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<p>In this paper, entropy and pressure are investigated for a random dynamical system <i>φ</i> over ℤ<sup><i>k</i></sup>-actions on a compact metric space. The pressure <i>P</i>(<i>φ</i>, <i>f</i>) of <i>φ</i> with respect to a random continuous function <i>f</i> and the measure-theoretic entropy <i>h</i><sub><i>μ</i></sub>(<i>φ</i>) for a <i>φ</i>-invariant measure <i>μ</i> are defined. A variational principle for pressure <i>P</i>(<i>φ</i>, <i>f</i>) is established, which states that <i>P</i>(<i>φ</i>, <i>f</i>) is the supremum of the sum of <i>h</i><sub><i>μ</i></sub>(<i>φ</i>) and the integral of <i>f</i> taken over all invariant measures <i>μ</i>. We also obtain some basic properties for equilibrium states.</p>

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On Entropy, Pressure and Variational Principle for Random Dynamical Systems over ℤk-Actions

  • 期刊论文

  • 发表日期：

    2025年10月15日

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  • Xinsheng Wang,
  • Ziyao Zhang,
  • Yujun Zhu

摘要

In this paper, entropy and pressure are investigated for a random dynamical system φ over ℤk-actions on a compact metric space. The pressure P(φ, f) of φ with respect to a random continuous function f and the measure-theoretic entropy hμ(φ) for a φ-invariant measure μ are defined. A variational principle for pressure P(φ, f) is established, which states that P(φ, f) is the supremum of the sum of hμ(φ) and the integral of f taken over all invariant measures μ. We also obtain some basic properties for equilibrium states.