On entropy, regularity and rigidity for convex representations of hyperbolic manifolds. (English) Zbl 1336.37024
The first result of the paper is the following:
Theorem A. Let \(\Gamma\) be a convex cocompact group of a \(\mathrm{CAT}(-1)\) space \(X\), \(\xi \in X\) and \(\rho: \Gamma \rightarrow \mathrm{PGL}(d,\mathbb R)\) be an irreducible convex representation with \(d\geq 3\). Then \(\alpha h_\rho \leq h_\Gamma\) and \(\alpha\mathbb H_\rho \leq h_\Gamma\), when \(\xi\) is \(\alpha\)-Hölder.
The main purpose of the paper is to extend the inequality \(\alpha h_{\rho_2} \leq h_{\rho_1}\) for convex representations. Here \(h_\rho\) is the entropy of \(\rho\) and \(\alpha\) is the exponent of the map \(\partial_\infty\Gamma \rightarrow \rho (R^d)\).
Theorem A. Let \(\Gamma\) be a convex cocompact group of a \(\mathrm{CAT}(-1)\) space \(X\), \(\xi \in X\) and \(\rho: \Gamma \rightarrow \mathrm{PGL}(d,\mathbb R)\) be an irreducible convex representation with \(d\geq 3\). Then \(\alpha h_\rho \leq h_\Gamma\) and \(\alpha\mathbb H_\rho \leq h_\Gamma\), when \(\xi\) is \(\alpha\)-Hölder.
The main purpose of the paper is to extend the inequality \(\alpha h_{\rho_2} \leq h_{\rho_1}\) for convex representations. Here \(h_\rho\) is the entropy of \(\rho\) and \(\alpha\) is the exponent of the map \(\partial_\infty\Gamma \rightarrow \rho (R^d)\).
Reviewer: Victor Sharapov (Dayton)
MSC:
| 37C45 | Dimension theory of smooth dynamical systems |
| 22A10 | Analysis on general topological groups |
| 57N15 | Topology of the Euclidean \(n\)-space, \(n\)-manifolds (\(4 \leq n \leq \infty\)) (MSC2010) |
| 37A35 | Entropy and other invariants, isomorphism, classification in ergodic theory |
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