[Paper Review] Smooth conjugacy of Anosov diffeomorphisms on higher dimensional tori

Andrey Gogolev|ArXiv.org|Apr 24, 2008

Mathematical Dynamics and Fractals10 references3 citations

TL;DR

This paper establishes $C^{1+ u}$ regularity of the conjugacy between Anosov diffeomorphisms on higher-dimensional tori ($d \geq 3$) that are $C^1$-close to an irreducible hyperbolic automorphism with simple real spectrum and share the same periodic eigenvalue data. It proves that under a transitivity assumption on intermediate foliations, such conjugacies are smoother than Hölder, resolving a long-standing question in smooth conjugacy theory for higher-dimensional Anosov systems.

ABSTRACT

Let $L$ be a hyperbolic automorphism of $\mathbb T^d$, $d\ge3$. We study the smooth conjugacy problem in a small $C^1$-neighborhood $\mathcal U$ of $L$. The main result establishes $C^{1+ν}$ regularity of the conjugacy between two Anosov systems with the same periodic eigenvalue data. We assume that these systems are $C^1$-close to an irreducible linear hyperbolic automorphism $L$ with simple real spectrum and that they satisfy a natural transitivity assumption on certain intermediate foliations. We elaborate on the example of de la Llave of two Anosov systems on $\mathbb T^4$ with the same constant periodic eigenvalue data that are only Hölder conjugate. We show that these examples exhaust all possible ways to perturb $C^{1+ν}$ conjugacy class without changing periodic eigenvalue data. Also we generalize these examples to majority of reducible toral automorphisms as well as to certain product diffeomorphisms of $\mathbb T^4$ $C^1$-close to the original example.

Motivation & Objective

• To determine whether the coincidence of periodic eigenvalue data implies smooth conjugacy for Anosov diffeomorphisms on higher-dimensional tori.
• To identify conditions under which $C^1$-close Anosov systems with identical periodic data admit $C^{1+\nu}$ conjugacies.
• To understand the role of foliation transitivity and invariant structures in enabling smooth conjugacy beyond Hölder regularity.
• To classify the full set of obstructions to $C^{1+\nu}$ conjugacy in a $C^1$-neighborhood of linear Anosov automorphisms.
• To generalize de la Llave's counterexample on $\mathbb{T}^4$ to characterize all possible $C^1$-perturbations that preserve periodic data but not smooth conjugacy.

Proposed method

• Analyzes the $C^1$-neighborhood $\mathcal{U}$ of a hyperbolic automorphism $L$ on $\mathbb{T}^d$, $d \geq 3$, with simple real spectrum and positive eigenvalues.
• Uses the invariant splitting into one-dimensional distributions $F_i^f$ and $E_j^f$ corresponding to eigenvalues $\mu_i < \cdots < 1 < \lambda_j$, which integrate uniquely into foliations $U_i^f$ and $V_j^f$.
• Applies a transitive point argument using a factor dynamics $\tilde{f}$ on the space of central leaves $\mathcal{T}$, which is conjugate to a linear automorphism $B$ on $\mathbb{T}^2$.
• Constructs an invariant measure $\mu$ via weak* limits of empirical measures $\mu_n = \frac{1}{n}\sum_{i=0}^{n-1} (f^i)_*\eta_0$, where $\eta_0$ has density proportional to the Jacobian ratio $\rho(x_0, \cdot)$ along weak unstable leaves.
• Establishes absolute continuity of the weak unstable foliation $W_f^{wu}$ with respect to $\mu$ using standard Hopf argument techniques.
• Proves $C^{1+\nu}$-smoothness of the conjugacy $h$ along $W_f^{wu}$ by showing that the conjugacy preserves the flag of intermediate foliations and using induction on the flag structure.

Experimental results

Research questions

• RQ1 Under what conditions is the conjugacy between two $C^1$-close Anosov diffeomorphisms on $\mathbb{T}^d$, $d \geq 3$, with identical periodic eigenvalue data, $C^{1+\nu}$-smooth?
• RQ2 Can the counterexample of de la Llave on $\mathbb{T}^4$—where two Anosov systems with identical periodic data are only Hölder conjugate—be generalized to other linear automorphisms?
• RQ3 What role does transitivity of the dynamics on intermediate foliations play in ensuring smooth conjugacy?
• RQ4 Are there additional invariants beyond periodic eigenvalue data that govern $C^1$ conjugacy classes near linear Anosov automorphisms?
• RQ5 How do reducible or product-type Anosov diffeomorphisms in $C^1$-neighborhoods of linear automorphisms affect the smoothness of conjugacies?

Key findings

• The conjugacy between two Anosov diffeomorphisms $f$ and $g$ in a $C^1$-neighborhood of an irreducible hyperbolic automorphism $L$ on $\mathbb{T}^d$, $d \geq 3$, with simple real spectrum and identical periodic eigenvalue data is $C^{1+\nu}$-smooth if the systems satisfy a transitivity assumption on intermediate foliations.
• The paper shows that de la Llave's example on $\mathbb{T}^4$—where two Anosov systems with identical periodic data are only Hölder conjugate—is exhaustive in the sense that all perturbations that preserve periodic data but break $C^{1+\nu}$ conjugacy are topologically equivalent to this example.
• The construction of a transitive invariant measure $\mu$ with absolutely continuous weak unstable foliation allows the application of the Hopf argument to prove $C^{1+\nu}$-regularity of the conjugacy along the weak unstable direction.
• The conjugacy preserves the flag of intermediate foliations $V_m^f$, and this preservation is established inductively using transitive point arguments and the structure of the factor dynamics on the central leaf space.
• The method generalizes de la Llave's example to a broad class of reducible toral automorphisms and to certain product diffeomorphisms on $\mathbb{T}^4$ that are $C^1$-close to the original counterexample.
• The key technical advance is proving that the central holonomy is smooth under the transitivity assumption, which enables the induction step in the proof of $C^{1+\nu}$-regularity of the conjugacy.