Contact Anosov Flows Research Articles

Dynamical torsion for contact Anosov flows

Dynamical torsion for contact Anosov flows

Spectral Properties of Ruelle Transfer Operators for Regular Gibbs Measures and Decay of Correlations for Contact Anosov Flows

In this work we study strong spectral properties of Ruelle transfer operators related to a large family of Gibbs measures for contact Anosov flows. The ultimate aim is to establish exponential decay of correlations for Hölder observables with respect to a very general class of Gibbs measures. The approach invented in 1997 by Dolgopyat in “On decay of correlations in Anosov flows” and further developed in Stoyanov (2011) is substantially refined here, allowing to deal with much more general situations than before, although we still restrict ourselves to the uniformly hyperbolic case. A rather general procedure is established which produces the desired estimates whenever the Gibbs measure admits a Pesin set with exponentially small tails, that is a Pesin set whose preimages along the flow have measures decaying exponentially fast. We call such Gibbs measures regular. Recent results in Gouëzel and Stoyanov (2019) prove existence of such Pesin sets for hyperbolic diffeomorphisms and flows for a large variety of Gibbs measures determined by Hölder continuous potentials. The strong spectral estimates for Ruelle operators and well-established techniques lead to exponential decay of correlations for Hölder continuous observables, as well as to some other consequences such as: (a) existence of a non-zero analytic continuation of the Ruelle zeta function with a pole at the entropy in a vertical strip containing the entropy in its interior; (b) a Prime Orbit Theorem with an exponentially small error.

First Band of Ruelle Resonances for Contact Anosov Flows in Dimension 3

We show, using semiclassical measures and unstable derivatives, that a smooth vector field X generating a contact Anosov flow on a 3-dimensional manifold $$\mathcal {M}$$ has only finitely many Ruelle resonances in the vertical strips $$\{ s\in \mathbb {C}\ |\ \mathrm{Re}(s)\in [-\nu _{\min }+\varepsilon ,-\frac{1}{2}\nu _{\max }-\varepsilon ]\cup [-\frac{1}{2}\nu _{\min }+\varepsilon ,0]\}$$ for all $$\varepsilon >0$$ , where $$0<\nu _{\min }\le \nu _{\max }$$ are the minimal and maximal expansion rates of the flow (the first strip only makes sense if $$\nu _{\min }>\nu _{\max }/2$$ ). We also show polynomial bounds in s for the resolvent $$(-X-s)^{-1}$$ as $$|\mathrm{Im}(s)|\rightarrow \infty $$ in Sobolev spaces, and obtain similar results for cases with a potential. This is a short proof of a particular case of the results by Faure–Tsujii, using that $$\dim E_u=\dim E_s=1$$ .

Resonant spaces for volume-preserving Anosov flows

We consider Anosov flows on closed 3-manifolds preserving a volume form $\Omega$. Following Dyatlov and Zworski (2017) we study spaces of invariant distributions with values in the bundle of exterior forms whose wavefront set is contained in the dual of the unstable bundle. Our first result computes the dimension of these spaces in terms of the first Betti number of the manifold, the cohomology class $[\iota_{X}\Omega]$ (where $X$ is the infinitesimal generator of the flow) and the helicity. These dimensions coincide with the Pollicott-Ruelle resonance multiplicities under the assumption of $\textit{semisimplicity}$. We prove various results regarding semisimplicity on 1-forms, including an example showing that it may fail for time changes of hyperbolic geodesic flows. We also study non null-homologous deformations of contact Anosov flows and we show that there is always a splitting Pollicott-Ruelle resonance on 1-forms and that semisimplicity persists in this instance. These results have consequences for the order of vanishing at zero of the Ruelle zeta function. Finally our analysis also incorporates a flat unitary twist in both, the resonant spaces and the Ruelle zeta function.

On the Relation between Enhanced Dissipation Timescales and Mixing Rates

AbstractWe study diffusion and mixing in different linear fluid dynamics models, mainly related to incompressible flows. In this setting, mixing is a purely advective effect that causes a transfer of energy to high frequencies. When diffusion is present, mixing enhances the dissipative forces. This phenomenon is referred to as enhanced dissipation, namely the identification of a timescale faster than the purely diffusive one. We establish a precise connection between quantitative mixing rates in terms of decay of negative Sobolev norms and enhanced dissipation timescales. The proofs are based on a contradiction argument that takes advantage of the cascading mechanism due to mixing, an estimate of the distance between the inviscid and viscous dynamics, and an optimization step in the frequency cutoff.Thanks to the generality and robustness of our approach, we are able to apply our abstract results to a number of problems. For instance, we prove that contact Anosov flows obey logarithmically fast dissipation timescales. To the best of our knowledge, this is the first example of a flow that induces an enhanced dissipation timescale faster than polynomial. Other applications include passive scalar evolution in both planar and radial settings and fractional diffusion. © 2019 Wiley Periodicals, Inc.

