich in the ergodic theory of linear flows (see below). We stress though that, while some chaotic properties like ergodicity depend on the orbits of the flow, others (like mixing and spectral properties) crucially depend on the time-parametrization of the orbits and require ad-hoc techniques (see Sects. 3.9and 3.14).

3.4 Linear flows and Teichmueller dynamics

The study of linear flows on translation flows and their ergodic properties has been a highly topical area of research for the past four decades (from the 1980s), in connection with the study of billiards in (rational) polygons, interval exchange transformations (or for short IETs) and Teichmueller dynamics, a research area which has benefited from the contribution of several Fields medallists (including Avila, Kontsevich, McMullen, Mirzakhani and Yoccoz).

In virtue of this flourishing activity, the ergodic and spectral properties of typical (in the measure theoretical sense) translation flows are by now well understood. Let us say that a property holds for a typical linear flow if it holds in a.e. direction on almost every translation surface with respect to a natural measure on translation surfaces known Masur–Veech measure.¹⁷ One of the first results, shown already in the 1970s by Keane [50], is that a typical linear flow is minimal. Moreover, it is uniquely ergodic (which implies that every infinite orbit is not only dense, but also equidistributed, see Sect. 2). Unique ergodicity of typical linear flows was known as Keane’s conjecture and proved independently in the seminal works by Masur and Veech [63, 87] through renormalization techniques which gave birth to the topical field of Teichmueller dynamics. On the other hand linear flows are never mixing, as proved by Katok [46] already in the 1980s, but they are neverthelss typically (in the above sense) weak-mixing (refer to Sect. 2 for definitions). a long-standing conjecture settled by Avila and Forni in [3].

From the spectral theory perspective, for typical translation flows, the nature of the spectrum (which turns out to be singular continuous) has been known since seminal work by Veech, see [87, 88]). Recently there have been also advances in the spectral theory of non generic (especially self-similar) translation flows and IETs, see for example [9,10,11].

We now discuss locally Hamiltonian flows, explaining how minimality and unique ergodicity can be understood reducing the study of (minimal components of) locally Hamiltonian flows to linear flows, while the classification of mixing properties (described in Sect. 3.8) has been based on geometric mixing mechanisms specific to smooth slowly parabolic flows (such as shearing, see Sect. 3.9) and the spectral theory is only now starting to be understood (refer to Sect. 3.14). We first specify (in the next Sect. 3.5) the topology and measure that we will use on the space of locally Hamiltonian flows.

3.5 Genericity notions for locally Hamiltonian flows

Let us define two natural ways of defining a notion of generic (or typical) locally Hamiltonian flow, one topological and the other measure-theoretical.

One can define a topology on locally Hamiltonian flows by considering perturbations of closed smooth 1-forms by (small) closed smooth 1-forms.¹⁸ We say that a condition is generic (in the sense of Baire) if it holds for flows described by an open and dense set of forms with respect to this topology. For example, asking that the 1-form $\eta$ is Morse, i.e. it is locally the differential of a Morse function (which has non-degenerate zeros) is a generic condition.

A measure-theoretical notion of typical is defined as follows by using the Katok fundamental class (introduced by Katok in [45], see also [64]), i.e. the cohomology class of the 1-form $\eta$ which defines the flow. Let $\Sigma$ be the set of fixed points of $\eta$ and let k be the cardinality of $\Sigma$. Let $\gamma _1, \ldots , \gamma _n$ be a base of the relative homology $H_1(S, \Sigma , \mathbb {R})$, where $n=2g+k-1$. The image of $\eta$ by the period map Per is $Per(\eta ) = (\int _{\gamma _1} \eta , \ldots , \int _{\gamma _n} \eta ) \in \mathbb {R}^{n}$. The pull-back $Per_* Leb$ of the Lebesgue measure class by the period map gives the desired measure class on closed 1-forms. When we use the expression typical below, we mean full measure with respect to this measure class.

