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.
Abstract An account of the earliest known works on outer billiards is accompanied by a reproduction of the summary of B. H. Neumann’s 1959 colloquium and some highlights from the theory.
We show that, under certain conditions, a strongly continuous semigroup admits an almost surely frequently hypercyclic random vector defined as a stochastic integral in Fréchet spaces with respect to the Brownian motion. Two criteria are given. We will apply the second criterion to three examples: translation semigroups on spaces of integrable functions, the exponential of weighted shifts, and the translation operators on the space of entire functions. This last example, with a stochastic approach, seems to be new in the literature. Some other examples are given.
We show that the canonical anticommutation relations (CAR) algebra admits a Cantor spectrum C & lowast; $\mathrm {C}<^>\ast $ normal upper C Superscript asterisk -diagonal that is not conjugate to the standard AF diagonal. We obtain this by classification theory of C & lowast; $\mathrm {C}<^>\ast $ normal upper C Superscript asterisk -algebras, and the diagonal arises by realizing the CAR algebra as the crossed product of a free minimal action on the Cantor space, where the acting group is the product of a locally finite group with the infinite dihedral group. The main ingredient in the construction is a binary subshift associated to the well-known regular paper-folding sequence. Moreover, we show that the CAR algebra in fact admits countably many, pairwise non-conjugate, Cantor spectrum diagonals which are distinguished by the different values of their diagonal dimension, as defined by Li, Liao and the second named author.
We show that the continuity property of Lyapunov exponents proved by Buzzi et al. [Continuity properties of Lyapunov exponents for surface diffeomorphisms. Invent. Math. 230(2) (2022), 767-849] for smooth surface diffeomorphisms extends to smooth interval maps in the case when the map only has non-flat critical points and the entropies converging to the topological entropy. The result we obtained is stronger than the continuity of Lyapunov exponents, in particular, we prove the uniform integrability of Lyapunov exponents over entropies.
We study the the asymptotic dynamics of elementary cellular automaton 18 through its limit set, generic limit set and $\mu $ -limit set. The dynamics of rule 18 are characterized by persistent local patterns known as kinks. We characterize the configurations of the generic limit set containing at most two kinks. As a corollary, we show that the three limit sets of rule 18 are distinct.
We introduce Feldman-Katok convergence for invariant measures of a topological dynamical system. This can be seen as a counterpart to the convergence with respect to the $\bar {f}$ -metric for finite-state stationary processes (shift-invariant measures on a symbolic space). Feldman-Katok convergence is based on a dynamically defined Feldman-Katok pseudometric. This convergence is stronger than weak $<^>*$ convergence. We prove that Feldman-Katok convergence preserves ergodicity and makes the Kolmogorov-Sinai entropy lower semicontinuous, thereby preserving zero entropy. We apply our findings to non-hyperbolic (having at least one vanishing Lyapunov exponent) ergodic measures constructed using the GIKN method as axiomatized by Bonatti, D & iacute;az and Gorodetski [Nonlinearity, 23 (2010), 687-705]. The GIKN method, originally introduced by Gorodetski, Ilyashenko, Kleptsyn and Nalsky [Functional Analysis and its Applications, 39 (2005), 21-30], has been widely adapted to produce non-hyperbolic ergodic measures for diffeomorphisms of compact manifolds. We prove that an ergodic measure satisfying the conditions provided by the axiomatized GIKN method is the Feldman-Katok limit of a sequence of periodic measures, which implies that it is either a periodic measure or a loosely Kronecker measure (a measure Kakutani equivalent to an aperiodic ergodic rotation on a compact group) and has zero entropy. This classifies all these measures up to Kakutani equivalence and confirms that geometric constructions of non-hyperbolic measures via periodic approximations based on the axiomatized GIKN method presented in Bonatti et al. [op. cit.] systematically produce zero-entropy systems.
Given a Fuchsian group $\Gamma $ , we introduce escape rate spectra associated with $\Gamma $ to investigate the transient behavior of the geodesic flow on the hyperbolic surface $\mathbb {D}/\Gamma $ . Our definition is motivated by Bishop's linear escape set. In the special case when $\mathbb {D}/\Gamma $ is a hyperbolic $\mathbb {Z}$ -covering of a surface uniformized by a generalized Schottky group, we completely determine the escape rate spectra by the convex conjugate of a generalized Poincar & eacute; exponent.
A power-bounded operator T satisfying supn(n )n|| T-n-Tn+1|| < infinity is a Ritt operator. For such operators, we study the generalized square function Q(T) alpha,s,r F = (Sigma(n) n(alpha)|T-n(I-T )(r )f|(s))(1/s).It is known that when T is a positive contraction and a Ritt operator on L-p, 1 < p < infinity, then for any integer r >= 1, the square function Q(2r-1,2,r)(T)f defines a bounded operator [17] on L-p. In this work, we extend the theory to the endpoint case p = 1. We show that if T is a Ritt operator on L-1, then the generalized square function Q(alpha,s,r )(T)f is bounded on L-1 whenever alpha + 1 < sr. In the particular setting where T is a convolution operator of the form T-mu = Sigma(k) mu(k)U(k)f, with mu a probability measure on Z and U the composition operator induced by an invertible measure-preserving transformation, we provide sufficient conditions on mu under which Q(T mu) (2r-1,2,r) f is of weak type (1, 1), for r > 0. We also establish bounds for variational and oscillation norms, || n(beta)T(n)(1-T )(R)|| (v(s)) and || n(beta)T(n)(1-T )(R) ||( o(s)), for Ritt operators, highlighting endpoint behavior.
