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Regularity of limit sets of Anosov representations

Tengren Zhang, Andrew Zimmer

Abstract

In this paper we establish necessary and sufficient conditions for the limit set of a projective Anosov representation to be a differentiable submanifold of projective space with Holder continuous derivatives. We also calculate the optimal value of the Holder constant in terms of the eigenvalue data of the Anosov representation.

Regularity of limit sets of Anosov representations

Abstract

In this paper we establish necessary and sufficient conditions for the limit set of a projective Anosov representation to be a differentiable submanifold of projective space with Holder continuous derivatives. We also calculate the optimal value of the Holder constant in terms of the eigenvalue data of the Anosov representation.

Paper Structure

This paper contains 42 sections, 51 theorems, 372 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Anosov representations
  3. A definition of Anosov representations
  4. Singular values and Anosov representations
  5. Properties of Anosov representations
  6. Gromov geodesic flow space
  7. Dominated Splittings
  8. $\rho$-controlled sets
  9. Properly convex domains
  10. The proof of Proposition \ref{['prop:rho_controlled_sets_projections']}
  11. Constructing an invariant properly convex domain
  12. Constructing a projective geodesic flow
  13. Finishing the proof of Proposition \ref{['prop:rho_controlled_sets_projections']}
  14. Sufficient conditions for differentiability of $\rho$-controlled subsets
  15. The relative exponential growth rate
  16. ...and 27 more sections

Key Result

Theorem 1.1

(Theorem thm:main_body) Suppose $\Gamma$ is a hyperbolic group, $\partial_\infty \Gamma$ is a topological $(m-1)$-manifold, and $\rho: \Gamma \rightarrow \mathop{\mathrm{PGL}}\nolimits_{d}(\mathop{\mathrm{\mathbb{R}}}\nolimits)$ is a $P_1$-Anosov representation. If then Moreover, $T_{\xi_\rho^1(x)} M = \xi_\rho^m(x)$ for any $x \in \partial_\infty \Gamma$.

Figures (2)

  • Figure 1: The projection $p_{x,y}$.
  • Figure 2: $M$ in the affine chart $\mathop{\mathrm{\mathbb{A}}}\nolimits_y$.

Theorems & Definitions (119)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3: Theorem 1.1 and Proposition 4.6 of Ben04
  • Remark 1.4
  • Theorem 1.5: Theorem 1.4 of L2006
  • Example 1.6
  • Example 1.7
  • Remark 1.8
  • Theorem 1.9
  • Remark 1.10
  • ...and 109 more