3.6 Periodic and minimal components

A generic locally Hamiltonian flow (in the sense of Baire category, with respect to the topology defined in the previous Sect. 3.5) has only non-degenerate fixed points, i.e. centers and simple saddles (see Fig. 2a, b), as opposed to degenerate multi-saddles (as in Fig. 2c). We call saddle connection a flow trajectory from a saddle to a saddle and a saddle loop a saddle connection from a saddle to the same saddle (see Fig. 3a). It can be shown that each center is contained in a disk filled with closed (i.e. periodic) trajectories and bounded by a saddle loop, called an island of periodic orbits, see Fig. 3a. Hence, in presence of centers, the flow $\varphi _{\mathbb R}$ is never minimal (since orbits in the complement of the island avoid the island and hence cannot be dense).

From the point of view of topological dynamics (as proved independently by Maier [60], Levitt [59] and Zorich [91]), each smooth area-preserving flow can be decomposed into subsurfaces (with boundary) on which the restriction of $\varphi _{\mathbb R}$ either foliates into closed (i.e. periodic) orbits and up to g subsurfaces (recall that g is the genus of S) on which (the restriction of) $\varphi _{\mathbb R}$ is minimal, i.e. every bi-infinite orbit is dense. The first ones are called periodic components and are either islands (as in Fig. 3a) or cylinders filled by periodic orbits and bounded by saddle loops, as in Fig. 3b. The latter are known as minimal components (see an example in Fig. 3c) and by topological reasons there cannot be more than g of them. The flows in Fig. 1, for example, can be decomposed, in the case of Fig. 1a, into two islands and one cylinder filled by closed orbits and two minimal components (one of of genus one and one of genus two), while, in the case of the flow on the torus in Fig. 1b, there is one island and one minimal component (the so-called Arnold flow).

Fig. 3

Examples of periodic and minimal components

Minimal components of a locally Hamiltonian flow (and in particular minimal such flows, for which S is in itself a minimal component), in suitably chosen coordinates, have the same orbits (up to time-reparametrization, see Sect. 1) than linear flows discussed in Sect. 3.3 (see e.g. [91]).

3.7 Minimality and ergodicity

To classify chaotic behaviour in locally Hamiltonian flows it is crucial to distinguish between two (complementary, up to measure zero) open sets (with respect to the topology described in Sect. 3.5): in the first open set, which we will denote by $\mathscr {U}_{min}$, the typical flow is minimal (in particular there are no centers and there is a unique minimal component). On the other open set that we will call $\mathscr {U}_{\lnot min}$ there are periodic components (bounded by saddle loops homologous to zero), but the typical flow is still minimal when restricted to each complementary (minimal) component.

Let us remark that if the flow $\varphi _{\mathbb R}$ given by a closed 1-form $\eta$ has a saddle loop homologous to zero (i.e. the saddle loop is a separating curve on the surface), then the saddle loop is persistent under small perturbations (see Section 2.1 in [91] or Lemma 2.4 in [73]). In particular, the set of locally Hamiltonian flows which have at least one saddle loop is open and gives the set denoted $\mathscr {U}_{\lnot min}$ above. The set $\mathscr {U}_{min}$ is given by the interior (which one can show to be non-empty) of the complement of $\mathscr {U}_{\lnot min}$, i.e. the set of locally Hamiltonian flows without saddle loops homologous to zero.¹⁹

The typical locally Hamiltonian flow (with respect to the measure defined in Sect. 3.5) is $\mathscr {U}_{\lnot min}$ is not only minimal, but also uniquely ergodic. For a typical flow in $\mathscr {U}_{\lnot min}$, the restriction of the flow on each minimal component is (uniquely) ergodic. Both results about minimality and ergodicity can be deduced from the classical results respectively by Keane and Masur and Veech (recalled in Sect. 3.4) respectively concerning of minimality and ergodicity of typical translation flows, by using that that the flow restricted to a minimal components is a time-change of a linear flow (see Sect. 3.3).²⁰

3.8 Mixing properties of locally Hamiltonian flows

As mentioned earlier, finer chaotic properties such as (weak) mixing and spectral properties, crucially depend also on the speed of motion along the orbits.

