Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Feldman-Katok pseudometric and the GIKN construction of nonhyperbolic ergodic measures

Dominik Kwietniak, Martha Łącka

TL;DR

The paper develops the Feldman-Katok (FK) convergence framework by introducing the Feldman-Katok pseudometric $\bar{\textit{fk}}$, a topological analogue of Feldman’s $\bar{f}$ metric, to study invariant measures of general dynamical systems. It proves that GIKN royal measures arise as $\bar{\textit{fk}}$-limits of periodic (and ergodic) measures, and that $\bar{\textit{fk}}$-limits of ergodic measures remain ergodic, with entropy lower semicontinuity guaranteeing zero entropy for royal measures. The work further shows that royal measures are necessarily loosely Kronecker, yielding a precise Kakutani-equivalence classification (to ergodic rotations on compact groups) for these measures. The results provide a unified, non-symbolic framework connecting the GIKN construction to ergodic properties, zero entropy, and Loosely Kronecker dynamics, with broad implications for nonhyperbolic invariant measures. The FK approach also paves the way for applications beyond royal measures, including entropy and dimension theory, via a robust notion of convergence stronger than weak$^*$.

Abstract

The GIKN construction was introduced by Gorodetski, Ilyashenko, Kleptsyn, and Nalsky in [Functional Analysis and its Applications, 39 (2005), 21--30]. It gives a nonhyperbolic ergodic measure which is a weak$^*$ limit of a special sequence of measures supported on periodic orbits. This method was later adapted by numerous authors and provided examples of nonhyperbolic invariant measures in various settings. We prove that the result of the GIKN construction is always a loosely Kronecker measure in the sense of Ornstein, Rudolph, and Weiss (equivalently, standard measure in the sense of Katok, another name is loosely Bernoulli measure with zero entropy). For a proof we introduce and study the Feldman-Katok pseudometric $\bar{F_{K}}$. The pseudodistance $\bar{F_{K}}$ is a topological counterpart of the $\bar f$ metric for finite-state stationary stochastic processes introduced by Feldman and, independently, by Katok, later developed by Ornstein, Rudolph, and Weiss. We show that every measure given by the GIKN construction is the $\bar{F_{K}}$-limit of a sequence of periodic measures. On the other hand we prove that a measure which is the $\bar{F_{K}}$-limit of a sequence of ergodic measures is ergodic and its entropy is smaller or equal than the lower limit of entropies of measures in the sequence. Furthermore we demonstrate that $\bar{F_{K}}$-Cauchy sequence of periodic measures tends in the weak$^*$ topology either to a periodic measure or to a loosely Kronecker measure.

Feldman-Katok pseudometric and the GIKN construction of nonhyperbolic ergodic measures

TL;DR

The paper develops the Feldman-Katok (FK) convergence framework by introducing the Feldman-Katok pseudometric , a topological analogue of Feldman’s metric, to study invariant measures of general dynamical systems. It proves that GIKN royal measures arise as -limits of periodic (and ergodic) measures, and that -limits of ergodic measures remain ergodic, with entropy lower semicontinuity guaranteeing zero entropy for royal measures. The work further shows that royal measures are necessarily loosely Kronecker, yielding a precise Kakutani-equivalence classification (to ergodic rotations on compact groups) for these measures. The results provide a unified, non-symbolic framework connecting the GIKN construction to ergodic properties, zero entropy, and Loosely Kronecker dynamics, with broad implications for nonhyperbolic invariant measures. The FK approach also paves the way for applications beyond royal measures, including entropy and dimension theory, via a robust notion of convergence stronger than weak.

Abstract

The GIKN construction was introduced by Gorodetski, Ilyashenko, Kleptsyn, and Nalsky in [Functional Analysis and its Applications, 39 (2005), 21--30]. It gives a nonhyperbolic ergodic measure which is a weak limit of a special sequence of measures supported on periodic orbits. This method was later adapted by numerous authors and provided examples of nonhyperbolic invariant measures in various settings. We prove that the result of the GIKN construction is always a loosely Kronecker measure in the sense of Ornstein, Rudolph, and Weiss (equivalently, standard measure in the sense of Katok, another name is loosely Bernoulli measure with zero entropy). For a proof we introduce and study the Feldman-Katok pseudometric . The pseudodistance is a topological counterpart of the metric for finite-state stationary stochastic processes introduced by Feldman and, independently, by Katok, later developed by Ornstein, Rudolph, and Weiss. We show that every measure given by the GIKN construction is the -limit of a sequence of periodic measures. On the other hand we prove that a measure which is the -limit of a sequence of ergodic measures is ergodic and its entropy is smaller or equal than the lower limit of entropies of measures in the sequence. Furthermore we demonstrate that -Cauchy sequence of periodic measures tends in the weak topology either to a periodic measure or to a loosely Kronecker measure.

Paper Structure

This paper contains 32 sections, 22 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Basic definitions and notation
  3. Dynamical systems (standing assumptions)
  4. Note on invertibility
  5. Measure-preserving systems
  6. Invariant measures, generic sequences
  7. Symbolic systems
  8. Processes
  9. Entropy
  10. Faithful coding
  11. Kakutani equivalence
  12. Loosely Bernoulli systems
  13. Feldman's $\bar{f}$ and Ornstein's $\bar{d}$ metrics for shift-invariant measures on $\mathscr A^\infty$
  14. Properties of $\bar{f}$
  15. Loosely Kronecker systems
  16. ...and 17 more sections

Key Result

Lemma 2.2

If $\underline{x}\in X^\infty$ is generic for $\mu\in\mathop{\mathrm{\mathcal{M}_T}}\nolimits(X)$ and $\mathcal{P}\in\mathbf{P}^k(X)$ is such that $\mu(\partial\mathcal{P})=0$, then $\omega=\mathcal{P}(\underline{x})\in\Omega_k$ is a generic point for the measure $\mu_{\mathcal{P}}$ on $\Omega_k$.

Theorems & Definitions (51)

  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Definition 3.1
  • Definition 3.2
  • Theorem 3.3
  • Definition 3.4
  • Theorem 3.5
  • Definition 4.1
  • ...and 41 more