Abstract For a boundary-preserving partially hyperbolic diffeomorphism with interval central leaves, we completely characterize the C k $C^k$ upper C Superscript k -robust transitivity ( k ≥ 2 ) $(k\geq 2)$ left parenthesis k greater than or equals 2 right parenthesis by boundary interconnection. As an application, if the boundary Sinai–Ruelle–Bowen (SRB) measures admit negative central Lyapunov exponents, then boundary interconnection also completely characterizes the phenomenon of robustly intermingled basins for boundary SRB measures.
Abstract In this article, we contribute to the study of the dynamics by conjugation on the space of subgroups of Baumslag–Solitar groups BS ( m , n ) $\mathrm {BS}(m,n)$ upper B upper S left parenthesis m comma n right parenthesis via the mixing properties of elements asymptotically produced by suitable random walks on the group. In an acylindrically hyperbolic context, Hull, Minasyan and Osin [Subgroup mixing and random walks in groups acting on hyperbolic spaces. Ergod. Th. & Dynam. Sys. 46 (10) (2026), 2505–2541] demonstrated strong mixing situations, namely topological μ $\mu $ mu -mixing, a strengthening of high topological transitivity. Regarding non-metabelian BS ( m , n ) $\mathrm {BS}(m,n)$ upper B upper S left parenthesis m comma n right parenthesis with | m | ≠ | n | $|m|\neq |n|$ StartAbsoluteValue m EndAbsoluteValue not equals StartAbsoluteValue n EndAbsoluteValue , we exhibit here a radically different situation on each of the pieces except one of the partition introduced by Carderi, Gaboriau, Le Maître and Stalder [On the space of subgroups of Baumslag–Solitar groups I: perfect kernel and phenotype. Rev. Mat. Iberoam. 41 (5) (2025), 1711–1758] (although it is highly topologically transitive on each piece). However, when | m | = | n | $|m|= |n|$ StartAbsoluteValue m EndAbsoluteValue equals StartAbsoluteValue n EndAbsoluteValue , we demonstrate the topological μ $\mu $ mu -mixing character of the action on each of the pieces.
Abstract This work is motivated by the study of continued fraction expansions of real numbers: we describe in dynamical terms their orbits under the multiplicative action by Z $\mathbb {Z}$ double struck upper Z and the projective action of PGL 2 ( Z ) $\operatorname {\mathrm {PGL}}_2(\mathbb {Z})$ upper P upper G upper L 2 left parenthesis double struck upper Z right parenthesis ; hence, of PGL 2 ( Q ) $\operatorname {\mathrm {PGL}}_2(\mathbb {Q})$ upper P upper G upper L 2 left parenthesis double struck upper Q right parenthesis . A real number gives rise to a Sturmian system encoding a rotation of the circle, which is a particular case of an interval exchange system. It is well known that PGL 2 ( Z ) $\operatorname {\mathrm {PGL}}_2(\mathbb {Z})$ upper P upper G upper L 2 left parenthesis double struck upper Z right parenthesis -equivalence of real numbers, characterized by the tails of their continued fraction expansions, amounts to flow equivalence of Sturmian systems. We show that the multiplicative action of m ∈ Z $m\in \mathbb {Z}$ m element of double struck upper Z on a real number corresponds to taking the m th-power followed by what we call an infinitesimal 2-asymptotic factor of its Sturmian system. This leads us to introduce the notion of isogeny between zero-dimensional systems: it combines flow equivalences, virtual conjugacies and infinitesimal asymptotic equivalences. We develop tools for classifying systems up to isogeny involving cohomological invariants and states. We then use this to give a complete description of PGL 2 ( Q ) $\operatorname {\mathrm {PGL}}_2(\mathbb {Q})$ upper P upper G upper L 2 left parenthesis double struck upper Q right parenthesis -equivalence of real numbers in terms of Sturmian systems. Considering powers of Sturmian systems leads naturally to the realm of