1 Correction to: Geom Dedicata (2017) 191:171–198 https://doi.org/10.1007/s10711-017-0251-z
The paper [2] studies the dynamics of a class of circle maps and their two-dimensional natural extensions built using the generators of a given cocompact and torsion-free Fuchsian group \(\Gamma \). If \(\mathbb D\) denotes the Poincaré unit disk model endowed with the standard hyperbolic metric, then \(\Gamma \backslash \mathbb D\) is a compact surface of constant negative curvature and of a certain genus \(g>1\). Most of the considerations and proofs in the paper were done for a special case of surface groups, those that admit a fundamental domain \(\mathcal {F}\) given by a regular\((8g-4)\)-sided polygon. On p. 172 in the Introduction, we neglected to explain in the paragraph below equation (1.2) that, although not all surface groups admit such a fundamental domain, it is possible to reduce the general case to this special situation without affecting the results of the paper (see also [1, Appendix A]).
More precisely, given \(\Gamma '\backslash \mathbb D\) a compact surface of genus \(g>1\), there exists a Fuchsian group \(\Gamma \) such that:
-
(i)
\(\Gamma \backslash \mathbb D\) is a compact surface of the same genus g;
-
(ii)
\(\Gamma \) has a fundamental domain \(\mathcal F\) given by a regular \((8g-4)\)-sided polygon;
-
(iii)
By the Fenchel–Nielsen theorem [3] there exists an orientation preserving homeomorphism h from \(\bar{\mathbb D}\) onto \(\bar{\mathbb D}\) such that \(\Gamma '=h\circ \Gamma \circ h^{-1}\).
One can now extend the considerations described in the introductory section of the paper to any compact surface \(\Gamma '\backslash \mathbb D\), using the orientation preserving homeomorphism h and the setting for \(\Gamma \backslash \mathbb D\). Let
Then the set \(\{T_i'\}\) satisfies relations (1.3)–(1.5) and the order of the points \(\{P'_i\}\cup \{Q'_i\}\) will be the same as for the set \(\{P_i\}\cup \{Q_i\}\). The geodesics \(P_i'Q_{i+1}'\) will produce the \((8g-4)\)-sided polygon \(\mathcal F'\) whose sides are identified by transformations \(T_i'\). Adler and Flatto [1, Appendix A] conclude that region \(\mathcal F'\) satisfies all the conditions of Poincaré’s theorem, hence it is the fundamental domain for \(\Gamma '\).
The main object of study in our paper is the generalized Bowen-Series circle map \(f_A:S\rightarrow S\) given by (1.8)
with the set of jump points \(\bar{A}=\{A_1,A_2,\dots , A_{8g-4}\}\) satisfying the condition that \(A_i\in (P_i,Q_i)\), \(1\le i\le 8g-4\). The corresponding two-dimensional extension map given by (1.9) is
Even though the main results of the paper (Theorems 1.2 and 1.3) were proved for the special situation of a genus g compact surface \(\Gamma \backslash \mathbb D\) that admits a regular \((8g-4)\)-sided fundamental region, the results remain true in full generality for an arbitrary genus g compact surface \(\Gamma '\backslash \mathbb D\) with the set of \((8g-4)\) generators \(\{T'_i\}\), the set of jump points \(\bar{A}'=\{A_1', A'_2, \dots , A'_{8g-4}\}\) with \(A'_i=h(A_i)\in (P'_i,Q'_i)\) and the corresponding maps:
The orientation preserving homeomorphism \(h:\bar{\mathbb D}\rightarrow \bar{\mathbb D}\) and the relations
allow us to conclude that:
-
(a)
A partition point \(A'_i\in (P'_i,Q'_i)\), \(1\le i\le 8g-4\), satisfies the cycle property, i.e., there exist positive integers \(m_i, k_i\) such that
$$\begin{aligned} f_{\bar{A}'}^{m_i}(T'_iA'_i)=f_{\bar{A}'}^{k_i}(T'_{i-1}A'_i) \end{aligned}$$if and only if the corresponding partition point \(A_i=h^{-1}(A'_i)\in (P_i,Q_i)\) satisfies the cycle property
$$\begin{aligned} f_{\bar{A}}^{m_i}(T_iA_i)=f_{\bar{A}}^{k_i}(T_{i-1}A_i). \end{aligned}$$ -
(b)
A partition point \(A'_i\) satisfies the short cycle property
$$\begin{aligned} f_{\bar{A}'}(T'_iA'_i)=f_{\bar{A}'}(T'_{i-1}A'_i) \end{aligned}$$if and only if the corresponding partition point \(A_i=h^{-1}(A'_i)\) satisfies the short cycle property:
$$\begin{aligned} f_{\bar{A}}(T_iA_i)=f_{\bar{A}}(T_{i-1}A_i). \end{aligned}$$ -
(c)
If \(\displaystyle \Omega _{\bar{A}}=\bigcap \nolimits _{n=0}^\infty F_{\bar{A}}^n({\mathbb S}\times {\mathbb S}{\setminus } \Delta )\) is the global attractor of the map \(F_{\bar{A}}\), then \(\Omega _{\bar{A}'}=(h\times h)(\Omega _{\bar{A}})\) is the global attractor of the map \(F_{\bar{A}'}\). Also, if \(\Omega _{\bar{A}}\) has finite rectangular structure, then \(\Omega _{\bar{A}'}\) has finite rectangular structure, since \(h\times h\) preserves horizontal and vertical lines.
We would like to use this opportunity to also correct some misprints: on p. 173, last paragraph, the text “of the fundamental domain \(\mathcal {F}\)” should read “of \(\mathbb D\)”; on p. 193, in equation (7.2), the term “\(A_i+1\)” should read “\(A_{i+1}\)”; on p. 193, Proposition 7.1, the relations “\(B_i = T_iA_i\), and \(C_i = T_{i-1}A_i\)” should read “\(B_i = T_{\sigma (i-1)}A_{\sigma (i-1)}\), and \(C_i = T_{\sigma (i+1)}A_{\sigma (i+1)+1}\).”
References
Adler, R., Flatto, L.: Geodesic flows, interval maps, and symbolic dynamics. Bull. Am. Math. Soc. 25(2), 229–334 (1991)
Katok, S., Ugarcovici, I.: Structure of attractors for boundary maps associated to Fuchsian groups. Geom. Dedicata 191, 171–198 (2017). https://doi.org/10.1007/s10711-017-0251-z
Tukia, P.: On discrete groups of the unit disk and their isomorphisms. Ann. Acad. Sci. Fenn. Ser. A I. Math. 504, 5–44 (1972)
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Katok, S., Ugarcovici, I. Correction to: Structure of attractors for boundary maps associated to Fuchsian groups. Geom Dedicata 198, 189–191 (2019). https://doi.org/10.1007/s10711-018-0336-3
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DOI: https://doi.org/10.1007/s10711-018-0336-3