114 research outputs found
Abrupt bifurcations in chaotic scattering : view from the anti-integrable limit
Bleher, Ott and Grebogi found numerically an interesting chaotic phenomenon in 1989 for the scattering of a particle in a plane from a potential field with several peaks of equal height. They claimed that when the energy E of the particle is slightly less than the peak height Ec there is a hyperbolic suspension of a topological Markov chain from which chaotic scattering occurs, whereas for E > Ec there are no bounded orbits. They called the bifurcation at E = Ec an abrupt bifurcation to chaotic scattering.
The aim of this paper is to establish a rigorous mathematical explanation for how chaotic orbits occur via the bifurcation, from the viewpoint of the anti-integrable limit, and to do so for a general range of chaotic scattering problems
Invariant manifolds and equilibrium states for non-uniformly hyperbolic horseshoes
In this paper we consider horseshoes containing an orbit of homoclinic
tangency accumulated by periodic points. We prove a version of the Invariant
Manifolds Theorem, construct finite Markov partitions and use them to prove the
existence and uniqueness of equilibrium states associated to H\"older
continuous potentials.Comment: 33 pages, 6 figure
Meanders: Exact Asymptotics
We conjecture that meanders are governed by the gravitational version of a
c=-4 two-dimensional conformal field theory, allowing for exact predictions for
the meander configuration exponent \alpha=\sqrt{29}(\sqrt{29}+\sqrt{5})/12, and
the semi-meander exponent {\bar\alpha}=1+\sqrt{11}(\sqrt{29}+\sqrt{5})/24. This
result follows from an interpretation of meanders as pairs of fully packed
loops on a random surface, described by two c=-2 free fields. The above values
agree with recent numerical estimates. We generalize these results to a score
of meandric numbers with various geometries and arbitrary loop fugacities.Comment: new version with note added in proo
Lee-Yang-Fisher zeros for DHL and 2D rational dynamics, II. Global Pluripotential Interpretation
In a classical work of the 1950's, Lee and Yang proved that for fixed
nonnegative temperature, the zeros of the partition functions of a
ferromagnetic Ising model always lie on the unit circle in the complex magnetic
field. Zeros of the partition function in the complex temperature were then
considered by Fisher, when the magnetic field is set to zero. Limiting
distributions of Lee-Yang and of Fisher zeros are physically important as they
control phase transitions in the model. One can also consider the zeros of the
partition function simultaneously in both complex magnetic field and complex
temperature. They form an algebraic curve called the Lee-Yang-Fisher (LYF)
zeros. In this paper we continue studying their limiting distribution for the
Diamond Hierarchical Lattice (DHL). In this case, it can be described in terms
of the dynamics of an explicit rational function R in two variables (the
Migdal-Kadanoff renormalization transformation). We study properties of the
Fatou and Julia sets of this transformation and then we prove that the
Lee-Yang-Fisher zeros are equidistributed with respect to a dynamical
(1,1)-current in the projective space. The free energy of the lattice gets
interpreted as the pluripotential of this current. We also prove a more general
equidistribution theorem which applies to rational mappings having
indeterminate points, including the Migdal-Kadanoff renormalization
transformation of various other hierarchical lattices.Comment: Continues arXiv:1009.4691. Final version. To appear in The Journal of
Geometric Analysis. (This is a tiny update to correct an error in the Latex
file from previous version.
Semiclassical approach to discrete symmetries in quantum chaos
We use semiclassical methods to evaluate the spectral two-point correlation
function of quantum chaotic systems with discrete geometrical symmetries. The
energy spectra of these systems can be divided into subspectra that are
associated to irreducible representations of the corresponding symmetry group.
We show that for (spinless) time reversal invariant systems the statistics
inside these subspectra depend on the type of irreducible representation. For
real representations the spectral statistics agree with those of the Gaussian
Orthogonal Ensemble (GOE) of Random Matrix Theory (RMT), whereas complex
representations correspond to the Gaussian Unitary Ensemble (GUE). For systems
without time reversal invariance all subspectra show GUE statistics. There are
no correlations between non-degenerate subspectra. Our techniques generalize
recent developments in the semiclassical approach to quantum chaos allowing one
to obtain full agreement with the two-point correlation function predicted by
RMT, including oscillatory contributions.Comment: 26 pages, 8 Figure
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