1,126 research outputs found
On conformal measures and harmonic functions for group extensions
We prove a Perron-Frobenius-Ruelle theorem for group extensions of
topological Markov chains based on a construction of -finite conformal
measures and give applications to the construction of harmonic functions.Comment: To appear in Proceedings of "New Trends in Onedimensional Dynamics,
celebrating the 70th birthday of Welington de Melo
Equilibrium states for potentials with \sup\phi - \inf\phi < \htop(f)
In the context of smooth interval maps, we study an inducing scheme approach
to prove existence and uniqueness of equilibrium states for potentials
with he `bounded range' condition \sup \phi - \inf \phi < \htop, first used
by Hofbauer and Keller. We compare our results to Hofbauer and Keller's use of
Perron-Frobenius operators. We demonstrate that this `bounded range' condition
on the potential is important even if the potential is H\"older continuous. We
also prove analyticity of the pressure in this context.Comment: Added Lemma 6 to deal with the disparity between leading eigenvalues
and operator norms. Added extra references and corrected some typo
Cellular automata and Lyapunov exponents
In this article we give a new definition of some analog of Lyapunov exponents
for cellular automata . Then for a shift ergodic and cellular automaton
invariant probability measure we establish an inequality between the entropy of
the automaton, the entropy of the shift and the Lyapunov exponent
Breaking Synchrony by Heterogeneity in Complex Networks
For networks of pulse-coupled oscillators with complex connectivity, we
demonstrate that in the presence of coupling heterogeneity precisely timed
periodic firing patterns replace the state of global synchrony that exists in
homogenous networks only. With increasing disorder, these patterns persist
until they reach a critical temporal extent that is of the order of the
interaction delay. For stronger disorder these patterns cease to exist and only
asynchronous, aperiodic states are observed. We derive self-consistency
equations to predict the precise temporal structure of a pattern from the
network heterogeneity. Moreover, we show how to design heterogenous coupling
architectures to create an arbitrary prescribed pattern.Comment: 4 pages, 3 figure
Complex maps without invariant densities
We consider complex polynomials for and
, and find some combinatorial types and values of such that
there is no invariant probability measure equivalent to conformal measure on
the Julia set. This holds for particular Fibonacci-like and Feigenbaum
combinatorial types when sufficiently large and also for a class of
`long-branched' maps of any critical order.Comment: Typos corrected, minor changes, principally to Section
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