Structure of proof of Theorem Sample Clauses

Structure of proof of Theorem. 3.3.1 Since we saw in Lemma 3.4.2 that the dimension of infinite entropy Bernoulli mea- sures is bounded above by s, we can now restrict our attention to finite entropy Bernoulli measures. Our proof of Theorem 3.3.1 is split into two parts. Firstly, in Chapter 4, we use a thermodynamic formalism approach to study the dimension of Bernoulli measures µp where p satisfies some conditions on its weights pn. Given a probability vector which satisfies these conditions, we can obtain the dimension of the corresponding Bernoulli measure as the derivative of an associated function βp. The problem then reduces to obtaining a ‘global bound’ on the convexity of βp, that is, a uniform lower bound on the second derivative of βp. This approach was proposed by Xxxxxx¨xxxxx, Xxxxxxxxx and Xxxxx´ski and outlined in a talk given by Xxxxxx¨xxxxx in [Ke]. In Chapter 5 we consider Xxxxxxxxx measures µp where p does not satisfy the conditions on the weights pn which were required for the above approach. By ‘re- distributing mass’ within such a Bernoulli measure µp, we can obtain an associated Bernoulli measure µp∗ which does satisfy the conditions. The problem then reduces to estimating the change in the entropy and Lyapunov exponent, in order to obtain an upper bound on dim µp in terms of dim µp∗ . In order to make this precise, we need to introduce some notation pertaining to various subsets of the simplex P.