Quadratic irrationals and linking numbers of modular knots
Dubi Kelmer
Journal of Modern Dynamics 2012, 6(4): 539-561 doi: 10.3934/jmd.2012.6.539
A closed geodesic on the modular surface gives rise to a knot on the 3-sphere with a trefoil knot removed, and one can compute the linking number of such a knot with the trefoil knot. We show that, when ordered by their length, the set of closed geodesics having a prescribed linking number become equidistributed on average with respect to the Liouville measure. We show this by using the thermodynamic formalism to prove an equidistribution result for a corresponding set of quadratic irrationals on the unit interval.
keywords: Closed geodesics Modular surface equidistribution. linking numbers
Quantum ergodicity for products of hyperbolic planes
Dubi Kelmer
Journal of Modern Dynamics 2008, 2(2): 287-313 doi: 10.3934/jmd.2008.2.287
For manifolds with geodesic flow that is ergodic on the unit tangent bundle, the Quantum Ergodicity Theorem implies that almost all Laplacian eigenfunctions become equidistributed as the eigenvalue goes to infinity. For a locally symmetric space with a universal cover that is a product of several upper half-planes, the geodesic flow has constants of motion so it cannot be ergodic. It is, however, ergodic when restricted to the submanifolds defined by these constants. Accordingly, we show that almost all eigenfunctions become equidistributed on these submanifolds.
keywords: hyperbolic plane. quantum ergodicity
Approximation of points in the plane by generic lattice orbits
Dubi Kelmer
Journal of Modern Dynamics 2017, 11(1): 143-153 doi: 10.3934/jmd.2017007

We give upper and lower bounds for Diophantine exponents measuring how well a point in the plane can be approximated by points in the orbit of a lattice $\Gamma < {\rm{S}}{{\rm{L}}_2}\left( {\mathbb{R}} \right)$ acting linearly on ${\mathbb{R}^2}$. Our method gives bounds that are uniform for almost all orbits.

keywords: Diophantine approximation lattice action shrinking targets
A quantitative Oppenheim theorem for generic ternary quadratic forms
Anish Ghosh Dubi Kelmer
Journal of Modern Dynamics 2018, 12(1): 1-8 doi: 10.3934/jmd.2018001

We prove a quantitative version of Oppenheim's conjecture for generic ternary indefinite quadratic forms. Our results are inspired by and analogous to recent results for diagonal quadratic forms due to Bourgain [3].

keywords: Values of quadratic forms Diophantine approximation flows on homogeneous spaces

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