360 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
Admissibility of kneading sequences and structure of Hubbard trees for quadratic polynomials
Hubbard trees are invariant trees connecting the points of the critical
orbits of postcritically finite polynomials. Douady and Hubbard \cite{Orsay}
introduced these trees and showed that they encode the essential information of
Julia sets in a combinatorial way. The itinerary of the critical orbit within
the Hubbard tree is encoded by a (pre)periodic sequence on \{\0,\1\} called
\emph{kneading sequence}.
We prove that the kneading sequence completely encodes the Hubbard tree and
its dynamics, and we show how to reconstruct the tree and in particular its
branch points (together with their periods, their relative posititions, their
number of arms and their local dynamics) in terms of the kneading sequence
alone.
Every kneading sequence gives rise to an abstract Hubbard tree, but not every
kneading sequence occurs in real dynamics or in complex dynamics. Milnor and
Thurston \cite{MT} classified which kneading sequences occur in real dynamics;
we do the same for complex dynamics in terms of a complex \emph{admissibility
condition}. This complex admissibility condition fails if and only if the
abstract Hubbard tree has a so-called \emph{evil} periodic branch point that is
incompatible with local homeomorphic dynamics on the plane.Comment: 26 pages, 5 figure
Topological invariance of the sign of the Lyapunov exponents in one-dimensional maps
We explore some properties of Lyapunov exponents of measures preserved by
smooth maps of the interval, and study the behaviour of the Lyapunov exponents
under topological conjugacy.Comment: 9 page
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
The topological entropy of Banach spaces
We investigate some properties of (universal) Banach spaces of real functions
in the context of topological entropy. Among other things, we show that any
subspace of which is isometrically isomorphic to contains a
functions with infinite topological entropy. Also, for any ,
we construct a (one-dimensional) Banach space in which any nonzero function has
topological entropy equal to .Comment: The paper is going to appear at Journal of Difference Equations and
Application
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