Nonconjugate pointed generalized solenoids with shift equivalent $\pi _1$-actions

Abstract:

Going back to the pioneering work of R. F. Williams, it has been widely believed that if elementary presentations $(K_i, f_i), i=1, 2$, satisfy the Williams axioms and fix the branch points, then there is a pointed conjugacy \[ \overline r:(\varprojlim (K_1,f_1),\overline {y}_1)\to (\varprojlim (K_2,f_2),\overline {y}_2)\] between the natural shift maps $\overline {f}_1$ and $\overline {f}_2$ if and only if the fundamental group homomorphisms $\pi _1(f_1, y_1)$ and $\pi _1(f_2, y_2)$ are shift equivalent. The “only if” direction is true. We prove that the “if” direction goal of this belief is false by exhibiting a family of counterexamples not previously analyzed. Of course, the associated hyperbolic attractors constructed by R. F. Williams cannot be conjugate in these exceptional cases.