We construct cross sections for the geodesic flow on the orbifolds $Γ\backslash H$ which are tailor-made for the requirements of transfer operator approaches to Maass cusp forms and Selberg zeta functions. Here, $H$ denotes the hyperbolic plane and $Γ$ is a nonuniform geometrically finite Fuchsian group (not necessarily a lattice, not necessarily arithmetic) which satisfies an additional condition of geometric nature. The construction of the cross sections is uniform, geometric, explicit and algorithmic.
We give a finitary criterion for the convergence of measures on non-elementary geometrically finite hyperbolic orbifolds to the unique measure of maximal entropy. We give an entropy criterion controlling escape of mass to the cusps of the orbifold. Using this criterion we prove new results on the distribution of collections of closed geodesics on such orbifold, and as a corollary we prove equidistribution of closed geodesics up to a certain length in amenable regular covers of geometrically finite orbifolds.
Let $Q$ be a closed manifold admitting a locally-free action of a compact Lie group $G$. In this paper we study the properties of geodesic flows on $Q$ given by Riemannian metrics which are invariant by such an action. In particular, we will be interested in the existence of geodesics which are closed up to the action of some element in the group $G$, since they project to closed magnetic geodesics on the quotient orbifold $Q/G$.
We prove that the geodesic flow on the unit tangent bundle to a hyperbolic 2-orbifold is left-handed if and only if the orbifold is a sphere with three conic points. As a consequence, on the unit tangent bundle to a 3-conic sphere, the lift of every finite collection of closed geodesics that is zero in integral homology is the binding of an open book decomposition.
We establish that, for every hyperbolic orbifold of type (2, q, $\infty$) and for every orbifold of type (2, 3, 4g+2), the geodesic flow on the unit tangent bundle is left-handed. This implies that the link formed by every collection of periodic orbits (i) bounds a Birkhoff section for the geodesic flow, and (ii) is a fibered link. We also prove similar results for the torus with any flat metric. Besides, we observe that the natural extension of the conjecture to arbitrary hyperbolic surfaces (with non-trivial homology) is false.
We construct a template with two ribbons that describes the topology of all periodic orbits of the geodesic flow on the unit tangent bundle to any sphere with three cone points with hyperbolic metric. The construction relies on the existence of a particular coding with two letters for the geodesics on these orbifolds.
In this paper, we study the topological behavior of elementary planes in the Apollonian orbifold $M_A$, whose limit set is the classical Apollonian gasket. The existence of these elementary planes leads to the following failure of equidistribution: there exists a sequence of closed geodesic planes in $M_A$ limiting only on a finite union of closed geodesic planes. This contrasts with other acylindrical hyperbolic 3-manifolds analyzed in [MMO1, arXiv:1802.03853, arXiv:1802.04423]. On the other hand, we show that certain rigidity still holds: the area of an elementary plane in $M_A$ is uniformly…
We describe a family of arithmetic cross-sections for geodesic flow on compact surfaces of constant negative curvature based on the study of generalized Bowen-Series boundary maps associated to cocompact torsion-free Fuchsian groups and their natural extensions, introduced in Katok-Ugarcovici's 2017 paper. If the boundary map satisfies the short cycle property, i.e., the forward orbits at each discontinuity point coincide after one step, the natural extension map has a global attractor with finite rectangular structure, and the associated arithmetic cross-section is parametrized by the attract…
We prove that the geodesic flow on the unit tangent bundle to every hyperbolic 2-orbifold that is a sphere with 3 or 4 singular points admits explicit genus one Birkhoff sections, and we determine the associated first return maps.
We calculate the asymptotic average rate at which a generic geodesic on a finite area hyperbolic 2-orbifold returns to a subsurface with geodesic boundary. As a consequence we get the average time a generic geodesic spends in such a subsurface. Related results are obtained for excursions into a collar neighborhood of a simple closed geodesic and the associated distribution of excursion depths.
We extend to orbifolds classical results on quantum ergodicity due to Shnirelman, Colin de Verdière and Zelditch, proving that, for any positive, first-order self-adjoint elliptic pseudodifferential operator P on a compact orbifold X with positive principal symbol p, ergodicity of the Hamiltonian flow of p implies quantum ergodicity for the operator P. We also prove ergodicity of the geodesic flow on a compact Riemannian orbifold of negative sectional curvature.
