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**Sample text**

Globally it may be a more complicated manifold formed by patching together several pieces of Rd , forming a torus, a cylinder, or some other manifold. When we need to stress that the dimension d of M is greater than one, we may refer to the point x ∈ M as xi where i = 1, 2, 3, . . , d. The evolution rule or dynamics f t : M → M that tells where a point x is in M after a time interval t. The pair (M, f ) is called a dynamical system. The dynamical systems we will be studying are smooth. This is expressed mathematically by saying that the evolution rule f t can be diﬀerentiated as many times as needed.

You might think that this strangeness has to do with contracting ﬂows only. Not at all - we chose this example as it is easier to visualise aperiodic dynamics when the ﬂow is contracting onto a lower-dimensional attracting set. As the next example we take a ﬂow that preserves phase space volumes. 13) with the 2D phase space coordinates x split into the conﬁguration space coordinates and the conjugate momenta of a Hamiltonian system with D degrees of sect. 1 freedom: x = (p, q) , q = (q1 , q2 , .

Cycle expansions oﬀer then an eﬃcient method for evaluating classical and quantum observables. The critical step in the derivation of the dynamical zeta function was the hyperbolicity assumption, that is the assumption of exponential shrinkage of all strips of the pinball repeller. 4), we have given up on any possibility of recovering the precise distribution of starting x (which should anyhow be impossible due to the exponential growth of errors), but in exchange we gain an eﬀective description of the asymptotic behavior of the system.