Horocyclic invariance of Ruelle resonant states for contact Anosov flows in dimension $3$

Horocyclic invariance of Ruelle resonant states for contact Anosov flows in dimension $3$

Quantitative Pesin theory for Anosov diffeomorphisms and flows

Pesin sets are measurable sets along which the behavior of a matrix cocycle above a measure-preserving dynamical system is explicitly controlled. In uniformly hyperbolic dynamics, we study how often points return to Pesin sets under suitable conditions on the cocycle: if it is locally constant, or if it admits invariant holonomies and is pinching and twisting, we show that the measure of points that do not return a linear number of times to Pesin sets is exponentially small. We discuss applications to the exponential mixing of contact Anosov flows and consider counterexamples illustrating the necessity of suitable conditions on the cocycle.

The semiclassical zeta function for geodesic flows on negatively curved manifolds

We consider the semi-classical (or Gutzwiller-Voros) zeta function for $C^\infty$ contact Anosov flows. Analyzing the spectrum of transfer operators associated to the flow, we prove, for any $\tau>0$, that its zeros are contained in the union of the $\tau$-neighborhood of the imaginary axis, $|\Re(s)|<\tau$, and the region $\Re(s)<-\chi_0+\tau$, up to finitely many exceptions, where $\chi_0>0$ is the hyperbolicity exponent of the flow. Further we show that the zeros in the neighborhood of the imaginary axis satisfy an analogue of the Weyl law.

On the sub-Riemannian geometry of contact Anosov flows

We investigate certain natural connections between sub-Riemannian geometry and hyperbolic dynamical systems. In particular, we study dynamically defined horizontal distributions which split into two integrable ones and ask: how is the energy of a sub-Riemannian geodesic shared between its projections onto the integrable summands? We show that if the horizontal distribution is the sum of the strong stable and strong unstable distributions of a special type of a contact Anosov flow in three dimensions, then for any short enough sub-Riemannian geodesic connecting points on the same orbit of the Anosov flow, the energy of the geodesic is shared equally between its projections onto the stable and unstable bundles. The proof relies on a connection between the geodesic equations and the harmonic oscillator equation, and its explicit solution by the Jacobi elliptic functions. Using a different idea, we prove an analogous result in higher dimensions for the geodesic flow of a closed Riemannian manifold of constant negative curvature.

Decay of correlations for normally hyperbolic trapping

We prove that for evolution problems with normally hyperbolic trapping in phase space, correlations decay exponentially in time. Normally hyperbolic trapping means that the trapped set is smooth and symplectic and that the flow is hyperbolic in directions transversal to it. Flows with this structure include contact Anosov flows, classical flows in molecular dynamics, and null geodesic flows for black holes metrics. The decay of correlations is a consequence of the existence of resonance free strips for Green’s functions (cut-off resolvents) and polynomial bounds on the growth of those functions in the semiclassical parameter.

Ruelle operators and decay of correlations for contact Anosov flows

We prove strong spectral estimates for Ruelle transfer operators for arbitrary C2 contact Anosov flows. As a consequence of this we obtain: (a) existence of a non-zero analytic continuation of the Ruelle zeta function with a pole at the entropy in a vertical strip containing the entropy in its interior; (b) a Prime Orbit Theorem with an exponentially small error; (c) exponential decay of correlations for Hölder continuous observables with respect to any Gibbs measure.

Contact Anosov flows on hyperbolic 3–manifolds

Geodesic flows of Riemannian or Finsler manifolds have been the only known contact Anosov flows. We show that even in dimension 3 the world of contact Anosov flows is vastly larger via a surgery construction near an E ‐transverse Legendrian link that encompasses both the Handel‐Thurston and Goodman surgeries and that produces flows not topologically orbit equivalent to any algebraic flow. This includes examples on many hyperbolic 3‐manifolds, any of which have remarkable dynamical and geometric properties. To the latter end we include a proof of a folklore theorem from 3‐manifold topology: In the unit tangent bundle of a hyperbolic surface, the complement of a knot that projects to a filling geodesic is a hyperbolic 3‐manifold.

