We study dimension theory for dissipative dynamical systems, proving a conditional variational principle for the quotients of Birkhoff averages restricted to the recurrent part of the system. On the other hand, we show that when the whole system is considered (and not just its recurrent part) the conditional variational principle does not necessarily hold. Moreover, we exhibit an example of a topologically transitive map having discontinuous Lyapunov spectrum. The mechanism producing all these pathological features on the multifractal spectra is transience, that is, the non-recurrent part of the dynamics.
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Godofredo Iommi, Thomas Jordan, Mike Todd, Transience and multifractal analysis. Ann. Inst. H. Poincaré Anal. Non Linéaire 34 (2017), no. 2, pp. 407–421DOI 10.1016/J.ANIHPC.2015.12.007