1,530 research outputs found
Sixty Years of Fractal Projections
Sixty years ago, John Marstrand published a paper which, among other things,
relates the Hausdorff dimension of a plane set to the dimensions of its
orthogonal projections onto lines. For many years, the paper attracted very
little attention. However, over the past 30 years, Marstrand's projection
theorems have become the prototype for many results in fractal geometry with
numerous variants and applications and they continue to motivate leading
research.Comment: Submitted to proceedings of Fractals and Stochastics
On homoclinic orbits to center manifolds of elliptic-hyperbolic equilibria in Hamiltonian systems
We consider a Hamiltonian system which has an elliptic-hyperbolic equilibrium
with a homoclinic loop. We identify the set of orbits which are homoclinic to
the center manifold of the equilibrium via a Lyapunov- Schmidt reduction
procedure. This leads to the study of a singularity which inherits certain
structure from the Hamiltonian nature of the system. Under non-degeneracy
assumptions, we classify the possible Morse indices of this singularity,
permitting a local description of the set of homoclinic orbits. We also
consider the case of time-reversible Hamiltonian systems
Eigenvalues of Products of Unitary Matrices and Lagrangian Involutions
This paper introduces a submanifold of the moduli space of unitary
representations of the fundamental group of a punctured sphere with fixed local
monodromy. The submanifold is defined via products of involutions through
Lagrangian subspaces. We show that the moduli space of Lagrangian
representations is a Lagrangian submanifold of the moduli of unitary
representations.Comment: 35 pages, 2 figures, to appear in Topolog
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