Time-one Map Research Articles

Contributions to the ergodic theory of hyperbolic flows: unique ergodicity for quasi-invariant measures and equilibrium states for the time-one map

Contributions to the ergodic theory of hyperbolic flows: unique ergodicity for quasi-invariant measures and equilibrium states for the time-one map

A dichotomy for measures of maximal entropy near time-one maps of transitive Anosov flows

We show that time-one maps of transitive Anosov flows of compact manifolds are accumulated by diffeomorphisms robustly satisfying the following dichotomy: either all of the measures of maximal entropy are non-hyperbolic, or there are exactly two ergodic measures of maximal entropy, one with a positive central exponent and the other with a negative central exponent. We establish this dichotomy for certain partially hyperbolic diffeomorphisms isotopic to the identity whenever both of their strong foliations are minimal. Our proof builds on the approach developed by Margulis for Anosov flows where he constructs suitable families of measures on the dynamical foliations.

Refinements of topological invariants of flows

<p style='text-indent:20px;'>We construct topological invariants, called abstract weak orbit spaces, of flows and homeomorphisms on topological spaces. In particular, the abstract weak orbit spaces of flows on topological spaces are refinements of Morse graphs of flows on compact metric spaces, Reeb graphs of Hamiltonian flows with finitely many singular points on surfaces, and the CW decompositions which consist of the unstable manifolds of singular points for Morse flows on closed manifolds. Though the CW decomposition of a Morse flow is finite, the intersection of the unstable manifold and the stable manifold of closed orbits need not consist of finitely many connected components. Therefore we study the finiteness. Moreover, we consider when the time-one map reconstructs the topology of the original flow. We show that the orbit space of a Hamiltonian flow with finitely many singular points on a compact surface is homeomorphic to the abstract weak orbit space of the time-one map by taking an arbitrarily small reparametrization and that the abstract weak orbit spaces of a Morse flow on a compact manifold and the time-one map are homeomorphic. In addition, we state examples whose Morse graphs are singletons but whose abstract weak orbit spaces are not singletons.</p>

Statistical properties of physical-like measures* *SG is supported by NSFC 11831001 and NSFC 11771025. JY is partially supported by CNPq, FAPERJ, and PRONEX of Brazil and NSFC 11871487 of China.

In this paper we consider the semi-continuity of the physical-like measures for diffeomorphisms with dominated splittings. We prove that any weak-* limit of physical-like measures along a sequence of C 1 diffeomorphisms {f n } must be a Gibbs F-state for the limiting map f. As a consequence, we establish the statistical stability for the C 1 perturbation of the time-one map of three-dimensional Lorenz attractors, and the continuity of the physical measure for the diffeomorphisms constructed by Bonatti and Viana.

A Subgradient Algorithm for Data-Rate Optimization in the Remote State Estimation Problem

In the remote state estimation problem, an observer tries to reconstruct the state of a dynamical system at a remote location, where no direct sensor measurements are available. The observer only has access to information sent through a digital communication channel with a finite capacity. The recently introduced notion of restoration entropy provides a way to determine the smallest channel capacity above which an observer can be designed that observes the system without a degradation of the initial observation quality. In this paper, we propose a subgradient algorithm to estimate the restoration entropy via the computation of an appropriate Riemannian metric on the state space, which allows us to determine the approximate value of the entropy from the time-one map (in the discrete-time case) or the generating vector field (for ODE systems), respectively.

A symmetric Random Walk defined by the time-one map of a geodesic flow

<p style='text-indent:20px;'>In this note we consider a symmetric Random Walk defined by a <inline-formula><tex-math id="M1">\begin{document}$ (f, f^{-1}) $\end{document}</tex-math></inline-formula> Kalikow type system, where <inline-formula><tex-math id="M2">\begin{document}$ f $\end{document}</tex-math></inline-formula> is the time-one map of the geodesic flow corresponding to an hyperbolic manifold. We provide necessary and sufficient conditions for the existence of an stationary measure for the walk that is equivalent to the volume in the corresponding unit tangent bundle. Some dynamical consequences for the Random Walk are deduced in these cases.

