Subcohomology and a Livsic Theorem for Zooming Systems

TL;DR

The paper extends Livsic-type results to the broad class of zooming systems, which generalize non-uniformly expanding dynamics and may include critical sets, by proving that any Hölder potential with nonnegative average over all invariant measures admits a global coboundary witness $\lambda_{0}$, with Hölder regularity when the average is strictly positive. The main tool is a constructive approach on the dense zooming set, producing a continuous $\lambda_{0}$ satisfying $\phi \geq \lambda_{0} - \lambda_{0} \circ f$, and achieving the exact coboundary identity $\phi = \lambda_{0} - \lambda_{0} \circ f$ in the zero-average, dense-periodic-points regime. A complementary result shows that the maximizing measure is generically unique for a residual class of Hölder potentials. The work unifies and extends classical Livsic-type theorems from circle expanding maps to a wide array of maps (Viana, Benedicks–Carleson, Rovella, local diffeomorphisms) and provides a versatile framework for hyperbolic times, expanding sets, and symbolic dynamics in the zooming context.

Abstract

In the context of continuous zooming systems $f:M \to M$ on a compact metric space $M$, which include the non-uniformly expanding ones, possibly with the presence of a critical set, with the zooming set dense in $M$, we prove that any Hölder potential $φ: M \to \mathbb{R}$ for which the integrals $\int φdμ\geq 0$ with respect to any $f$-invariant probability $μ$, admits a continuous function $λ_{0} : M \to \mathbb{R}$ (which can be Hölder if some integral is positive) such that \[ φ\geq λ_{0}- λ_{0} \circ f. \] This extends a result in [9] for $C^{1}$-expanding maps on the circle $\mathbb{T} = \mathbb{R}/\mathbb{Z}$ to important classes of maps as uniformly expanding, local diffeomorphisms with non-uniform expansion, Viana maps, Benedicks-Carleson maps and Rovella maps. We also give an example beyond the exponential contractions context. Moreover, in the case of the integrals $\int φdμ= 0$ with respect to any $f$-invariant probability $μ$ and the set of periodic points to be dense in $M$, we obtain a version of the Livsic Theorem, that is, the functions $λ_{0}$ can be taken such that \[ φ= λ_{0}- λ_{0} \circ f. \] Additionally, we also prove that the measure which maximizes the integrals is unique for a residual set of potentials.

Theorems & Definitions (28)

• Theorem 1.1: Livsic Theorem
• Theorem 1.2: Livsic Theorem
• Theorem 1.3
• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Remark 1
• Definition 5
• Theorem A