Denjoy systems, and we classify Denjoy systems up to isogenies preserving this class via the action of PGL 2 ( Q ) $\operatorname {\mathrm {PGL}}_{2}(\mathbb {Q})$ upper P upper G upper L 2 left parenthesis double struck upper Q right parenthesis on their invariants. We also investigate eventual flow equivalence and show that it implies topological conjugacy for Sturmian and Denjoy systems. For this, we prove the following novel arithmetic result that we believe is of independent interest. Consider α , β ∈ R ∖ Q $\alpha ,\beta \in \mathbb {R}\setminus \mathbb {Q}$ alpha comma beta element of double struck upper R minus double struck upper Q ; if for all but finitely many prime integers n ∈ P $n\in \mathbb {P}$ n element of double struck upper P the numbers n α , n β $n\alpha ,n\beta $ n alpha comma n beta are PGL 2 ( Z ) $\operatorname {\mathrm {PGL}}_2(\mathbb {Z})$ upper P upper G upper L 2 left parenthesis double struck upper Z right parenthesis -equivalent, then α ≡ ± β mod Z $\alpha \equiv \pm \beta \bmod {\mathbb {Z}}$ alpha identical to plus or minus beta mod double struck upper Z . In another direction, we consider interval exchanges satisfying Keane’s condition. We characterize flow equivalence in terms of interval-induced subsystems (or the tails of their paths in the bilateral Rauzy induction diagram). We describe their 2-AI factors in terms of minimal models. Finally, we find rational invariants for isogeny involving the length modules and SAF invariants of the associated ergodic measures. This leads to a conjecture for their classification up to isogeny, which we prove in the totally ergodic case.
Abstract It is known that for a uniform morphic sequence u = ⟨ u n ⟩ n = 0 ∞ $\boldsymbol u = \langle u_n\rangle _{n=0}^\infty $ bold italic u equals left angle bracket u Subscript n Baseline right angle bracket Subscript n equals 0 Superscript infinity and an algebraic number β $\beta $ beta such that | β | > 1 $|\beta |>1$ StartAbsoluteValue beta EndAbsoluteValue greater than 1 , the number [ [ u ] ] β := ∑ n = 0 ∞ ( u n / β n ) $[\![ \boldsymbol {u} ]\!] _\beta :=\sum _{n=0}^\infty ({u_n}/{\beta ^n})$ left bracket left bracket bold italic u right bracket right bracket Subscript beta Baseline colon equals sigma summation Underscript n equals 0 Overscript infinity Endscripts left parenthesis u Subscript n Baseline divided by beta Superscript n Baseline right parenthesis either lies in Q ( β ) $\mathbb Q(\beta )$ double struck upper Q left parenthesis beta right parenthesis or is transcendental. In this paper, we show a similar rational–transcendental dichotomy for sequences defined by irreducible Pisot morphisms on binary alphabets. Subject to the Pisot conjecture (an irreducible Pisot morphism has pure discrete spectrum), we generalise the latter result to arbitrary finite alphabets. In certain cases, we are able to show transcendence of [ [ u ] ] β $[\![ \boldsymbol {u}]\!] _{\beta }$ left bracket left bracket bold italic u right bracket right bracket Subscript beta outright. In particular, for k ≥ 2 $k\geq 2$ k greater than or equals 2 , if u $\boldsymbol u$ bold italic u is the k -Bonacci word, then [ [ u ] ] β $[\![ \boldsymbol {u}]\!] _{\beta }$ left bracket left bracket bold italic u right bracket right bracket Subscript beta is transcendental.
Abstract Let Mod ( S ) $\operatorname {\mathrm {Mod}}(S)$ upper M o d left parenthesis upper S right parenthesis be the mapping class group of a closed orientable surface S of genus g ≥ 2 $g \geq 2$ g greater than or equals 2 . Let G be a non-elementary subgroup of Mod ( S ) $\operatorname {\mathrm {Mod}}(S)$ upper M o d left parenthesis upper S right parenthesis such that the associated Bowen–Margulis measure is finite. In this paper, we give an asymptotic growth formula for G with respect to the Teichmüller metric.
