For a given expanding d-fold covering transformation of the one-dimensional torus, the notion of weak Gibbs measure is defined by a natural generalization of the classical Gibbs property. For these measures, we prove that the singularity spectrum and the L q-spectrum form a Legendre transform pair. The main difficulty comes from the possible existence of first-order phase transition points, that is, points where the L q-spectrum is not differentiable. We give examples of weak Gibbs measure with phase transition, including the so-called Erdös measure.

We introduce a multifractal formalism for potentials defined on shift systems. We prove that the multifractal spectra are a Legendre transform of thermodynamic functions involving the potentials studied. We obtain the fractal distribution of pointwise dimension for g-measures. Such measures are equilibrium states of potentials not necessarily Hölder continuous and generalize Gibbs measures. In connection with phase transition, we also give examples of potentials with a non-unique equilibrium state and non-analytic multifractal spectra.

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We consider infinitely convolved Bernoulli measures (or simply Bernoulli convolutions) related to the β-numeration. A matrix decomposition of these measures is obtained in the case when β is a PV number. We also determine their Gibbs properties for β being a multinacci number, which makes the multifractal analysis of the corresponding Bernoulli convolution possible.

We consider the variational principle for dimension on compact subsets of the 2-torus which are invariant under a non-conformal expanding diagonal endomorphism. Condition (H) ensures that the invariant measures with full dimension are the equilibrium states of some potential function. This result applies to the problem of uniqueness of the measure with full dimension on the sofic affine-invariant sets.