Continuity of the Hausdorff dimension for invariant subsets of interval maps.

Raith, P.

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If $f : [0, 1] \to ℝ$ is strictly increasing and continuous define $T_{f} x = f (x) (mod 1)$ . A transformation $\tilde{T} : [0, 1] \to [0, 1]$ is called $ε$ -close to $T_{f}$ , if $\tilde{T} x = \tilde{f} (x) (mod 1)$ for a strictly increasing and continuous function $\tilde{f} : [0, 1] \to ℝ$ with $∥ \tilde{f} {- f ∥}_{\infty} < ε$ . It is proved that the topological pressure $p (T_{f}, g)$ is lower semi-continuous, and an upper bound for the jumps up is given. Furthermore the continuity of the maximal measure is shown, if a certain condition is satisfied. Then it is proved that the topological pressure is upper semi-continuous for every continuous function $g : [0, 1] \to ℝ$ , if and...

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