(a) Explain the arguments used in the definition of the

Lorenz-like families with criticalities a defined below in (8:35): See [Luzzatto & Viana (2000)]:

(b) Show that inequality (8:36) implies that given any y with |y|

(

x

2

, a

]

, there exists a unique x [a, a] such that a

(x) = y, and x and

y have apposite signs, and hence |x|

2

:

(c) Show that the properties L1L5 are satisfied by a nonempty open

set of one-parameter families, where openness is meant with respect to the

C

2

topology in the space of real maps :

(d) Show that for any a [a1, c] , one has that [a, a] is forward invariant

and

[a,a]

is transitive and (8:37) holds for all x [a, a] such that

j

a

(x) = 0 for every j = 0, 1, :::, n 1: See [Luzzatto & Viana (2000)]: