Mini-Workshop on Logic, Computability and Dynamical Systems

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Mini-Workshop on Logic, Computability and Dynamical Systems
9-10 December 2010
Abdus Salam International Centre for Theoretical Physics (ICTP), 
Trieste, Italy

Thursday 9 December 2010

14.00 Cristobal Rojas (Fields Institute, Toronto)

Introduction to Computability in Dynamics.

Abstract:  According to Turing, a real number x is computable if there 
exists an algorithm which, upon input n, produces a rational number  at 
distance no more than 2^-n from x.  This notion has been extended to 
treat computability of infinite objects over general metric spaces, and 
given rise to the theory of Computable Analysis. In this talk we will 
introduce these abstract notions of computability,  and will illustrate 
them via some examples taken from  the theory of dynamical systems: 
invariant sets, invariant measures and generic points.


15.15 Stefano Luzzatto (ICTP)

Finite Resolution Dynamics

We develop a new mathematical model for describing a dynamical system at 
limited resolution (or finite scale), and we give precise meaning to the 
notion of a dynamical system having some property at finite resolution. 
Open covers are used to approximate the topology of the phase space in a 
finite way,
and the dynamical system is represented by means of a combinatorial 
multivalued map. We formulate notions of transitivity and mixing in the 
finite resolution setting in a computable and consistent way. Moreover, 
we formulate equivalent conditions for these properties in terms of 
graphs, and provide effective algorithms for their verification. As an 
application we show that the H\'enon attractor is mixing at all 
resolutions coarser than 10^-5.


Friday 10 December

10.30 Giovanni Panti (Udine)

The automorphism group of product logic.

Abstract: Product logic is a many-valued propositional logic whose set 
of truth-values is the half-open interval (0,1], and whose conjunction 
connective is ordinary multiplication. As an equational theory, it has 
free structures, and it turns out that the automorphism groups of these 
structures have a rich and interesting dynamics. We survey a few results 
in this area, and discuss some open problems.


11.45 Stefano Galatolo (Pisa)

Logarithm laws and decay of correlations in dynamical systems

Abstract:
We will consider the behavior of the time which is needed for a typical 
point to enter in a sequence of decreasing targets. In several systems 
this time increases (having the same scaling behavior) as the inverse of 
the measure of the targets. We will see that a general condition for 
this to happen is superpolynomial decay of correlations. We will also 
see some applications, on the geometric Lorenz flow and geodesic flows 
in variable negative curvature. On the other side there are translations 
of the torus having particular arithmetical properties where the time 
needed to enter in a given sequence of balls increases much faster than 
the inverse of the ball’s measure. By a construction of Fayad, moreover 
we can also show smooth, mixing examples of the same kind.



Venue: Luigi Stasi Seminar Room, ICTP Leonardo Building, first floor.