3,857 research outputs found
Coexistence of bounded and unbounded geometry for area-preserving maps
The geometry of the period doubling Cantor sets of strongly dissipative
infinitely renormalizable H\'enon-like maps has been shown to be unbounded by
M. Lyubich, M. Martens and A. de Carvalho, although the measure of unbounded
"spots" in the Cantor set has been demonstrated to be zero.
We show that an even more extreme situation takes places for infinitely
renormalizable area-preserving H\'enon-like maps: bounded and unbounded
geometries coexist with both phenomena occuring on subsets of positive measure
in the Cantor sets
Automatic Filters for the Detection of Coherent Structure in Spatiotemporal Systems
Most current methods for identifying coherent structures in
spatially-extended systems rely on prior information about the form which those
structures take. Here we present two new approaches to automatically filter the
changing configurations of spatial dynamical systems and extract coherent
structures. One, local sensitivity filtering, is a modification of the local
Lyapunov exponent approach suitable to cellular automata and other discrete
spatial systems. The other, local statistical complexity filtering, calculates
the amount of information needed for optimal prediction of the system's
behavior in the vicinity of a given point. By examining the changing
spatiotemporal distributions of these quantities, we can find the coherent
structures in a variety of pattern-forming cellular automata, without needing
to guess or postulate the form of that structure. We apply both filters to
elementary and cyclical cellular automata (ECA and CCA) and find that they
readily identify particles, domains and other more complicated structures. We
compare the results from ECA with earlier ones based upon the theory of formal
languages, and the results from CCA with a more traditional approach based on
an order parameter and free energy. While sensitivity and statistical
complexity are equally adept at uncovering structure, they are based on
different system properties (dynamical and probabilistic, respectively), and
provide complementary information.Comment: 16 pages, 21 figures. Figures considerably compressed to fit arxiv
requirements; write first author for higher-resolution version
The quest for the ultimate anisotropic Banach space
We present a new scale (with and ) of
anisotropic Banach spaces, defined via Paley-Littlewood, on which the transfer
operator associated to a hyperbolic dynamical system has good spectral
properties. When and is an integer, the spaces are analogous to the
"geometric" spaces considered by Gou\"ezel and Liverani. When and
, the spaces are somewhat analogous to the geometric
spaces considered by Demers and Liverani. In addition, just like for the
"microlocal" spaces defined by Baladi-Tsujii, the spaces are
amenable to the kneading approach of Milnor-Thurson to study dynamical
determinants and zeta functions.
In v2, following referees' reports, typos have been corrected (in particular
(39) and (43)). Section 4 now includes a formal statement (Theorem 4.1) about
the essential spectral radius if (its proof includes the content of
Section 4.2 from v1). The Lasota-Yorke Lemma 4.2 (Lemma 4.1 in v1) includes the
claim that is compact.
Version v3 contains an additional text "Corrections and complements" showing
that s> t-(r-1) is needed in Section 4.Comment: 31 pages, revised version following referees' reports, with
Corrections and complement
Differentials and Distances in Probabilistic Coherence Spaces
In probabilistic coherence spaces, a denotational model of probabilistic functional languages, morphisms are analytic and therefore smooth. We explore two related applications of the corresponding derivatives. First we show how derivatives allow to compute the expectation of execution time in the weak head reduction of probabilistic PCF (pPCF). Next we apply a general notion of "local" differential of morphisms to the proof of a Lipschitz property of these morphisms allowing in turn to relate the observational distance on pPCF terms to a distance the model is naturally equipped with. This suggests that extending probabilistic programming languages with derivatives, in the spirit of the differential lambda-calculus, could be quite meaningful
Quantum theory in finite dimension cannot explain every general process with finite memory
Arguably, the largest class of stochastic processes generated by means of a
finite memory consists of those that are sequences of observations produced by
sequential measurements in a suitable generalized probabilistic theory (GPT).
These are constructed from a finite-dimensional memory evolving under a set of
possible linear maps, and with probabilities of outcomes determined by linear
functions of the memory state. Examples of such models are given by classical
hidden Markov processes, where the memory state is a probability distribution,
and at each step it evolves according to a non-negative matrix, and hidden
quantum Markov processes, where the memory state is a finite dimensional
quantum state, and at each step it evolves according to a completely positive
map. Here we show that the set of processes admitting a finite-dimensional
explanation do not need to be explainable in terms of either classical
probability or quantum mechanics. To wit, we exhibit families of processes that
have a finite-dimensional explanation, defined manifestly by the dynamics of
explicitly given GPT, but that do not admit a quantum, and therefore not even
classical, explanation in finite dimension. Furthermore, we present a family of
quantum processes on qubits and qutrits that do not admit a classical
finite-dimensional realization, which includes examples introduced earlier by
Fox, Rubin, Dharmadikari and Nadkarni as functions of infinite dimensional
Markov chains, and lower bound the size of the memory of a classical model
realizing a noisy version of the qubit processes.Comment: 18 pages, 0 figure
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