The question of mixing in locally Hamiltonian flows was motivated by Arnold’s conjecture in the 1990s. In contrast with translation flows, which are never mixing (see Sect. 3.4), Arnold in the 1990s noticed a geometric phenomenon (explained in Sect. 3.8) which could produce mixing in locally Hamiltonian flows on the torus with one minimal component (those which we nowadays call Arnold flows, see Sect. 3.6 and in particular footnote 16). His intuition was proved to be correct shortly after by Sinai and Khanin [51], for a full measure set (later improved by Kocergin [53]) of such flows on tori. The question of whether mixing is typical also for flows on higher genus is much more delicate, and stayed open for 2 decades.

It turns out that mixing depends crucially on the type of singularities of the flow. When there are degenerate saddles (i.e. multi saddles with $k \ge 6$ prongs, as in Fig. 2c), mixing had been proved already in the 1970s (by Kochergin in [53]) since in this case the saddles have a much stronger effect.²¹ In the case of non-degenerate saddles (which, we recall, is generic case, but much more delicate to treat), one has very different results in the open sets $\mathscr {U}_{min}$ and $\mathscr {U}_{\lnot min}$ introduced in the previous Sect. 3.7. The full classification of mixing properties has been a central part of my past research achievements [84,85,86]. The two following two results now give a complete picture:

Theorem 1

(U’ [85, 86]²²)

In $\mathscr {U}_{min}$, the typical locally Hamiltonian flow is weakly mixing, but it is not mixing.

Theorem 2

(U’ [84], Ravotti [73]²³) In $\mathscr {U}_{\lnot min}$, the restriction of the typical locally Hamiltonian flow $\varphi _\mathbb {R}$ on each of its minimal components is mixing.

Let us remark even though the typical flow in $\mathscr {U}_{min}$ is not mixing, there exists exceptional non-mixing flow in this open set as it was shown by Chaika and Wright [14]. Examples of mild mixing (which is an intermediate property between weak mixing and mixing) were also built in [41] by Kanigowski and Kułaga-Przymus, using the former work [60] of the latter, but again are non typical (and one might conjecture that mild mixing is indeed non typical).

Furthermore, there are also quantitative results on the speed of mixing (when there is mixing) which show that it happens (as expected in a parabolic flow) very slowly. More precisely, for a typical $\varphi _{\mathbb R}$ in $\mathscr {U}_{\lnot min}$, restricted to a a minimal component (which is mixing and hence display decay of correlations, refer to Sect. 2 for definitions), the speed of decay of correlations (also sometimes called speed of mixing) is sub-polynomial (in accordance to what we expect for a slowly chaotic flow) and actually logarithmic, namely for every pair f, g of smooth observables there exists constants $c>0, \alpha >0$ such that $|C_{f,g}(t)|\le c \log t ^\alpha$ (as shown by Ravotti in [73]).

3.9 The role of shearing in slow mixing

The parabolic nature of locally Hamiltonian flows is entirely due to the presence of the saddles, which split nearby trajectories (as shown in Fig. 6) and are responsible for the slow divergence of nearby trajectories through a geometric phenomenon called shearing (pictured in Fig. 4). The butterfly effect in locally Hamiltonian flows happens indeed in a special way. In presence of a Hamiltonian saddle, the closer a trajectory is to a saddle point, the more motion along the trajectory is slowed down. Thus, if we consider a small arc $\gamma$ transversal to the trajectories of the flow $\varphi _{\mathbb R}$, so that when flowing it, $\varphi _t(\gamma )$ passes nearby a saddle separatrix without hitting the saddle point (as shown in Fig. 4), the different deceleration rates of points cause $\varphi _t(\gamma )$ to shear in the direction of the flow (see Fig. 4a). This description also evidences the slow nature of the butterfly effect in this case: points on nearby trajectories diverge from each other since, even though they travel on nearby trajectories, one travels faster than the other. Therefore the speed of divergence is in this case only as fast as the speed of shearing.

The shearing accumulated can be later destroyed when the $\varphi _t(\gamma )$ passes near the other side of a saddle (see Fig. 4b). The presence of a saddle loop, though, (as in Fig. 4c) typically creates an asymmetry (this was the key intuition of Arnold that had motivated his conjecture on mixing) by producing stronger shearing on one side and hence, in this case, the accumulation of shearing predominantly in one direction produces global shearing.

