LxDS People Research Seminars Events Opportunities


MAY 22, 2019


14:00 Silvius Klein (Pontifíia Universidade Católica do Rio de Janeiro)
Uniform convergence rate for Birkhoff means of certain uniquely ergodic toral maps.

The first goal of this talk is to present a Fourier analytical proof of (an extension of) the Denjoy-Koksma inequality, as well as an example showing its sharpness. While more technical than the original proof of Hermann, our approach has the advantage of being easily adapted to other related contexts. Thus we obtain an estimate on the uniform convergence rate of the Birkhoff averages of a higher dimensional torus translation given by a frequency satisfying a generic arithmetic condition and a continuous observable. This convergence rate depends explicitly on the modulus of continuity of the observable and on the arithmetic properties of the frequency. Furthermore, we obtain similar results for affine skew product toral transformations. (Joint work with Xiao-Chuan Liu and Aline Melo.)

15:30 Jaqueline Siqueira (Universidade Federal do Rio de Janeiro)
Equilibrium states for non-uniformly hyperbolic maps: statistical properties, joint continuity and analyticity of the potential.

We consider a wide family of non-uniformly hyperbolic maps and hyperbolic potentials and prove that the unique equilibrium state associated to each element of the family is given by the eigen-measure and the eigenfunction of the transfer operator (both having the spectral radius as an eigenvalue). We prove that the transfer operator has the spectral gap property in the space of Holder continuous observables. From this we derive that the unique equilibrium state satisfies a central limit theorem and that it has exponential decay of correlations. Moreover, we prove continuity and analyticity with respect to the potential. (Based on various joint works with S. Afonso, J. Alves, V. Ramos.)

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Room C6.2.33 Faculdade de Ciências
Universidade de Lisboa
Campo Grande, Edifício C6, Piso 2
P-1749-016 Lisboa


João Lopes Dias (ISEG, CEMAPRE)
Pedro Duarte (FCUL, CMAF)
José Pedro Gaivão (ISEG, CEMAPRE)
Telmo Peixe (ISEG, CEMAPRE)


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