Stefano Isola

On the spectrum of Farey and Gauss maps

In this paper we introduce Hilbert spaces of holomorphic functions given by generalized Borel and Laplace transforms which are left invariant by the transfer operators of the Farey map and its induced transformation, the Gauss map, respectively. By means of a suitable operator-valued power series we are able to study simultaneously the spectrum of both these operators along with the analytic properties of associated dynamical ζ-functions. This construction establishes an explicit connection between previously unrelated results of Mayer and Rugh.

Recurrence and algorithmic information

In this paper, we initiate a somewhat detailed investigation of the relationships between quantitative recurrence indicators and algorithmic complexity of orbits in weakly chaotic dynamical systems. We mainly focus on examples.

On the Essential Spectrum of the Transfer Operator for Expanding Markov Maps

The essential spectrum of the transfer operator for expanding markov maps of the interval is studied in detail. To this end we construct explicityly an infinite set of eigenfunctions which allows us to prove that the essential spectrum inCk is a disk whose radius is related to the free energy of the Liapunov exponent.

Spectral analysis of transfer operators associated to Farey fractions

The spectrum of a one-parameter family of signed transfer operators associated to the Farey map is studied in detail. We show that when acting on a suitable Hilbert space of analytic functions they are selfadjoint and exhibit absolutely continuous spectrum and no non-zero point spectrum. Polynomial eigenfunctions when the parameter is a negative half-integer are also discussed.

Statistics of energy levels and zero temperature dynamics for deterministic spin models with glassy behaviour

We consider the zero-temperature dynamics for the infinite-range, non translation invariant one-dimensional spin model introduced by Marinari, Parisi and Ritort to generate glassy behaviour out of a deterministic interaction. It is argued that there can be a large number of metastable (i.e., one-flip stable) states with very small overlap with the ground state but very close in energy to it, and that their total number increases exponentially with the size of the system.We consider the zero-temperature dynamics for the infinite-range, non translation invariant one-dimensional spin model introduced by Marinari, Parisi and Ritort to generate glassy behaviour out of a deterministic interaction. It is argued that there can be a large number of metastable (i.e., one-flip stable) states with very small overlap with the ground state but very close in energy to it, and that their total number increases exponentially with the size of the system.

RETURN TIMES AND MIXING PROPERTIES

We briefly dwell upon possible relationships between mixing properties and regenerative behaviour of ergodic dynamical systems.

On the leading eigenvalue of transfer operators of the Farey map with real temperature

Abstract We study the spectral properties of a family of generalised transfer operators associated to the Farey map. We show that when acting on a suitable space of holomorphic functions, the operators are self-adjoint and the positive dominant eigenvalue can be approximated by means of the matrix expression of the operators.

Mean Field Behaviour of Spin Systems with Orthogonal Interaction Matrix

For the long-range deterministic spin models with glassy behaviour of Marinari, Parisi and Ritort we prove weighted factorization properties of the correlation functions which represent the natural generalization of the factorization rules valid for the Curie–Weiss case.

Parabolic dynamical systems and inducing

In these notes we shall try to sketch a somewhat general procedure to approach spectral properties of dynamical systems in situations where usual hyperbolicity assumptions are partially relaxed, such as parabolic rational maps of the Riemann sphere. Our basic setting will be a triple (X,T,m) where T is a noninvertible map from a space X to itself and m is a reference measure. We shall assume that T is nonsingular w.r.t. m, i.e. m(T (A)) = 0 whenever m(A) = 0, and locally invertible, i.e. there is a countable partition of X such that T is one-to-one on each element of it. This implies that JT = dm ◦ T/dm, its Jacobian w.r.t. m, exists and is > 0 m-a.e. The transfer operator P , which acts on a function f : X → I C as