Foliations and conjugacy, II: the Mendes conjecture for time-one maps of flows

AbstractA diffeomorphism of the plane is Anosov if it has a hyperbolic splitting at every point of the plane. In addition to linear hyperbolic automorphisms, translations of the plane also carry an Anosov structure (the existence of Anosov structures for plane translations was originally shown by White). Mendes conjectured that these are the only topological conjugacy classes for Anosov diffeomorphisms in the plane. Very recently, Matsumoto gave an example of an Anosov diffeomorphism of the plane, which is a Brouwer translation but not topologically conjugate to a translation, disproving Mendes’ conjecture. In this paper we prove that Mendes’ claim holds when the Anosov diffeomorphism is the time-one map of a flow, via a theorem about foliations invariant under a time-one map. In particular, this shows that the kind of counterexample constructed by Matsumoto cannot be obtained from a flow on the plane.

Classification of partially hyperbolic diffeomorphisms under some rigid conditions

AbstractConsider a three-dimensional partially hyperbolic diffeomorphism. It is proved that under some rigid hypothesis on the tangent bundle dynamics, the map is (modulo finite covers and iterates) an Anosov diffeomorphism, a (generalized) skew-product or the time-one map of an Anosov flow, thus recovering a well-known classification conjecture of the second author to this restricted setting.

Martingale approximations and anisotropic Banach spaces with an application to the time-one map of a Lorentz gas

In this paper, we show how the Gordin martingale approximation method fits into the anisotropic Banach space framework. In particular, for the time-one map of a finite horizon planar periodic Lorentz gas, we prove that Hölder observables satisfy statistical limit laws such as the central limit theorem and associated invariance principles. Previously, these properties were known only for a restricted class of observables, excluding for instance velocity.

Periodic measures and partially hyperbolic homoclinic classes

In this paper, we give a precise meaning to the following fact, and we prove it: $C^1$-open and densely, all the non-hyperbolic ergodic measures generated by a robust cycle are approximated by periodic measures. We apply our technique to the global setting of partially hyperbolic diffeomorphisms with one-dimensional center. When both strong stable and unstable foliations are minimal, we get that the closure of the set of ergodic measures is the union of two convex sets corresponding to the two possible $s$-indices; these two convex sets intersect along the closure of the set of non-hyperbolic ergodic measures. That is the case for robustly transitive perturbations of a time-one map of a transitive Anosov flow, or of the skew product of an Anosov torus diffeomorphism by a rotation of the circle.

Measure and Capacity of Wandering Domains in Gevrey Near-integrable Exact Symplectic Systems

A wandering domain for a diffeomorphism is an open connected set whose iterates are pairwise disjoint. We endow A^n = T^n x R^n with its usual exact symplectic structure. An integrable diffeomorphism Φ^h, i.e. the time-one map of a Hamiltonian h which depends only on the action variables, has no nonempty wandering domains. The aim of this paper is to estimate the size (measure and Gromov capacity) of wandering domains in the case of an exact symplectic perturbation of Φ^h , in the analytic or Gevrey category. Upper estimates are related to Nekhoroshev theory, lower estimates are related to examples of Arnold diffusion. This is a contribution to the quantitative Hamiltonian perturbation initiated in previous works on the optimality of long term stability estimates and diffusion times; our emphasis here is on discrete systems because this is the natural setting to study wandering domains. We first prove that the measure (or the capacity) of these wandering domains is exponentially small, with an upper bound of the form exp(-c e^(-1/2nα)), where e is the size of the perturbation, α is the Gevrey exponent (α is at least 1, α=1 for analytic systems) and c is some positive constant depending mildly on h. This is obtained as a consequence of an exponential stability theorem for near-integrable exact symplectic maps, in the analytic or Gevrey category, for which we give a complete proof based on the most recent improvements of Nekhoroshev theory for Hamiltonian flows, and which requires the development of specific Gevrey suspension techniques. The second part of the paper is devoted to the construction of near-integrable Gevrey systems possessing wandering domains, for which the capacity (and thus the measure) can be estimated from below. We suppose that n is at least 2, essentially because KAM theory precludes Arnold diffusion in too low a dimension. For any α>1, we produce examples with lower bounds of the form exp(-c e^(-1/2(n-1)(α-1))). This is done by means of a coupling technique, involving rescaled standard maps possessing wandering discs in A and near-integrable systems possessing periodic domains of arbitrarily large periods in A^(n-1). The most difficult part of the construction consists in obtaining a perturbed pendulum-like system on A with periodic islands of arbitrarily large periods, whose areas are explicitly estimated from below. Our proof is based on a version due to Herman of the translated curve theorem.

