[Paper Review] Critical exponent for geodesic currents

Olivier Glorieux|arXiv (Cornell University)|Apr 21, 2017

Mathematical Dynamics and Fractals5 references3 citations

TL;DR

This paper defines a critical exponent for geodesic currents on a hyperbolic surface by measuring the exponential growth rate of the number of closed curves whose intersection with a given current is bounded by R. It proves that for filling currents, this critical exponent equals the exponential growth rate of the intersection function with closed curves, generalizing the classical result for hyperbolic metrics and establishing a metric structure via a quasi-metric associated to the current.

ABSTRACT

For any geodesic current we associated a quasi-metric space. For a subclass of geodesic currents, called filling, it defines a metric and we study the critical exponent associated to this space. We show that is is equal to the exponential growth rate of the intersection function for closed curves.

Motivation & Objective

• To define a critical exponent for geodesic currents using the intersection function with closed curves.
• To establish a metric structure on the space of geodesic currents by associating a quasi-metric to each current.
• To prove that for filling currents, the critical exponent equals the exponential growth rate of the intersection function with closed curves.
• To generalize classical results on critical exponents in negatively curved geometry to the broader setting of geodesic currents.

Proposed method

• Define the geodesic critical exponent δ₉^η as the limsup of (1/R) log(Card{c ∈ C | i(η,c) ≤ R}) as R → ∞.
• Associate a quasi-metric dₙ to each geodesic current η via the intersection function i(η,c) for closed curves c.
• Use the action of the fundamental group Γ on the boundary of the hyperbolic plane to normalize axes of group elements.
• Apply normalization via a fixed hyperbolic element r ∈ Γ to control the position of axes relative to a fixed compact set N.
• Use covering and translation length estimates to bound the number of group elements with bounded intersection, leveraging the triangle inequality in the quasi-metric.
• Prove that elements whose axes avoid a fixed neighborhood pair are rare, enabling comparison between the counting function and the critical exponent.

Experimental results

Research questions

• RQ1 Does the critical exponent defined via the intersection function with closed curves coincide with the exponential growth rate of the counting function for filling geodesic currents?
• RQ2 Can a metric structure be naturally associated to a geodesic current via its intersection function?
• RQ3 How does the critical exponent for a Liouville current relate to the classical critical exponent of a hyperbolic metric?
• RQ4 What is the asymptotic behavior of the number of closed curves with bounded intersection against a given geodesic current?
• RQ5 Is the critical exponent invariant under the choice of normalization or basepoint in the construction?

Key findings

• For any filling geodesic current η, the geodesic critical exponent δ₉^η is equal to the exponential growth rate of the number of closed curves c with i(η,c) ≤ R.
• The critical exponent δ₉^η is well-defined and finite for all geodesic currents, and equals the limsup of (1/R) log(Card{c ∈ C | i(η,c) ≤ R}) as R → ∞.
• The construction yields a quasi-metric space from any geodesic current, and a metric space when the current is filling.
• The number of group elements γ ∈ Γ with i(η, [γ]) ≤ R grows exponentially at rate δ₉^η, and this rate is sharp.
• The proof establishes that elements whose axis avoids a fixed pair of neighborhoods are negligible in number, enabling the comparison of counting functions.
• The result generalizes the classical equality δ = δ₉ for hyperbolic metrics to the setting of geodesic currents, including Liouville currents and those from AdS quasi-Fuchsian manifolds.