In this paper we give a geometric interpretation of the renormalization algorithm and of the continued fraction map that we introduced in arxiv:0905.0871 to give a characterization of symbolic sequences for linear flows in the regular octagon. We interpret this algorithm as renormalization on the Teichmueller disk of the octagon and explain the relation with Teichmueller geodesic flow. This connection is analogous to the classical relation between Sturmian sequences, continued fractions and geodesic flow on the modular surface. We use this connection to construct the natural extension and the …
The planar $N$-centre problem describes the motion of a particle moving in the plane under the action of the force fields of $N$ fixed attractive centres: \[ \ddot{x}(t)=\sum_{j=1}^N\nabla V_j(x-c_j). \] In this paper we prove symbolic dynamics at slightly negative energy for an $N$-centre problem where the potentials $V_j$ are positive, anisotropic and homogeneous of degree $-α_j$: \[ V_j(x)=|x|^{-α_j}V_j\left(\frac{x}{|x|}\right). \] The proof is based on a broken geodesics argument and trajectories are extremals of the Maupertuis' functional. Compared with the classical $N$-centre problem w…
Using Hecke triangle surfaces of finite and infinite area as examples, we present techniques for thermodynamic formalism approaches to Selberg zeta functions with unitary finite-dimensional representations $(V,χ)$ for hyperbolic surfaces (orbifolds) $Γ\backslash\mathbb{H}$ as well as transfer operator techniques to develop period-like functions for $(Γ,χ)$-automorphic cusp forms. This leads to several natural conjectures. We further show how to extend these results to the billiard flow on the underlying triangle surfaces, and study the convergence of transfer operators along sequences of Hecke…
The fact that the modular template coincides with the Lorenz template, discovered by Ghys, implies modular knots have very peculiar properties. We obtain a generalization of these results to other Hecke triangle groups. In this context, the geodesic flow can never be seen as a flow on a subset of $S^3$, and one is led to consider embeddings into lens spaces. We will geometrically construct homeomorphisms from the unit tangent bundles of the orbifolds into the lens spaces, elliminating the need for elliptic functions. Finally we will use these homeomorphisms to compute templates for the geodesi…
We construct Markov partitions for non-invertible and/or singular nonuniformly hyperbolic systems defined on higher dimensional Riemannian manifolds. The generality of the setup covers classical examples not treated so far, such as geodesic flows in closed manifolds, multidimensional billiard maps, and Viana maps, and includes all the recent results of the literature. We also provide a wealth of applications.
In this paper we study the geodesic flow for a particular class of Riemannian non-compact manifolds with variable pinched negative sectional curvature. For a sequence of invariant measures we are able to prove results relating the loss of mass and bounds on the measure entropies. We compute the entropy contribution of the cusps. We develop and study the corresponding thermodynamic formalism. We obtain certain regularity results for the pressure of a class of potentials. We prove that the pressure is real analytic until it undergoes a phase transition, after which it becomes constant. Our techn…
In this survey based on the book by the authors [BPP], we recall the Patterson-Sullivan construction of equilibrium states for the geodesic flow on negatively curved orbifolds or tree quotients, and discuss their mixing properties, emphazising the rate of mixing for (not necessarily compact) tree quotients via coding by countable (not necessarily finite) topological shifts. We give a new construction of numerous nonuniform tree lattices such that the (discrete time) geodesic flow on the tree quotient is exponentially mixing with respect to the maximal entropy measure: we construct examples who…
We construct symbolic dynamics on sets of full measure (w.r.t. an ergodic measure of positive entropy) for $C^{1+ε}$ flows on compact smooth three-dimensional manifolds. One consequence is that the geodesic flow on the unit tangent bundle of a compact $C^\infty$ surface has at least const $\times(e^{hT}/T)$ simple closed orbits of period less than $T$, whenever the topological entropy $h$ is positive -- and without further assumptions on the curvature.
