Abstract In this paper, we study (Sinai–Ruelle–Bowen) SRB measures for random dynamical systems and their associated skew-product transformations. Extending Burguet’s variational method [SRB measures for C ∞ $C^\infty $ upper C Superscript infinity surface diffeomorphisms. Invent. Math. 235 (3) (2024), 1019–1062], we establish the existence of random SRB measures under a fiber leaf condition, which ensures recurrent hyperbolic behavior along unstable leaves within generic fibers. The construction is based on geometric-time techniques and Følner averaging, leading to invariant measures that satisfy the relative Pesin entropy formula. We further identify a spectral-gap condition relating the Lyapunov spectra of the base and fiber dynamics, under which the random SRB measure coincides with an SRB measure for the smooth skew-product system.
Abstract We consider a partially hyperbolic diffeomorphism f : M → M $f: M \to M$ f colon upper M right arrow upper M without periodic points on a closed manifold M . We prove that f is accessible when M is a 3-manifold with non-virtually-solvable fundamental group π 1 ( M ) $\pi _1(M)$ pi 1 left parenthesis upper M right parenthesis . In the case where dim E c = 1 $\dim E^c = 1$ dimension upper E Superscript c Baseline equals 1 , we demonstrate that the center bundle E c $E^c$ upper E Superscript c is uniquely integrable if f lacks accessibility. Furthermore, we provide a complete characterization of accessibility classes for such systems with one-dimensional center bundles.
Abstract Non-statistical (or historical) dynamics refer to systems where a set of points with positive measure, relative to a reference probability measure (typically the Lebesgue measure on a manifold), fails to exhibit a convergent sequence of empirical measures. There are various strategies to identify maps with non-statistical behavior. In the first part of this paper, we use a strategy related to statistical instability to demonstrate that a Baire generic map in the space of Anosov–Katok diffeomorphisms of the annulus exhibits non-statistical behavior and even maximal oscillation. The notion of statistical instability was first formalized in its general format by Talebi [Non-statistical rational maps. Math. Z. 302 (1) (2022), 589–608], where its close connection to the existence of non-statistical dynamics was established. In the second part, we extend this framework by introducing analogous formalization and corresponding results in the case of pointwise convergence of empirical measures. Additionally, we propose a general strategy to identify non-statistical dynamics within other families of dynamical systems.
Abstract We construct a birational map of P d $\mathbb {P}^d$ double struck upper P Superscript d ( d ≥ 6 $d\geq 6$ d greater than or equals 6 ) whose intermediate dynamical degrees are all transcendental.
Abstract Consider the four-punctured sphere S 4 2 $\mathbb {S}_4^2$ double struck upper S 4 squared . Each choice of four traces, one for each puncture, determines a relative character variety for the representations of the fundamental group of S 4 2 $\mathbb {S}_4^2$ double struck upper S 4 squared in SL 2 ( C ) ${\textsf {SL}}_2(\mathbb {C})$ sans serif SL Subscript 2 Baseline left parenthesis double struck upper C right parenthesis . We classify the stationary probability measures for the action of the mapping class group Mod ( S 4 2 ) ${\textsf {Mod}}(\mathbb {S}_4^2)$ sans serif Mod left parenthesis double struck upper S 4 squared right parenthesis on these character varieties.