Fig. 4

Shearing mechanism in locally Hamiltonian flows

This geometric shearing phenomenon is a crucial ingredient in the proofs of the mixing results in Sect. 3.8 (in particular Theorem 2, but also to prove mixing in the exceptional examples in [14]) as it allows to deduce mixing from ergodicity (i.e. equidistribution of flow trajectories, recall Sect. 2). For large times $t\gg 0$, segments which do not hit the singularities will be so sheared along the flow, to be well approximated by long flow trajectories. Thus, for any given measurable set $A\subset X$, for every large t one can cover an arbitrarily large proportion $A_t\subset A$ of with a collection of short transversal segments $\{ \gamma _\alpha , \alpha \in \mathscr {A}_t \}$, each of which, after time t, shadows a long trajectory of $\varphi _{\mathbb R}$, which is (close to) equidistributed by ergodicity. One can hence show that each $\varphi _t(\gamma _\alpha )$ is also (close to) equidistributed. Thus, since

$$\begin{aligned} \varphi _t (A_t)\cap B = \cup _{\alpha \in \mathscr {A}_t} \left( \varphi _t(\gamma _\alpha )\cap B\right) \end{aligned}$$

by a Fubini argument, one can deduce equidistribution of $\varphi _t(A)$ (i.e. mixing, see Sect. 2) from equidistribution of each $\varphi _t(\gamma _\alpha )$ (which follows as we said from shearing and unique ergodicity). Furthermore, the speed of mixing (or equivalently the speed of decay of correlation) depends on the speed of shearing, which is slow (namely subpolynomial in this case).

This mechanism for mixing via shearing seem to be a very common phenomen in slowly chaotic dynamics. The few of the early results on horocycle flows (such as Marcus proof of mixing in [61] or Ratner’s results, see Sect. 3.10) exploit that small segments of geodesics curves, pushed by the horocycle flow, are sheared in the horocycle direction.²⁴ Furthermore, since this is essentially a geometric mechanism for explaining mixing, this phenomenon persists under perturbation and hence can be used also for time-changes (see [29, 61], where we prove quantitative mixing results and show polynomial estimates on the decay of correlations for smooth time-changes of the horocycle flow).

A similar mechanism, namely shearing of segments of a suitable foliation (but with the difference that the direction of shearing is not global but depends on the segment considered) was also exploited in [20] to prove mixing in some (exceptional) elliptic flows²⁵ and in the context of nilflows: while nilflows are never mixing (see footnote 14), in suitable classes of smooth time-changes one can implement this mechanism to prove mixing using shearing, see [4, 5, 74].

Finally, the complementary results on absence of mixing (see Theorem 1) involve showing absence of shearing.²⁶ Indeed, a criterion for absence of mixing already formulated by Kocergin in [52] shows that (at least for typical) locally Hamiltonian flows mixing via shearing is essentially the only possible way of achieving mixing.

3.10 Beyond mixing, exploting shearing: Ratner’s work

Whether one can deduce stronger and finer ergodic and spectral properties from shearing, in the context of flows with singularities, has been an open problem for decades, which has seen advances only very recently (see Sect. 3.12). A great example of the fine and deep results on finer ergodic properties and rigidity phenomena that one can obtain from shearing is given by the celebrated works by Marina Ratner on the horocycle flow (and more generally unipotent flows in homogeneous dynamics) [68,69,70,71]. Her work, and more in general the rigidity theory for unipotent flows, developed by Dani, Margulis and many others, has found breakthrough applications and has led to the solutions of important problems in number theory (such as the Oppenheim conjecture) and mathematical physics (such as the Bolztmann-grad limit for the Lorentz-gas).

Shearing is at the at the heart of Ratner’s work and the above mentioned rigidity results. A crucial ingredient in her work, indeed, is a technical property introduced in [70] (that she calls property H), nowadays known as the Ratner property (see [82]). It is this property, that Ratner verified for horocycle flows, that is used to deduce some of the main rigidity properties of horocycle flows (such as joinings and measure rigidity).