\textsc{J. Hadamard} studied the geometric properties of geodesic flows on surfaces of negative curvature, thus initiating "Symbolic Dynamics". In this article, we follow the same geometric approach to study the geodesic trajectories of billiards in "rational polygons" on the hyperbolic plane. We particularly show that the billiard dynamics resulting thus are just 'Subshifts of Finite Type' or their dense subsets. We further show that 'Subshifts of Finite Type' play a central role in subshift dynamics and while discussing the topological structure of the space of all subshifts, we demonstrate …
We describe arithmetic cross-sections for geodesic flow on compact surfaces of constant negative curvature using generalized Bowen-Series boundary maps and their natural extensions associated to cocompact torsion-free Fuchsian groups. If the boundary map parameters are extremal, that is, each is an endpoint of a geodesic that extends a side of the fundamental polygon, then the natural extension map has a domain with finite rectangular structure, and the associated arithmetic cross-section is parametrized by this set. This construction allows us to represent the geodesic flow as a special flow …
Let S be an ergodic measure-preserving automorphism on a non-atomic probability space, and let T be the time-one map of a topologically weak mixing suspension flow over an irreducible subshift of finite type under a Holder ceiling function. We show that if the measure-theoretic entropy of S is strictly less than the topological entropy of T, then there exists an embedding from the measure-preserving automorphism into the suspension flow. As a corollary of this result and the symbolic dynamics for geodesic flows on compact surfaces of negative curvature developed by Bowen and Ratner, we also ob…
We consider a geodesic flow on a compact manifold endowed with a Riemannian (or Finsler, or Lorentz) metric satisfying some generic, explicit conditions. We couple the geodesic flow with a time-dependent potential, driven by an external flow on some other compact manifold. If the external flow satisfies some very general recurrence condition, and the potential satisfies some explicit conditions that are also very general, we show that the coupled system has orbits whose energy grows at a linear rate with respect to time. This growth rate is optimal. We also show the existence of symbolic dynam…
Extreme value theorem for geodesic flow on the quotient of the theta group
HUN-REN Alfréd Rényi Institute of Mathematics（HUN-REN 阿尔弗雷德·雷尼数学研究所）；Department of Mathematical Sciences, Seoul National University（首尔大学数学科学系）；Research Institute of Mathematics, Seoul National University（首尔大学数学研究所）
We consider a restricted $(N+1)$-body problem, with $N \geq 3$ and homogeneous potentials of degree $-\a<0$, $\a \in [1,2)$. We prove the existence of infinitely many collision-free periodic solutions with negative and small Jacobi constant and small values of the angular velocity, for any initial configuration of the centres. We will introduce a Maupertuis' type variational principle in order to apply the broken geodesics technique developed in the paper "N. Soave and S. Terracini. Symbolic dynamics for the $N$-centre problem at negative energies. Discrete and Cont. Dynamical Systems A, 32 (2…
Two flows are topologically almost commensurable if, up to removing finitely many periodic orbits and taking finite coverings, they are topologically equivalent. We prove that all suspensions of automorphisms of the 2-dimensional torus and all geodesic flows on unit tangent bundles to hyperbolic 2-orbifolds are pairwise almost commensurable.
We prove and explore a family of identities relating lengths of curves and orthogeodesics of hyperbolic surfaces. These identities hold over a large space of metrics including ones with hyperbolic cone points, and in particular, show how to extend a result of the first author to surfaces with cusps. One of the main ingredients in the approach is a partition of the set of orthogeodesics into sets depending on their dynamical behavior, which can be understood geometrically by relating them to geodesics on orbifold surfaces. These orbifold surfaces turn out to be exactly on the boundary of the sp…
As a final work to establish that frame flows for geometrically finite hyperbolic manifolds of arbitrary dimensions are exponentially mixing with respect to the Bowen-Margulis-Sullivan measure, this paper focuses on the case with cusps. To prove this, we utilize the countably infinite symbolic coding of the geodesic flow of Li-Pan and perform a frame flow version of Dolgopyat's method à la Sarkar-Winter and Tsujii-Zhang. This requires the local non-integrability condition and the non-concentration property but the challenge in the presence of cusps is that the latter holds only on a large prop…
We describe geometric algorithms that generalize the classical continued fraction algorithm for the torus to all translation surfaces in hyperelliptic components of translation surfaces. We show that these algorithms produce all saddle connections which are best approximations in a geometric sense, which generalizes the notion of best approximation for the classical continued fraction. In addition, they allow to list all systoles along a Teichmueller geodesic and all bispecial words which appear in the symbolic coding of linear flows. The elementary moves of the described algorithms provide a …
We examine the asymptotics of the number of the closed trajectories $γ$ of hyperbolic flows $ϕ_t$ whose primitive periods $T_γ$ lie in exponentially shrinking intervals $(x - e^{-δx}, x + e^{-δx}),\:δ> 0,\: x \to + \infty.$ Our results holds for hyperbolic dynamical systems having a symbolic model with a non-lattice roof function $f$ under the assumption that the corresponding Ruelle operator related to $f$ satisfies strong spectral estimates. In particular, our analysis works for open billiard systems and for the geodesics flow on manifolds with constant negative curvature.
We present two proofs of the fact, originally due to Reiner Martin, that any fully irreducible hyperbolic element of $Out(F_N)$ acts on the projectivized space of geodesic currents $\mathbb{P}Curr(F_N)$ with uniform north-south dynamics. The first proof, using purely train-track methods, provides an elaborated and corrected version of Reiner Martin's original approach. The second proof uses the geometric intersection form of Kapovich and Lustig and relies on unique ergodicity results from symbolic dynamics.