We characterize dynamics of every distortion element in the group of diffeomorphisms of the 2-sphere that has at least two fixed points and another recurrent point. The key result is that if f is such a diffeomorphism, then the homeomorphism $\check {f}_{\mathrm { ann}}$ , which is a lift of the homeomorphism of the closed annulus $\overline {\mathcal {A}}$ obtained from $\mathbb {S}<^>2$ by blowing up two fixed points of f to the universal covering space of $\overline {\mathcal {A}}$ , has a unique rotation number. Moreover, we find the differential of such a distortion element in the group of diffeomorphisms of the 2-sphere at each fixed point up to conjugacy.
We show that, under certain conditions, a strongly continuous semigroup admits an almost surely frequently hypercyclic random vector defined as a stochastic integral in Fr & eacute;chet spaces with respect to the Brownian motion. Two criteria are given. We will apply the second criterion to three examples: translation semigroups on spaces of integrable functions, the exponential of weighted shifts, and the translation operators on the space of entire functions. This last example, with a stochastic approach, seems to be new in the literature. Some other examples are given.
Let X=G/Gamma be the quotient of a semisimple Lie group G by its non-cocompact arithmetic lattice. Let H be a reductive algebraic subgroup of G acting on X. We give several equivalent algebraic conditions on H for the existence of a fixed compact set in X intersecting every H-orbit. This generalizes previous results concerning certain special reductive group action on X in this setting. When G is of real rank one, Gamma is a non-cocompact lattice of G, and H
It is well known that the dynamical behavior of a rational map $f:\widehat {\mathbb C}\to \widehat {\mathbb C}$ is governed by the forward orbits of the critical points of f. The map f is said to be postcritically finite if every critical point has finite forward orbit, or equivalently, if every critical point eventually maps into a periodic cycle of f. We encode the orbits of the critical points of f with a finite directed graph called a ramification portrait. In this article, we study which graphs arise as ramification portraits. We prove that every abstract polynomial ramification portrait is realized as the ramification portrait of a postcritically finite polynomial and classify which abstract polynomial ramification portraits can only be realized by unobstructed maps.
An avoshift is a subshift where for each set C from a suitable family of subsets of the shift group, the set of all possible valid extensions of a globally valid pattern on C to the identity element is determined by a bounded subpattern. This property is shared (for various families of sets C) by, for example, cellwise quasigroup shifts, totally extremally permutive (TEP) subshifts, and subshifts of finite type (SFTs) with a safe symbol. In this paper, we concentrate on avoshifts on polycyclic groups, when the sets C are what we call 'inductive intervals'. We show that then, avoshifts are a recursively enumerable subset of subshifts of finite type. Furthermore, we can effectively compute lower-dimensional projective subdynamics and certain factors (avofactors), and we can decide equality and inclusion for subshifts in this class. These results were previously known for group shifts, but our class also covers many non-algebraic examples as well as many SFTs without dense periodic points. The theory also yields new proofs of decidability of inclusion for SFTs on free groups, and SFTness of subshifts with the topological strong spatial mixing property.
Polynomial Julia sets with tree structure, typically Hubbard trees, play an important role in holomorphic dynamics. In this paper, we study the dynamics of ${f_\alpha (z)=z<^>2+\alpha \bar {z}}$ for $\alpha $ being real and the Julia sets being trees. We show that all such $\alpha $ form an interval $[1,4]$ . This answers a question of G. Sienra. We further show that $f_\alpha $ exhibits non-trivial dynamics on the Fatou set which equals to the escaping set.
In this article, we prove that the set of well-approximable points W-phi(z)={x is an element of X: d(f (n )x, z) < phi(n)for infinite n is an element of N+}in the shrinking targets problems is distributional chaotic of type 1 for systems with a weak form of the exponential specification property. We apply it to transitive Anosov systems,beta-shifts, etc.
We study the ergodic properties of the translation surface $X_{\unicode{x3bb} ,\mu }$ formed by gluing two flat tori along a slit with holonomy $(\unicode{x3bb} ,\mu ) \in \mathbb {R}<^>2$ . Extending the dichotomy result of Cheung, Hubert, and Masur for the case $\mu = 0$ , we prove the following: for slits not parallel to any absolute homology class, the Hausdorff dimension of the set $\operatorname {\mathrm {NE}}(X_{\unicode{x3bb} ,\mu },\omega )$ of non-ergodic directions is either $0$ or $\frac 12$ . This dichotomy is completely characterized by the P & eacute;rez-Marco condition expressed in terms of best approximation denominators. As a corollary, we obtain that the P & eacute;rez-Marco condition for best approximation denominators is norm-independent.
An avoshift is a subshift where for each set C from a suitable family of subsets of the shift group, the set of all possible valid extensions of a globally valid pattern on C to the identity element is determined by a bounded subpattern. This property is shared (for various families of sets C) by for example cellwise quasigroup shifts, TEP subshifts, and subshifts of finite type with a safe symbol. In this paper we concentrate on avoshifts on polycyclic groups, when the sets C are what we call “inductive intervals”. We show that then avoshifts are a recursively enumerable subset of subshifts of finite type. Furthermore, we can effectively compute lower-dimensional projective subdynamics and certain factors (avofactors), and we can decide equality and inclusion for subshifts in this class. These results were previously known for group shifts, but our class also covers many non-algebraic examples as well as many SFTs without dense periodic points. The theory also yields new proofs of decidability of inclusion for SFTs on free groups, and SFTness of subshifts with the topological strong spatial mixing property.
