Wild attractors and thermodynamic formalism

Weighted Thermodynamic Formalism and Applications

Let (X, T ) and (Y, S) be two subshifts so that Y is a factor of X. For any asymptotically sub-additive potential Φ on X and a = (a, b) ∈ R with a > 0, b ≥ 0, we introduce the notions of a-weighted topological pressure and a-weighted equilibrium state of Φ. We setup the weighted variational principle. In the case that X, Y are full shifts with one-block factor map, we prove the uniqueness and G...

A Micro-Thermodynamic Formalism

We consider the micro-canonical ensemble of a classical Hamiltonian dynamical system, the Hamiltonian being parameter dependent and in the possible presence of other first integrals. We describe a thermodynamic formalism in which a 1st law of thermodynamics, or fundamental relation, is based upon the bulk-entropy, SΩ. Under an ergodic hypothesis, SΩ is shown to be an adiabatic invariant. Expres...

C∗-Algebras and Thermodynamic Formalism

We show a relation of the KMS state of a certain C∗Algebra U with the Gibbs state of Thermodynamic Formalism. More precisely, we consider here the shift T : X → X acting on the Bernoulli space X = {1, 2, ..., k} and μ a Gibbs state defined by a Holder continuous potential p : X → R, and L(μ) the associated Hilbert space. Consider the C∗-Algebra U = U(μ), which is a sub-C∗-Algebra of the C∗-Alge...

Thermodynamic Formalism of Neural Computing

Neural networks are systems of interconnected processors mimicking some of the brain functions. After a rapid overview of neural computing, the thermodynamic formalism of the learning procedure is introduced. Besides its use in introducing eecient stochas-tic learning algorithms, it gives an insight in terms of information theory. Main emphasis is given in the information restitution process ; ...

Thermodynamic Formalism, Large Deviation, and Multifractals