Title:	On the maximal eigenspace of the Ruelle Operator
Authors:	Melo, Leonardo Cavalcanti de
Orientador(es)::	Cioletti, Leandro Martins
Assunto::	Operador de Ruelle Operador de transferência Teorema de Ruelle-Perron-Frobenius Análise espectral Funções harmônicas Funções invariantes Teoria Ergódica Processos de Markov Modelo de campo médio
Issue Date:	29-Oct-2020
Data de defesa::	27-Jul-2020
Citation:	MELO, Leonardo Cavalcanti de. On the maximal eigenspace of the Ruelle Operator. 2020. 94 f., il. Tese (Doutorado em Matemática)—Universidade de Brasília, Brasília, 2020.
Abstract:	In this PhD dissertation we analyze the spectral data of the Ruelle operator L and its extension to L1(ν), here denoted by L. To do so, we use as the main route the L1 theory of Markov processes introduced by Eberhard Hopf. This technique provides us with sufficient tools to extract information on the eigenspace of the operator, even for low- regularity potentials presenting phase transition. The theory we develop here comprises compact metric alphabets. In this setting and for an arbitrary continuous potential, we show that the eigenspace of L associated with its spectral radius is at most one- dimensional and, when there is a continuous maximal eigenfunction, it must have a definite sign. These properties are known to hold for finite alphabets and some classes of regular potentials, such as those fulfilling the hypothesis of the Ruelle-Perron- Frobenius Theorem. We demonstrate that those properties are only related to the positivity of the operator and full support of the eigenmeasures and do not depend on the finite alphabet or on the regularity of the potential. As for the extension L to L1(ν), we give conditions on ν to have a well-posed extension. When the chosen ν is a conformal measure (maximal eigenmeasure), we prove that ν is fully supported if and only if the a priori measure p is fully supported. In this case, we demonstrate that the dimension of the maximal eigenspace of the extended operator is upper bounded by the number of extreme measures whose convex combination yields ν. This gives us a new criterion for phase transition, since a multidimensional maximal eigenspace can only emerge in the case of multiple extreme conformal measures. We also construct an example inspired on the Currie-Weiss model that exhibits phase transition with a bi-dimensional maximal eigenspace.
metadata.dc.description.unidade:	Instituto de Ciências Exatas (IE) Departamento de Matemática (IE MAT)
Description:	Tese (doutorado)—Universidade de Brasília, Instituto de Ciências Exatas, Departamento de Matemática, 2020.
metadata.dc.description.ppg:	Programa de Pós-Graduação em Matemática
Licença::	A concessão da licença deste item refere-se ao termo de autorização impresso assinado pelo autor com as seguintes condições: Na qualidade de titular dos direitos de autor da publicação, autorizo a Universidade de Brasília e o IBICT a disponibilizar por meio dos sites www.bce.unb.br, www.ibict.br, http://hercules.vtls.com/cgi-bin/ndltd/chameleon?lng=pt&skin=ndltd sem ressarcimento dos direitos autorais, de acordo com a Lei nº 9610/98, o texto integral da obra disponibilizada, conforme permissões assinaladas, para fins de leitura, impressão e/ou download, a título de divulgação da produção científica brasileira, a partir desta data.
Appears in Collections:	Teses, dissertações e produtos pós-doutorado

Items in DSpace are protected by copyright, with all rights reserved, unless otherwise indicated.