Local inverse measure-theoretic entropy for endomorphisms
Eugen Mihailescu, Radu B. Munteanu
TL;DR
This work introduces local inverse measure-theoretic entropy for non-invertible dynamics by examining backward trajectories through the inverse limit $\widehat{X}$ and inverse Bowen balls. It defines two complementary notions, inverse metric entropy and inverse partition entropy, and establishes their connections to folding entropy $F_f(\mu)$ and forward entropy $h_f(\mu)$, yielding the principal identity $h^-_f(\mu)=h_f(\mu)-F_f(\mu)$ under natural hypotheses. The paper then applies these concepts to hyperbolic and Anosov endomorphisms, proving entropy rigidity for special Anosov endomorphisms on $\mathbb{T}^2$ (classification up to a smooth conjugacy by the pair $(h_f(\mu_f^+), h_f^-(\mu_f^-))$) and developing partial/full variational principles linking inverse entropy to generalized inverse topological entropy. A diverse set of examples, including expanding maps, linear toral endomorphisms, fat baker maps, and Tsujii endomorphisms, illustrate the theory and compute inverse entropies for SRB and inverse-SRB measures. Overall, the framework provides new invariants to distinguish endomorphisms beyond forward entropy and offers tools for detailed analysis of non-invertible dynamics via backward trajectories and prehistories.
Abstract
We introduce a new notion of local inverse metric entropy along backward trajectories for ergodic measures preserved by endomorphisms (non-invertible maps) on a compact metric space. A second notion of inverse measure entropy is defined by using measurable partitions. Our notions have several useful applications. Inverse entropy can distinguish between isomorphism classes of endomorphisms on Lebesgue spaces, when they have the same forward measure-theoretic entropy. In a general setting we prove that the local inverse entropy of an ergodic measure μis equal to the forward entropy minus the folding entropy. The inverse entropy of hyperbolic measures on compact manifolds is explored, focusing on their negative Lyapunov exponents. We compute next the inverse entropy of the inverse SRB measure on a hyperbolic repellor. We prove an entropy rigidity result for special Anosov endomorphisms of \mathbb T^2, namely that they can be classified up to smooth conjugacy by knowing the entropy of their SRB measure and the inverse entropy of their inverse SRB measure. Next we study the relations between our inverse measure-theoretic entropy and the generalized topological inverse entropy on subsets of prehistories. In general we establish a Partial Variational Principle for inverse entropy. We obtain also a Full Variational Principle for inverse entropy in the case of special TA-covering maps on tori. In the end, several examples of endomorphisms are studied, such as fat baker transformations, fat solenoidal attractors, special Anosov endomorphisms, toral endomorphisms, and the local inverse entropy is computed for their SRB measures.