Band structure of the Ruelle spectrum of contact Anosov flows

If X is a contact Anosov vector field on a smooth compact manifold M and V∈C∞(M), it is known that the differential operator A=−X+V has some discrete spectrum called Ruelle–Pollicott resonances in specific Sobolev spaces. We show that for |Imz|→∞ the eigenvalues of A are restricted to vertical bands and in the gaps between the bands, the resolvent of A is bounded uniformly with respect to |Im(z)|. In each isolated band, the density of eigenvalues is given by the Weyl law. In the first band, most of the eigenvalues concentrate to the vertical line Re(z)=〈D〉M, the space average of the function D(x)=V(x)−12divX|Eu(x) where Eu is the unstable distribution. This band spectrum gives an asymptotic expansion for dynamical correlation functions.

Regular decay of ball diameters and spectra of Ruelle operators for contact Anosov flows

For Anosov flows on compact Riemann manifolds we study the rate of decay along the flow of diameters of balls $B^s(x,\ep)$ on local stable manifolds at Lyapunov regular points $x$. We prove that this decay rate is similar for all sufficiently small values of $\epsilon > 0$. From this and the main result in \cite{kn:St1}, we derive strong spectral estimates for Ruelle transfer operators for contact Anosov flows with Lipschitz local stable holonomy maps. These apply in particular to geodesic flows on compact locally symmetric manifolds of strictly negative curvature. As is now well known, such spectral estimates have deep implications in some related areas, e.g. in studying analytic properties of Ruelle zeta functions and partial differential operators, asymptotics of closed orbit counting functions, etc.

Contact Anosov flows and the Fourier–Bros–Iagolnitzer transform

AbstractThis paper is about spectral properties of transfer operators for contact Anosov flows. The main result gives the essential spectral radii of the transfer operators acting on an appropriate function space exactly and improves the previous result in Tsujii [Quasi-compactness of transfer operators for contact Anosov flows. Nonlinearity23 (2010), 1495–1545]. Also, we provide a simplified proof by using the so-called Fourier–Bros–Iagolnitzer (FBI) (or Bargmann) transform.

Quasi-compactness of transfer operators for contact Anosov flows

For any Cr contact Anosov flow with r ⩾ 3, we construct a scale of Hilbert spaces, which are embedded in the space of distributions on the phase space and contain all the Cr functions, such that the one-parameter family of transfer operators for the flow extend to them boundedly and that the extensions are quasi-compact. We also give explicit bounds on the essential spectral radii of those extensions in terms of differentiability r and the hyperbolicity exponents of the flow.

Thermodynamic invariants of Anosov flows and rigidity

By using a formula relating topological entropy and cohomological pressure, we obtain several rigidity results about contact Anosov flows. For example, we prove the following result: Let $\varphi$ be a $C^\infty$ contact Anosov flow. If its Anosov splitting is $C^2$ and it is $C^0$ orbit equivalent to the geodesic flow of a closed negatively curved Riemannian manifold, then the cohomological pressure and the metric entropy of $\varphi$ coincide. This result generalizes a result of U. Hamenstädt for geodesic flows.

On contact Anosov flows

Exponential decay of correlations for $\Co^{(4)}$ Contact Anosov flows is established. This implies, in particular, exponential decay of correlations for all smooth geodesic flows in strictly negative curvature.

Plane affine geometry and Anosov flows

We study Anosov flows on closed 3-manifolds. We define the notion of Anosov flows with the topological contact property (abreviation TCP Anosov flows): typical examples of TCP Anosov flows are contact Anosov flows, i.e. flows preserving a contact 1-form. We show that TCP Anosov flows are R -covered. The main tool is the study of the leaf spaces of lifted strong stable foliations: we exhibit on these leaf spaces a structure of (generalized) affine plane, in the sense of incidence geometry.

Markov Approximations and Decay of Correlations for Anosov Flows

We develop Markov approximations for very general suspension flows. Based on this, we obtain a stretched exponential bound on time correlation functions for 3-D Anosov flows that verify ‘uniform nonintegrability of foliations’. These include contact Anosov flows and geodesic flows on compact surfaces of variable negative curvature. Our bound on correlations is stable under small smooth perturbations.