Topological entropy on points without physical-like behaviour

We study a class of asymptotically entropy-expansive $C^1$ diffeomorphisms with dominated splitting on a compact manifold $M$, that satisfy the specification property. This class includes, in particular, transitive Anosov diffeomorphisms and time-one maps of transitive Anosov flows. We consider the nonempty set of physical-like measures that attracts the empirical probabilities (i.e. the time averages) of Lebesgue-almost all the orbits. We define the set $I_f \cap \Gamma_f \subset M$ of irregular points without physical-like behaviour. We prove that, if not all the invariant measures of $f$ satisfy Pesin Entropy Formula (for instance in the Anosov case), then $I_f \cap \Gamma_f$ has full topological entropy. We also obtain this result for some class of asymptotically entropy-expansive continuous maps on a compact metric space, if the set of physical-like measures are equilibrium states with respect to some continuous potential. Finally, we prove that also the set $(M \setminus I_f) \cap \Gamma_f$ of regular points without physical-like behaviour, has full topological entropy.

On symplectic dynamics

This paper contributes to the foundation of various Hofer-like topologies of a closed symplectic manifold (M,ω). Here, considering an arbitrarily linear section of the natural projection of the space of closed 1-forms onto the first de Rham's group, we study various Hofer-like metrics [2,7]. The outcome is that different splitting of the space of closed 1-forms lead to similar “symplectic analogues” of some results from Hamiltonian dynamics: Without appealing to the positivity result of any symplectic displacement energy, we point out an impact of the L∞-Hofer-like lengths in the investigation of the symplectic nature of the uniform limits of sequences of symplectic diffeomorphisms isotopic to the identity map: This provides various symplectic analogues of a result that was proved by Hofer–Zehnder [10] (for compactly supported Hamiltonian diffeomorphisms on R2n); and reformulated by Oh–Müller [13] for Hamiltonian diffeomorphisms in general. Moreover, we show that Polterovich's regularization method for Hamiltonian paths extends over the whole group of symplectic paths (no matter the choice of the above section), and use this to prove the equality between the two versions of the corresponding Hofer-like metrics defined on the group of time-one maps of all symplectic isotopies: The symplectic analogues of the uniqueness result of Hofer's geometry [14]. This includes some condition(s) that make the latter uniqueness result continues holds in the context of Bus–Leclercq [7], and a short proof of the non-degeneracy of various Hofer-like metrics. Finally, an alternate proof (without appealing to Floer's theory) of a result from flux geometry found by McDuff–Salamon [12] is given, and various symplectic analogues of some approximation results found by Oh–Müller [13] are provided.

A multidimensional Birkhoff theorem for time-dependent Tonelli Hamiltonians

Let M be a closed and connected manifold, $$H:T^*M\times {{\mathbb {R}}}/\mathbb {Z}\rightarrow \mathbb {R}$$ a Tonelli 1-periodic Hamiltonian and $${\mathscr {L}}\subset T^*M$$ a Lagrangian submanifold Hamiltonianly isotopic to the zero section. We prove that if $${\mathscr {L}}$$ is invariant by the time-one map of H, then $${\mathscr {L}}$$ is a graph over M. An interesting consequence in the autonomous case is that in this case, $${\mathscr {L}}$$ is invariant by all the time t maps of the Hamiltonian flow of H.

