×

Entropie d’une dynamique. (Entropy of a dynamic). (French) Zbl 0723.54020

The authors develop a notion of entropy H(R) of a given relation R on a compact metric space X. They also define the entropy of a semigroup \(R_ t\) of relations on X. Let T be a map of an interval into itself and R be the inverse relation (i.e. yRx if and only if \(y\in T^{-1}(x))\). Suppose T is piecewise linear and the slope of each linear piece has absolute value greater than or equal to 1, then the authors show that \(H(R)=0\). Let X be the unit tangent bundle of a manifold S. On X a semigroup \(R_ t\) of relations is defined as follows: \((y,w)R_ t(x,v)\) if and only if there exists a \(C^ 1\) geodesic path \(\gamma\) of length less than or equal to t, with modulus bounded by a constant c such that \(\gamma (0)=x\), \(\gamma (t)=y\), \(\gamma '(0)=\| \gamma '(0)\| v\), and \(\gamma '(t)=\| \gamma '(t)\| w\). The authors announce that if S has negative curvature and if c is small enough then \(H(R_ t)>0\).

MSC:

54C70 Entropy in general topology
54H20 Topological dynamics (MSC2010)
28D20 Entropy and other invariants
PDFBibTeX XMLCite