Fig. 5

The Ratner property describing quantitative slow shearing

The Ratner property encodes a quantitative property of controlled divergence of nearby trajectories in the flow direction (illustrated in Fig. 5). Heuristically, it requires that for most pairs of nearby points $x,x'$, the orbits of $x,x'$ split in the flow direction (say at time $t_1$) by a definite amount, called the shift and then realign, say by $\pm 1$ time-unit²⁷ so that now $\varphi _{t_1}(x)$ and the time-shifted orbit point $\varphi _{t_1\pm 1}(x')$ are close; then one requires the two orbits, $\left( \varphi _{t_1+t}(x)\right) _{t\ge 0}$ and the time-shifted orbit $\left( \varphi _{t+1\pm 1+t}(x')\right) _{t\ge 0}$, to still stay close (see Fig. 5) for a fixed proportion $\kappa$ of the time $t_1$ it took to see the shift, namely for most times $t \in [t_1 ,t_1+\kappa \, t_1]$. One can see that this type of phenomenon is possible only for parabolic systems, in which orbits of nearby points diverge with polynomial or subpolynomial speed.

3.11 Searching for Ratner properties beyond unipotent flows

Since the Ratner property describes a form of divergence of nearby trajectories (or butterfly effect) which is peculiar to parabolic flows, it is reasonable to expect that some quantitative form of parabolic divergence similar to the Ratner property should hold and be crucial in proving analogous rigidity properties for other classes of parabolic flows. Even more, since there is no formal definition for a system to be parabolic, one might even hope that the Ratner property could be taken as one of the characteristics making a system parabolic.

The natural question hence arose whether the Ratner property might hold for smooth flows on surfaces of higher genus. For a long time, though, there were no known examples of systems with the Ratner property beyond horocycle flows and their (smooth) time changes. This changed drastically in the last decade. The first examples outside the homogeneous world were given by Frączek and Lemańczyk in [33,34,35] (in the setting of special flows). The two authors could also show in [33] that a variant of Ratner’s property hold for some surface flows, more precisely in a class of flows on genus one tori known as von Neumann flows²⁸ (for non generic flows, corresponding to a measure zero set of frequencies). However, the flows in [33] are not (globally) smooth.

Fig. 6

Splitting of trajectories of a locally Hamiltonian flow near a saddle

The difficulty in treating smooth flows on higher genus surfaces is given by the presence of singularities (which are unavoidable when $g\ge 2$, see Sect. 3.2), which introduce discontinuities and destroy the slow form of divergence a la Ratner: essentially, as soon as two nearby trajectories are separated by hitting a saddle (see Fig. 6), one drastically looses control of the divergence. The Ratner property in its classical form (as well as the weaker versions defined in [33, 34]) is expected to fail for of smooth area-preserving flows with non-degenerate fixed points.²⁹

3.12 The Switchable Ratner property in locally Hamiltonian flows

The possibility of pushing our understanding of smooth flows on surfaces, using techniques loosely inspired by Ratner’s work, emerged only recently, in virtue of the recent developments in the field. A key breakthrough was achieved recently by Fayad and Kanigowski, who, in [18], introduced a new modification of the Ratner property, the so called Switchable Ratner property (or SR-property). According to this variation, it is sufficient to see the Ratner divergence of orbits for most pairs of initial conditions (x, y) either in the future (for $t>0$) or in the past (for $t<0$), depending on the pair of initial points. Thus, if one pair of nearby trajectories is separated by hitting a singularity (as shown in Fig. 6), and hence their distance explodes in an uncontrolled manner, one can still hope to be able to prove the Ratner slow form of divergence when flowing backward in time.

Let us remark that above mentioned variations Ratner property (thus in particular also the switchable Ratner property) were defined in order to have the same strong dynamical consequences of the original Ratner property. In particular, all variants of the Ratner property, as the original Ratner property does, imply a rigidity-type result on joinings³⁰ results (by restricting the type of self-joinings that the flow can have³¹) and allow to enhance mixing properties (see for example Corollary 1).