Generalizations of the Action Function in Symplectic Geometry

In symplectic geometry, the action function is a classical object defined on the set of contractible fixed points of the time-one map of a Hamiltonian isotopy. On closed aspherical surfaces, we give a dynamical interpretation of this function, which permits us to generalize it to the case of a diffeomorphism that is isotopic to identity and preserves a Borel finite measure of rotation vector zero. We define a boundedness property on the contractible fixed points set of the time-one map of an identity isotopy. We generalize the classical action function to any Hamiltonian homeomorphism, provided that the proposed boundedness condition is satisfied. We prove that the generalized action function only depends on the time-one map but not on the isotopy. Finally, we define the action spectrum and show that it is invariant under conjugation by an orientation and measure preserving homeomorphism.

The topological entropy of non-dense orbits and generalized Schmidt games

We generalize the notion of Schmidt games to the setting of the general Caratheódory construction. The winning sets for such generalized Schmidt games usually have large corresponding Caratheódory dimensions (e.g., Hausdorff dimension and topological entropy). As an application, we show that for every $C^{1+\unicode[STIX]{x1D703}}$-partially hyperbolic diffeomorphism $f:M\rightarrow M$ satisfying certain technical conditions, the topological entropy of the set of points with non-dense forward orbits is bounded below by the unstable metric entropy (in the sense of Ledrappier–Young) of certain invariant measures. This also gives a unified proof of the fact that the topological entropy of such a set is equal to the topological entropy of $f$, when $f$ is a toral automorphism or the time-one map of a certain non-quasiunipotent homogeneous flow.

Rigidity for partially hyperbolic diffeomorphisms

In this work we completely classify$C^{\infty }$conjugacy for smooth conservative (pointwise) partially hyperbolic diffeomorphisms homotopic to a linear Anosov automorphism on the 3-torus by its center foliation behavior. We prove that the uniform version of absolute continuity for the center foliation is the natural hypothesis to obtain$C^{\infty }$conjugacy to its linear Anosov automorphism. Avila, Viana and Wilkinson [Absolute continuity, Lyapunov exponents and rigidity I: Geodesic flows.J. Eur. Math. Soc. (JEMS)17(6) (2015), 1435–1462] proved that for a perturbation in the volume preserving case of the time-one map of an Anosov flow absolute continuity of the center foliation implies smooth rigidity. The absolute version of absolute continuity is the appropriate scenario for our context since it is not possible to obtain a result analogous to that of Avila, Viana and Wilkinson for our class of maps, for absolute continuity alone fails miserably to imply smooth rigidity for our class of maps. Our theorem is a global rigidity result as we do not assume the diffeomorphism to be at some distance from the linear Anosov automorphism. We also do not assume ergodicity. In particular, a metric condition on the center foliation implies ergodicity and$C^{\infty }$center foliation.

Continuity of topological entropy for perturbation of time-one maps of hyperbolic flows

We consider a C 1 neighborhood of the time-one map of a hyperbolic flow and prove that the topological entropy varies continuously for diffeomorphisms in this neighborhood. This shows that the topological entropy varies continuously for all known examples of partially hyperbolic diffeomorphisms with one-dimensional center bundle.

Absolute continuity, Lyapunov exponents and rigidity I: geodesic flows

We consider volume-preserving perturbations of the time-one map of the geodesic flow of a compact surface with negative curvature. We show that if the Liouville measure has Lebesgue disintegration along the center foliation then the perturbation is itself the time-one map of a smooth volume-preserving flow, and that otherwise the disintegration is necessarily atomic.

On embeddability of automorphisms into measurable flows from the point of view of self-joining properties

We compare self-joining- and embeddability properties. In particular, we prove that a measure preserving flow $(T_t)_{t\in\mathbb{R}}$ with $T_1$ ergodic is 2-fold quasi-simple (2-fold distally simple) if and only if $T_1$ is 2-fold quasi-simple (2-fold distally simple). We also show that the Furstenberg-Zimmer decomposition for a flow $(T_t)_{t\in\mathbb{R}}$ with $T_1$ ergodic with respect to any flow factor is the same for $(T_t)_{t\in\mathbb{R}}$ and for $T_1$. We give an example of a 2-fold quasi-simple flow disjoint from simple flows and whose time-one map is simple. We describe two classes of flows (flows with minimal self-joining property and flows with the so-called Ratner property) whose time-one maps have unique embeddings into measurable flows. We also give an example of a 2-fold simple flow whose time-one map has more than one embedding.