Erratum to: Spectral Triple and Sinai–Ruelle–Bowen Measures

1 Erratum to: Complex Anal. Oper. Theory DOI 10.1007/s11785-014-0355-5

We correct an error in the proof of Theorem (4.1).

The proof of Theorem (4.1) exploited the fact that the set of locally constant functions forms a dense sub algebra in the space of complex-valued continuous functions defined on the Julia set, ${\mathcal {J}}_{c}$, denoted by ${\mathcal {C}} ({\mathcal {J}}_{c}, {\mathbb {C}})$. We must have used the dense sub algebra of polynomial functions in stead of the locally constant functions. In particular, we re-write the paragraph following the Eq. (4.5) in the proof of Theorem (4.1).

Theorem 4.1

$({\mathcal {H}}, {\mathcal {A}}, D_{\varphi })$ is a spectral triple.

Let $f \in \mathcal {A} = \mathcal {C} (\mathcal {J}_{c}, \mathbb {C})$ be a polynomial function. We write $P (\mathcal {J}_{c})$ for the set of all polynomials in $\mathcal {J}_{c}$. $P (\mathcal {J}_{c})$ is a dense subalgebra in $\mathcal {A}$. Further, given any $\epsilon > 0$, there exists a $N \in \mathbb {Z}^{+}$ such that for any generic $z_{1}, z_{2} \in \mathcal {J}_{c}$ and $n \ge N$, we have

$$\begin{aligned} \left| f \left( \left\{ P_{c}^{-n} z_{1} \right\} \cap C \right) - f \left( \left\{ P_{c}^{-n} z_{2} \right\} \cap C \right) \right| <\epsilon ,\ \ \ \ \forall C \in \mathfrak {P}_{(n)}. \end{aligned}$$