Fayad and Kanigowski could prove in [18] that this variation of the Ratner property holds for some smooth surface flows in genus one, more precisely for typical Arnold flows (see Sect. 3.6, Fig. 1b) as well as for (a measure zero class of) torus flows with one degenerate (or fake) singularity (sometimes known as Kocergin flows). Let us recall that in higher genus ($g\ge 2$) it is important to distinguish between the two open sets $\mathscr {U}_{min}$ and $\mathscr {U}_{\lnot min}$ (see Sect. 3.7) of locally Hamiltonian flows with non-degenerate singularities. In [41], the SR-property was proved for some (measure zero set of) minimal smooth flows³² in $\mathscr {U}_{min}$. It is likely that to prove a form of Ratner properties for other flows (hopefully a full measure set) in $\mathscr {U}_{min}$ will require introducing yet another variant of the Ratner property, one which could take into consideration average shearing and thus will require new ideas.

The result in [41], on the other hand, shows that the switchable Ratner property holds for (the minimal component of) the typical (Arnold) flow in $\mathscr {U}_{\lnot min}$ when $g=1$ and the flow has only one simple saddl (and center). In joint work with Kanigowski and Kułaga-Przymus [42], we could prove that the switchable version of the Ratner property is typical among mixing (components of) locally Hamiltonian flows in $\mathscr {U}_{\lnot min}$ for any genus $g\ge 1$ (thus extending to more singularities³³ and generalizing to higher genus $g\ge 2$ the result by [18]):

Theorem 3

(Kanigowski, Kułaga-Przymus and U’ [42]) For any $g\ge 1$, a typical locally Hamiltonian flow $\varphi _{\mathbb R}$ in $\mathscr {U}_{\lnot min}$, restricted on any of its mininal component, has the switchable Ratner form of shearing.

This result hence imply a rigidity type result for the classification of joinings (see footnote 31) and in particular allowed us to upgrade mixing to a stronger property, namely mixing of all orders (see Sect. 2 and (4) for the definition).

Corollary 1

(KKU) For any $g\ge 1$, the restriction of a typical locally Hamiltonian flow $\varphi _{\mathbb R}$ in $\mathscr {U}_{\lnot min}$ on any of its miminal components is mixing of all orders.

Thus, the Corollary show that Rohlin’s conjecture (see Sect. 2) holds for these class of smooth flows.

Further recent works also show that Ratner properties also hold for other classes of slowly chaotic flows. For example the Switchable Ratner property holds for a class of time changes of constant type Heisenberg nilflows, see the recent work [18] by Forni and Kanigowski.

3.13 Disjointness of rescalings

Advances in our understanding of disjointness properties became possible building on the switchable Ratner property [6, 19, 29]. The notion of disjointness³⁴ was introduced in the 1970s by Furstenberg (see in particular [37]); two disjoint flows are in particular not isomorphic.³⁵

A disjointness property which has received a lot of attention recently (in particular as a tool in connection with Sarnak’s conjecture on Moebius orthogonality, see below) is disjointness of rescalings. Given a real number $\kappa >0$, by the $\kappa$-rescaling of $\varphi _{\mathbb R}$ we simply mean the flow $\varphi ^\kappa _{\mathbb R}:=(\varphi _{\kappa t})_{t\in {\mathbb R}}$ (in which the time is rescaled by the factor $\kappa$).³⁶ Thus, a rescaling is a special type of time-reparametrization of a flow, given by a linear time-change. We say that $\varphi _{\mathbb R}$ has disjoint rescalings if for all (or all but finitely many) $p,q>0$, the rescalings $\varphi _{\mathbb R}^p$ and $\varphi _{\mathbb R}^q$, where $p,q>0$ and $p\ne q$, are disjoint (in the sense of Furstenberg). Disjointness of rescalings has played a key role in proving some of the first instances of Sarnak’s conjecture [77] of orthogonality of the Moebius function³⁷ in number theory with entropy zero dynamical systems (as a tool to prove the conjecture via the so called Katai orthogonality criterion, see for example [8, 25] and more in general the survey [21]).

In recent joint work with Kanigowski and Lemańczyk [40], we introduced a new tool to study disjointness phenomena for smooth surface flows, namely a disjointness criterion based on the switchable Ratner property. The criterion was devised and formulated so that it can be applied to prove disjointness of two flows which both have the switchable Ratner property, so that in both flows one can observe a controlled form of divergence of nearby trajectories (for example polynomial divergence), but the speed of divergence for the two flows is different (for example for one flow is it linear, in the other quadratic).³⁸ Exploiting this criterium, we were able to show that disjointness of rescalings is typical among Arnold flows (see Sect. 3.6, Fig. 1b).

