On the existence of polynomials with chaotic behaviour

We establish a general result on the existence of hypercyclic (resp., transitive, weakly mixing, mixing, frequently hypercyclic) polynomials on locally convex spaces. As a consequence we prove that every (real or complex) infinite-dimensional separable Frèchet space admits mixing (hence hypercyclic) polynomials of arbitrary positive degree. Moreover, every complex infinite-dimensional separable Banach space with an unconditional Schauder decomposition and every complex Frèchet space with an unconditional basis support chaotic and frequently hypercyclic polynomials of arbitrary positive degree. We also study distributional chaos for polynomials and show that every infinite-dimensional separable Banach space supports polynomials of arbitrary positive degree that have a dense distributionally scrambled linear manifold. © 2013 Nilson C. Bernardes Jr. and Alfredo Peris.The present work was done while the first author was visiting the Departament de Matematica Aplicada at Universitat Politecnica de Valencia (Spain). The first author is very grateful for the hospitality. The first author was supported in part by CAPES: Bolsista, Project no. BEX 4012/11-9. The second author was supported in part by MEC and FEDER, Project MTM2010-14909, and by GVA, Projects PROMETEO/2008/101 and PROMETEOII/2013/013.Bernardes, NC.; Peris Manguillot, A. (2013). On the existence of polynomials with chaotic behaviour. Journal of Function Spaces and Applications. 2013(320961). https://doi.org/10.1155/2013/320961S2013320961Bayart, F., & Matheron, E. (2009). Dynamics of Linear Operators. doi:10.1017/cbo9780511581113Grosse-Erdmann, K.-G., & Peris Manguillot, A. (2011). Linear Chaos. Universitext. doi:10.1007/978-1-4471-2170-1Rolewicz, S. (1969). On orbits of elements. Studia Mathematica, 32(1), 17-22. doi:10.4064/sm-32-1-17-22Herzog, G. (1992). 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Multiple integral representation for functionals of Dirichlet processes

We point out that a proper use of the Hoeffding--ANOVA decomposition for symmetric statistics of finite urn sequences, previously introduced by the author, yields a decomposition of the space of square-integrable functionals of a Dirichlet--Ferguson process, written $L^2(D)$, into orthogonal subspaces of multiple integrals of increasing order. This gives an isomorphism between $L^2(D)$ and an appropriate Fock space over a class of deterministic functions. By means of a well-known result due to Blackwell and MacQueen, we show that each element of the $n$th orthogonal space of multiple integrals can be represented as the $L^2$ limit of $U$-statistics with degenerate kernel of degree $n$. General formulae for the decomposition of a given functional are provided in terms of linear combinations of conditioned expectations whose coefficients are explicitly computed. We show that, in simple cases, multiple integrals have a natural representation in terms of Jacobi polynomials. Several connections are established, in particular with Bayesian decision problems, and with some classic formulae concerning the transition densities of multiallele diffusion models, due to Littler and Fackerell, and Griffiths. Our results may also be used to calculate the best approximation of elements of $L^2(D)$ by means of $U$-statistics of finite vectors of exchangeable observations.Comment: Published in at http://dx.doi.org/10.3150/07-BEJ5169 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm