Vaughn Climenhaga
Intrinsic ergodicity beyond specification: beta-shifts, S-gap shifts, and their factors.
Mon, 11/15/2010 - 16:01 — SomNambulAuthors: Vaughn Climenhaga, Daniel J. Thompson
Subjects: Dynamical Systems
AbstractWe give sufficient conditions for a shift space $(\Sigma,\sigma)$ to be
intrinsically ergodic, along with sufficient conditions for every subshift
factor of $\Sigma$ to be intrinsically ergodic. As an application, we show that
every subshift factor of a beta-shift is intrinsically ergodic, which answers
an open question included in Mike Boyle's article "Open problems in symbolic
dynamics''. We obtain the same result for S-gap shifts, and describe an
application of our conditions to more general coded systems.Multifractal formalism derived from thermodynamics.
Thu, 02/04/2010 - 16:01 — SomNambulAuthors: Vaughn Climenhaga
Subjects: Dynamical Systems
AbstractWe show that under quite general conditions, various multifractal spectra may
be obtained as Legendre transforms of functions $T\colon \RR\to \RR$ arising in
the thermodynamic formalism. We impose minimal requirements on the maps we
consider, and obtain partial results for any continuous map $f$ on a compact
metric space. In order to obtain complete results, the primary hypothesis we
require is that the functions $T$ be continuously differentiable.Bowen's equation in the non-uniform setting.
Mon, 08/31/2009 - 17:16 — SomNambulAuthors: Vaughn Climenhaga
Subjects: Dynamical Systems
AbstractWe show that Bowen's equation, which characterises the Hausdorff dimension of
certain sets in terms of the topological pressure of an expanding conformal
map, applies in greater generality than has been heretofore established. In
particular, the property of uniform expansion may be significantly weakened to
positivity of the Lyapunov exponent. Among other things, this allows us to
compute the dimension spectrum for Lyapunov exponents for maps with parabolic
periodic points.
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