Imre Péter Tóth (BME MI)
About the complexity of the singularity set in planar dispersing
billiards: the complexity is typically finite.
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Abstract:
In the study of statistical properties of hyperbolic dynamical systems
with singularities, understanding the singularity structure plays a
decisive role. Under complexity of the singularities we mean the function,
which tells that if we run the dynamics for time t, what is the maximum
number of component into which a small neighbourhood of a phase point can
break
up. Estimating this function is necessary to prove one of the most
important statistical properties, exponential decay of correlations (with
the existing techniques). In dispersing billiards this is a serious
problem, because -- while in the simplest planar case the complexity is
known to grow at most linearly -- in dimensions greater than two, no
usable estimate is known. Although it is natural to conjecture that in
a _typical_ configuration the complexity is finite, because every phase
point can be singular at most a finite number of times, during its entire
trajectory.
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In the talk I prove this conjecture for the simplest possible case: planar
dispersing billiards with finite horizon, with typicality meant in the C^3
topological sense (as holding on a residual set). Even this is not easy:
one has to fight the problem of "recollisions", that the effect on the
trajectory of a change in the configuration is hard to follow, because the
trajectory, during its history, can return several times to the
configuration point effected by the change. In the proof I present, the
way out is to study the effect of perturbations in a dynamical way, making
use of the hyperbolicity of the system.