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showed that 58 per cent of UK consumers would be more likely to try a new product from a brand they knew, versus only 3 per cent for a new brand Lower cost: the extension advantages of awareness and reputation mean that you do tend to get more bang for your buck compared to new brands Studies show that cost per unit of trial is 36 per cent lower and that repurchase is also higher (1) (See Table 11)

Theorem 161 The dynamical system = (T2 , , ) is ergodic if and only if at least one of the following conditions is ful lled: (i) Any measurable set E T2 , which is -invariant (ie, 1 E = E ), is such that (E ) = 0 or (T2 \ E ) = 0 (ii) If f L (T2 , ), that is, is bounded almost everywhere with respect to , is -invariant, that is, f = f , then it is constant almost everywhere with respect to If is ergodic, then the set of periodic orbits of the map is -negligible Within this ergodicity context, one notices that the hyperbolic automorphims (164) are mixing Generally, a map is mixing if for all f , g L2 (T2 , ) the following condition is ful lled: lim f ( k (x)) g (x) (dx) = ( f ) ( g )

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In particular, by choosing for f and g the characteristic functions f = 2 E 2 of two measurable sets E 1 , E 2 T , a mixing map veri es

k E 1 E 2 = (E 1 ) (E 2 )

84 86

(167)

In other words, the part of the set k E 1 intersecting E 2 has a measure that is asymptotically proportional to the measure of E 2 , which means that E 1 (if (E 1 ) > 0) spreads uniformly in T2

To set up the quantum states on the torus, let us rst recall two main features of quantum states on the plane viewed as a phase space: (i) The (pure) quantum states of a particle having the line R as a con guration space and so the plane R2 as a phase space are most generally tempered distributions S (R), or simply S , for example, plane waves or Dirac distributions, and, when normalizable, are elements of the Hilbert space L2 (R), the latter being viewed as a subspace of S (R) (ii) The translations acting on the classical phase space realize, on the quantum level, as transformations of L2 (R) that belong to a unitary irreducible representation of the Weyl Heisenberg group Weyl Heisenberg group ( , q, p) U ( , q, p) = e

pQ qP =D(z)

Table 11: Success rates for extensions versus new brands Trial (index) New brands Extensions 100 123 Cost per unit of trial (index) 100 64 Repeat/loyalty (index) 100 161

90 91

(168)

Fig 911 Temporal behavior of the average position for the asymmetric P schl Teller potential ( , ) = (4, 8) with n max = 10, for (a) J = 20 and (b) J = 50 Source Antoine et al [112] (reprinted with permission from [Antoine, J-P, Gazeau, JP, Monceau, P, Klauder JR, Pernon KA, Temporally stable coherent states for in nite well, J Math Phys, 42, p 2349, 2001], American Institute of Physics)

The speci c choices we have made here for the set of coherent states are based on two additional guiding principles besides continuity and resolution of unity, as was proposed in [58] The rst of these is temporal stability , which in words asserts that the temporal evolution of any coherent state always remains a coherent state The second of these, referred to as the action identity in [58], chooses variables for the coherent state labels that have as close a connection as possible to classical action angle variables In particular, for a single degree of freedom, the label pair ( J, ) is used to identify the coherent state | J, Temporal stability means that, under the dynamics chosen, temporal evolution proceeds according to | J, + t , for some xed parameter To ensure that ( J, ) describes action angle variables, it is suf cient to require that the symplectic potential induced by the coherent states themselves is of Darboux form, or speci cally that i J, | d | J, = J d ,

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