864 research outputs found
Transience and thermodynamic formalism for infinitely branched interval maps
We study a one-parameter family of countably piecewise linear interval maps,
which, although Markov, fail the `large image property'. This leads to
conservative as well as dissipative behaviour for different maps in the family
with respect to Lebesgue. We investigate the transition between these two
types, and study the associated thermodynamic formalism, describing in detail
the second order phase transitions (i.e. the pressure function is but not
at the phase transition) that occur in transition to dissipativity. We
also study the various natural definitions of pressure which arise here,
computing these using elementary recurrence relations.Comment: Corrected the proof of the lower bound in Proposition 2 in the case
$\lambda\leq 1/2
Wild attractors and thermodynamic formalism
Fibonacci unimodal maps can have a wild Cantor attractor, and hence be
Lebesgue dissipative, depending on the order of the critical point. We present
a one-parameter family of countably piecewise linear unimodal
Fibonacci maps in order to study the thermodynamic formalism of dynamics where
dissipativity of Lebesgue (and conformal) measure is responsible for phase
transitions. We show that for the potential ,
there is a unique phase transition at some , and the pressure
is analytic (with unique equilibrium state) elsewhere. The pressure
is majorised by a non-analytic curve (with all derivatives equal to
0 at ) at the emergence of a wild attractor, whereas the phase
transition at can be of any finite order for those for
which is Lebesgue conservative. We also obtain results on the
existence of conformal measures and equilibrium states, as well as the
hyperbolic dimension and the dimension of the basin of
Equilibrium states, pressure and escape for multimodal maps with holes
For a class of non-uniformly hyperbolic interval maps, we study rates of
escape with respect to conformal measures associated with a family of geometric
potentials. We establish the existence of physically relevant conditionally
invariant measures and equilibrium states and prove a relation between the rate
of escape and pressure with respect to these potentials. As a consequence, we
obtain a Bowen formula: we express the Hausdorff dimension of the set of points
which never exit through the hole in terms of the relevant pressure function.
Finally, we obtain an expression for the derivative of the escape rate in the
zero-hole limit.Comment: Minor edits. To appear in Israel J. Mat
Hitting times and periodicity in random dynamics
We prove quenched laws of hitting time statistics for random subshifts of
finite type. In particular we prove a dichotomy between the law for periodic
and for non-periodic points. We show that this applies to random Gibbs
measures
Markov extensions and lifting measures for complex polynomials
For polynomials on the complex plane with a dendrite Julia set we study
invariant probability measures, obtained from a reference measure. To do this
we follow Keller in constructing canonical Markov extensions. We discuss
``liftability'' of measures (both -invariant and non-invariant) to the
Markov extension, showing that invariant measures are liftable if and only if
they have a positive Lyapunov exponent. We also show that -conformal
measure is liftable if and only if the set of points with positive Lyapunov
exponent has positive measure.Comment: Some changes have been made, in particular to Sections 2 and 3, to
clarify the exposition. Typos have been corrected and references update
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