Theorem 4

(Kanigowski–Lemanczyk-U’, [40]) A typical Arnold flow has disjoint rescalings. In particular, if $\varphi _{\mathbb R}$ is the restriction of the unique minimal component of an Arnold flow, there exists only two values³⁹ of the form q, 1/q such that $\varphi _{\mathbb R}$ and $\varphi _{\mathbb R}^p$ are disjoint for any positive $p\notin \{1,q,1/q\}$.

As a Corollary, Sarnak’s Moebius disjointness conjecture holds for these flows (see [40] for details).

We believe disjoitness of rescalings should also hold for typical locally Hamiltonian flows in higher genus, but this is currently an open problem. Preliminary work seems to indicate that, despite some technical additional difficulties, the techniques used to prove Theorem 4 should allow to prove disjointness of rescalings for all minimal components of typical flows in the open set $\mathscr {U}_{\lnot min}$ (refer to Sect. 3.7 for the definition of the open set $\mathscr {U}_{\lnot min}$ ). The recent work [6] by Berk and Kanigowski, even though it does not apply to surface flows directly, gives a good indication that disjointness of rescalings could also hold for typical flows (under a suitable full measure Diophantine-type condition) in the complementary set $\mathscr {U}_{\lnot min}$.

It is natural to ask whether disjoitness of rescalings could actually be a widespread feature of slowly chaotic systems. The new disjointness criterion is also used in [40] to prove disjointness of rescalings for (a class of) smooth (non-trivial) time-changes of the horocycle flow (see also [25], where the result is proved for a more general class of time changes with different methods), answering in particular a question of Marina Ratner. Notice that two different rescalings of the (classical, non time-changed) horocycle flow $h_{\mathbb R}$ are never disjoint.⁴⁰ Thus, it seems that (non trivial) time-changes of the horocycle flow are in some sense better behaved and display chaotic features that the horocycle flow itself lacks, due to its homogeneous and self-similar nature. Perhaps unfortunately, the most studied and best understood model of a parabolic flow, the horocycle flow, may have not provided the most significant example in terms of generic chaotic properties. Hence, the importance of better understanding new and larger classes of slowly chaotic systems and their typical chaotic features.

The new criterion for disjointness introduced in [40] has already proved useful in different contexts, see for example the recent works [16, 28] where it is applied to study disjointess phenomena respectively for Heisenberg nilflows in [28], for von Neumann flows in genus one in [16]. Finally, the disjointness criterion is used in [40] also to show that a typical Arnold flow is disjoint from any smooth time change of the horocycle flow (and in particular from the classical horocycle flow itself), thus showing that these two classes of parabolic flows are truly distinct.

3.14 Spectral theory of locally Hamiltonian flows

The last aspect we want to discuss is the spectral theory of smooth area-preserving flows (introduced in Sect. 2). The spectral properties (and in particular what is the spectral type, see Sect. 3.14 for definitions) of locally Hamiltonian flows is a natural question, which has been lingering for decades (see e.g. [49, Section 6] and [58]). While the classification of mixing properties of locally Hamiltoninan flows is essentially complete (as summarized in Sect. 3.8), very little is known on the spectral properties of these flows beyond the case of genus one.

One of the few results in the literature concerning on the nature of the spectrum of some area-preserving surface flows was proved by Frączek and Lemańczyk in [32], who showed singularity of the spectrum for Blokhin examples⁴¹ (which were the first examples of ergodic flows on surfaces, see [7]). This gives examples of locally Hamiltonian flows on surfaces of any genus $\ge 1$ with singular continuous spectrum (see [30, Theorem 1]), but these are essentially built glueing genus one flows and thus, they are highly non typical.

It turns out that the geometric approach to (quantitative) mixing through shearing can sometimes be pushed to provide also spectral information on parabolic flows. For example, in joint work with Forni [29] we were able to show that from shearing estimates on can also access information about the spectrum of time-changes of the horocycle flow, in particular settling in particular a conjecture of Katok-Thouvenot (see also [83] where Tiedra de Aldecoa gave simultaneously a different proof by operator methods (see also a previous result by Kuschnirenko [61]). Further developments based on our approach were achieved also for unipotent flows by Simonelli [79].

