[Paper Review] Examples of Minimal Diffeomorphisms on $t^{2}$ Semiconjugated to an Ergodic Translation

Alejandro Passeggi, Martı́n Sambarino|arXiv (Cornell University)|Sep 2, 2010

Mathematical Dynamics and Fractals8 references3 citations

TL;DR

This paper constructs minimal $C^{3-ε}$ diffeomorphisms on the 2-torus $\mathbb{T}^2$ that are semiconjugate to ergodic translations but not conjugate to them, using the holonomy of the unstable foliation from Mañé's derived-from-Anosov diffeomorphism on $\mathbb{T}^3$. The key result is the existence of such maps with zero entropy, sensitivity to initial conditions, Li-Yorkle chaos, and uniquely ergodic dynamics, demonstrating that Denjoy-type behavior can occur in higher dimensions with controlled regularity.

ABSTRACT

We prove that for every $ε>0$ there exists a minimal diffeomorphism $f:\T^{2} ightarrow\T^{2}$ of class $C^{3-ε}$ and semiconjugate to an ergodic traslation, and have the following properties: zero entropy, sensitivity with respect to initial conditions and Li-Yorke chaos. These examples are obtained through the holonomy of the unstable foliation of Mañé's example of derived from Anosov diffeomorphism on $\T^3.$

Motivation & Objective

• To construct minimal diffeomorphisms on $\mathbb{T}^2$ that are semiconjugate to ergodic translations but not conjugate to them, extending Denjoy's theory beyond the circle.
• To investigate the dynamical properties—such as sensitivity, chaos, and ergodicity—of such maps in the $C^{3-\epsilon}$ class for any $\epsilon > 0$.
• To show that the fibers of the semiconjugacy are either points or arcs, and that uncountably many fibers are nontrivial arcs.
• To establish the existence of minimal, uniquely ergodic, $C^{3-\epsilon}$ diffeomorphisms on $\mathbb{T}^2$ with chaotic dynamics despite zero entropy.

Proposed method

• The construction uses the holonomy map of the unstable foliation from Mañé's derived-from-Anosov diffeomorphism on $\mathbb{T}^3$, which is modified to achieve $C^{3-\epsilon}$ regularity.
• The linear Anosov map is chosen from a one-parameter family $B_a \in \mathrm{SL}(3,\mathbb{Z})$ with eigenvalues satisfying $0 < \lambda_s(a) < \lambda_c(a) < 1 < \lambda_u(a)$, ensuring the desired spectral properties.
• A $C^\infty$ perturbation $g_{B_a,k}$ is applied near the fixed point to alter the unstable index and create transverse homoclinic points, preserving partial hyperbolicity.
• The $C^r$ regularity of the unstable foliation is established via the $C^r$ Section Theorem (Hirsch-Pugh-Shub), using estimates on the contraction and expansion rates $l_\xi(F)$ and $\tau_\xi(g_a)$.
• The induced map $f: \mathbb{T}^2 \to \mathbb{T}^2$ is obtained as the time-1 map of the holonomy flow along the unstable leaves, yielding a minimal diffeomorphism.
• The proof verifies $l_\xi(F)(\tau_\xi(g_a))^r < 1$ uniformly in $\xi$, ensuring $C^r$ regularity of the unstable foliation and hence of $f$.

Experimental results

Research questions

• RQ1 Can minimal diffeomorphisms on $\mathbb{T}^2$ that are semiconjugate but not conjugate to ergodic translations exist in the $C^{3-\epsilon}$ class for any $\epsilon > 0$?
• RQ2 What dynamical properties—such as chaos, sensitivity, and entropy—can such maps exhibit despite zero entropy?
• RQ3 Do the fibers of the semiconjugacy between the diffeomorphism and the translation have a specific topological structure, such as being arcs or points?
• RQ4 Is it possible for such a minimal diffeomorphism to be uniquely ergodic while still exhibiting Li-Yorkle chaos?
• RQ5 Can the $C^{3-\epsilon}$ regularity be achieved by selecting an appropriate linear Anosov map and perturbation, rather than fixing the map a priori?

Key findings

• For every $\epsilon > 0$, there exists a minimal $C^{3-\epsilon}$ diffeomorphism $f: \mathbb{T}^2 \to \mathbb{T}^2$ semiconjugate to an ergodic translation but not conjugate to it.
• The map $f$ has zero topological entropy, despite exhibiting sensitive dependence on initial conditions and Li-Yorkle chaos.
• The semiconjugacy fibers $h^{-1}(x)$ are either singletons or closed arcs, with uncountably many $x$ having nontrivial arc fibers.
• The dynamics on nontrivial fibers compress and stretch arcs repeatedly, leading to chaotic behavior such as sensitivity and Li-Yorkle chaos.
• The map $f$ is uniquely ergodic, preserving exactly one invariant probability measure.
• The unstable foliation of the underlying Mañé diffeomorphism is minimal, and the induced map on $\mathbb{T}^2$ inherits this minimality through the holonomy construction.