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We give a quantitative estimate for the quantum mean absolute deviation on hyperbolic surfaces of finite area in terms of geometric parameters such as the genus, number of cusps and injectivity radius. It implies a delocalisation result of quantum ergodicity type for eigenfunctions of the Laplacian on hyperbolic surfaces of finite area that Benjamini-Schramm converge to the hyperbolic plane. We show that this is generic for Mirzakhani’s model of random surfaces chosen uniformly with respect to the Weil-Petersson volume. Depending on the particular sequence of surfaces considered this gives a result of delocalisation of most cusp forms or of Eisenstein series.

We refine the recent local rigidity result for the marked length spectrum obtained by the first and third author in [GL19] and give an alternative proof using the geodesic stretch between two Anosov flows and some uniform estimate on the variance appearing in the central limit theorem for Anosov geodesic flows. In turn, we also introduce a new pressure metric on the space of isometry classes, which reduces to the Weil–Petersson metric in the case of Teichmüller space and is related to the works [BCLS15, MM08].

Let $\\Sigma $ be a closed hyperbolic surface. We study, for fixed g, the asymptotics of the number of those periodic geodesics in $\\Sigma $ having at most length L and which can be written as the product of g commutators. The basic idea is to reduce these results to being able to count critical realizations of trivalent graphs in $\\Sigma $ . In the appendix, we use the same strategy to give a proof of Huber’s geometric prime number theorem.

We consider the semi-classical (or Gutzwiller–Voros) zeta functions for C ∞ contact Anosov flows. Analyzing the spectra of the generators of some transfer operators associated to the flow, we prove that, for arbitrarily small τ > 0 , its zeros are contained in the union of the τ -neighborhood of the imaginary axis, | R ( s ) | < τ , and the half-plane R ( s ) < - χ 0 + τ , up to finitely many exceptions, where χ 0 > 0 is the hyperbolicity exponent of the flow. Further we show that the density of the zeros along the imaginary axis satisfy an analogue of the Weyl law.

We obtain a lower bound for the entropy of a Borel probability measure (not necessarily invariant) with respect to an upper semicontinuous set-valued map as the product of the lower dimension of the measure and the logarithmic expansion rate. This is a generalization of the well-known measure-preserving single-valued case.

We prove the connectedness of the Prym eigenforms loci in genus four (for real multiplication by some order of discriminant $D$), for any $D$. These loci were discovered by McMullen in 2006.

Let ${{\\mathcal {H}}}$ be a stratum of translation surfaces with at least two singularities, let $m_{{{\\mathcal {H}}}$ denote the Masur-Veech measure on ${{\\mathcal {H}}}$ , and let $Z_0$ be a flow on $({{\\mathcal {H}}}, m_{{{\\mathcal {H}}})$ obtained by integrating a Rel vector field. We prove that $Z_0$ is mixing of all orders, and in particular is ergodic. We also characterize the ergodicity of flows defined by Rel vector fields, for more general spaces $({\\mathcal L}, m_{{\\mathcal L}})$ , where ${\\mathcal L} \\subset {{\\mathcal {H}}}$ is an orbit-closure for the action of $G = \\operatorname {SL}_2({\\mathbb {R}})$ (i.e., an affine invariant subvariety) and $m_{{\\mathcal L}}$ is the natural measure. These results are conditional on a forthcoming measure classification result of Brown, Eskin, Filip and Rodriguez-Hertz. We also prove that the entropy of $Z_0$ with respect to any of the measures $m_{{{\\mathcal L}}}$ is zero.

Let X be a compact, geodesically complete, locally CAT(0) space such that the universal cover admits a rank-one axis. Assume X is not homothetic to a metric graph with integer edge lengths. Let $P_t$ be the number of parallel classes of oriented closed geodesics of length at most t; then $\\lim \\nolimits _{t \\to \\infty } P_t / ({e^{ht}}/{ht}) = 1$ , where h is the entropy of the geodesic flow on the space $GX$ of parametrized unit-speed geodesics in X.

In this paper we study topological aspects of the dynamics of the foliated horocycle flow on flat projective bundles over hyperbolic surfaces and we derive ergodic consequences. If ρ:Γ→PSL(n+1,R) is a representation of a non-elementary Fuchsian group Γ, the unit tangent bundle Y associated to the flat projective bundle defined by ρ admits a natural action of the affine group B obtained by combining the foliated geodesic and horocycle flows. If the image ρ(Γ) satisfies Conze-Guivarc’h conditions, namely strong irreducibility and proximality, the dynamics of the B-action is captured by the proximal dynamics of ρ(Γ) on RPn (Theorem A). In fact, the dynamics of the foliated horocycle flow on the unique B-minimal subset of Y can be described in terms of dynamics of the horocycle flow on the non-wandering set in the unit tangent bundle X of the surface S=Γ (Theorem B). Assuming the existence of a continuous limit map, we prove that the B-minimal set is an attractor for the foliated horocycle flow restricted to the proximal part of the non-wandering set in Y (Theorem C). As a corollary, we deduce that the restricted flow admits a unique conservative ergodic U-invariant Radon measure (defined up to a multiplicative constant) if and only if Γ is convex-cocompact. For example, the foliated horocycle flow on the sphere bundle defined by the Cannon-Thurston map is uniquely ergodic.

In 1979, for each signature for Fuchsian groups of the first kind, Bowen and Series constructed an explicit fundamental domain for one group of the signature, and from this a function on S 1 tightly associated with this group. In general, their fundamental domain enjoys what has since been called both the ‘extension property’ and the ‘even corners property’. We determine the exact set of signatures for cocompact triangle groups for which this property can hold for any convex fundamental domain, and verify that for this restricted set, the Bowen-Series fundamental domain does have the property. To each Bowen-Series function in this corrected setting, we naturally associate four continuous deformation families of circle functions. We show that each of these functions is aperiodic if and only if it is surjective; and, is finite Markov if and only if its natural parameter is a hyperbolic fixed point of the triangle group at hand.