A breakthrough on the spectral side was achieved recently in [19] for a special class of smooth area preserving flows with degenerate singularities when the genus of the underlying surface is one, sometimes known as Kochergin flows (since Kochergin [53] proved their mixing, for any $g\ge 1$). These are minimal flows on the torus with one stopping point (see Fig. 7a), also called fake singularity (this point can be seen as a degenerate fixed point with only $k=2$ prongs). Taking as a starting point the idea used by Forni an myself in [29] to prove absolute continuity of the spectrum for time changes of horocycle flows, Forni, Fayad and Kanigowski, proved in [20] that, if the degenerate singularity is sufficiently strong⁴², the spectrum is absolutely continuous (and actually Lebesgue).

Theorem 5

(Fayad, Forni, Kanigowski, [19]) A locally Hamiltonian flow in genus one with only one sufficiently strong degenerate singularity as fixed point has countable Lebesgue spectrum.

Countable Lebesgue spectrum is a strong spectral result, which implies in particular that the spectrum is absolutely continuous (see Sect. 2 for the definition). The result provides the first example of such a strong chaotic property in an entropy zero and low dimensional smooth system. We remark that stopping points (and more in general degenerate fixed points) are known to produce shearing and hence mixing [53] (at rates which are expected to be polynomial, see e.g. [17]). The absolute continuity of the spectrum is essentially⁴³ based on a decay of correlations which is square-summable.

Fig. 7

The locally Hamiltonian flows with absolutely continuous and singular spectra respectively in Theorems 5 and 6

A recent spectral breakthrough, which goes in the opposite direction, concerns the nature of the spectrum of locally Hamiltonian flows on genus two surfaces, and, to the best of our knowledge, is the first general spectral result for surfaces of higher genus, namely $g \ge 2$.

Theorem 6

(Chaika–Fraczek–Kanigowski-U’, [13]) A typical locally Hamiltonian flow on a genus two surface with two isomorphic simple saddles has purely singular spectrum.

This result in genus two was inspired by the singularity result proved by Fraczek and Lemańczyk (for special flows over rotations) in [30]. Their result indeed shows that, when one can prove absence of mixing and some form of (partial) rigidity, it might be possible to deduce singularity of the spectrum. Theorem 6 strengthens one of the early results on absence of mixing, i.e. the absence of mixing for typical flows in the same class ($g=2$, two isomorphic saddles) proved by Scheglov [78] (which is a special case of Theorem 1). As in [78], the assumptions are crucial since the underlying surface has an inner symmetry⁴⁴ which plays a crucial role in the proof. Nevertheless, we believe it should be possible to extend the result on higher genus exploiting the same singularity criterion used in [13] (which is an extension of the criterion used in [30] as well as [31]), coupled with the delicate estimates for absence of shearing proved in [86] to prove Theorem 1.

The nature of the spectrum for other classes of locally Hamiltonian flows is unknown. It might be conjectured, in view of Theorem 5 for Kocergin flows, that also in higher genus, in presence of sufficiently strong degenerate singular points, the spectrum is also absolutely continuous (and even countable Lebesgue), essentially thanks to a strong quantitative control of decay of correlations. It is not clear what to expect when the degenerate singularity is not sufficiently strong.⁴⁵

At the heart of our proof of Theorem 6, on the other hand, is a strengthening of results on absence of mixing (in particular of the works [30, 31, 78]). As already mentioned, we hope that the techniques of [86] might be pushed to allow to apply the singularity criterium for typical flows in the open set $\mathscr {U}_{min}$ of minimal, uniquely ergodic (weakly mixing) but not mixing locally Hamiltonian flows.

In the open set $\mathscr {U}_{\lnot min}$, which consists of flows with non-degenerate singularities that are not minimal, but have several minimal components, the nature of the spectrum (for the restriction of a typical flow to a minimal component) is unclear. These flows are indeed mixing, but with sub-polynomial rate (see [73], which provides logarithmic upper bounds) and it is not clear whether to expect singularity or absolute continuity of the spectrum.