In this paper we study rigidity properties of abelian \hyphenation{break-able}actions with weak or no hyperbolicty. We introduce a general strategy for proving $C^\infty$ local rigidity of algebraic actions. As a consequence, we show $C^\infty$ local rigidity for a broad class of parabolic algebraic actions on homogeneous spaces of semisimple Lie groups. This is the first time in the literature that (strong) local rigidity for these actions is addressed.
We construct a renormalization operator which acts on analytic circle maps whose critical exponent $α$ is not necessarily an odd integer $2n+1$, $n\in\mathbb N$. When $α=2n+1$, our definition generalizes cylinder renormalization of analytic critical circle maps. In the case when $α$ is close to an odd integer, we prove hyperbolicity of renormalization for maps of bounded type. We use it to prove universality and $C^{1+α}$-rigidity for such maps.
We prove that every smooth action of Z^k, k>1, on the (k+1)-dimensional torus homotopic to an action by hyperbolic linear maps preserves an absolutely continuous measure. This is a first known result concerning abelian groups of diffeomorphisms where existence of an invariant geometric structure is obtained from homotopy data.
We consider Abelian-by-cyclic groups for which the cyclic factor acts by hyperbolic automorphisms on the Abelian subgroup. We show that if such a group acts faithfully by $C^1$ diffeomorphisms of the closed interval with no global fixed point at the interior, then the action is topologically conjugated to that of an affine group. Moreover, in case of non-Abelian image, we show a rigidity result concerning the multipliers of the homotheties, despite the fact that the conjugacy is not necessarily smooth. Some consequences for non-solvable groups are proposed. In particular, we give new proofs/ex…
Every volume-preserving centre-bunched fibred partially hyperbolic system with 2-dimensional centre either (1) has two distinct centre Lyapunov exponents, or (2) exhibits an invariant continuous line field (or pair of line fields) tangent to the centre leaves, or (3) admits a continuous conformal structure on the centre leaves invariant under both the dynamics and the stable and unstable holonomies. The last two alternatives carry strong restrictions on the topology of the centre leaves: (2) can only occur on tori, and for (3) the centre leaves must be either tori or spheres. Moreover, under s…
In this paper, based on the known rigidity theorems of Newton maps ([DS], [RYZ]) and real polynomials ([KSS1], [CvS]), we prove the density of hyperbolicity in the family of real Newton maps of degree d>2 with all free critical points real.
The famous MLC Conjecture states that the Mandelbrot set is locally connected, and it is considered by many to be the central conjecture in one-dimensional complex dynamics. Among others, it implies density of hyperbolicity in the quadratic family $\{z^2+c\}_{c\in\mathbb{C}}$. We describe recent advances on MLC and the relations between MLC, the Density of Hyperbolicity Conjecture, the Rigidity Conjecture, the No Invariant Line Fields Conjecture, and the Triviality of Fibers Conjecture. We treat families of unicritical polynomials and rational maps as well as the exponential family and familie…
We show that an exponential map $f_c(z)=e^z+c$ whose singular value $c$ is combinatorially non-recurrent and non-escaping is uniquely determined by its combinatorics, i.e. the pattern in which its dynamic rays land together. We do this by constructing puzzles and parapuzzles in the exponential family. We also prove a theorem about hyperbolicity of the postsingular set in the case that the singular value is non-recurrent. Finally, we show that boundedness of the postsingular set implies combinatorial non-recurrence if $c$ is in the Julia set.
We consider four long-standing Rigidity Conjectures about synchrony and phase patterns for hyperbolic periodic orbits of admissible ODEs for networks. Proofs of stronger local versions of these conjectures, published in 2010-12, are now known to have a gap, but remain valid for a broad class of networks. Using different methods we prove local versions of the conjectures under a stronger condition, `strong hyperbolicity', which is related to a network analogue of the Kupka-Smale Theorem. Under this condition we also deduce global versions of the conjectures and an analogue of the $H/K$ Theorem …
We prove the density of the sets of the form ${λ_1^m μ_1^n ξ_1 +...+λ_k^m μ_k^n ξ_k : m,n \in \mathbb N}$ modulo one, where $λ_i$ and $μ_i$ are multiplicatively independent algebraic numbers satisfying some additional assumptions. The proof is based on analysing dynamics of higher-rank actions on compact abelean groups.
An algebraic $Z^{d}$-action is an action of $Z^{d}$ on a compact abelian group $X$ by automorphisms of $X$. We prove that for $d \ge 8$, there exist mixing zero entropy algebraic $Z^{d}$-actions which do not exhibit isomorphism rigidity property.
We investigate rigidity of measurable structure for higher rank abelian algebraic actions. In particular, we show that ergodic measures for these actions fiber over a 0 entropy measure with Haar measures along the leaves. We deduce various rigidity theorems for isomorphisms and joinings as corollaries.
We show that all $C^\infty$ Anosov $Z^r$-actions on tori and nilmanifolds without rank-one factor actions are, up to $C^\infty$ conjugacy, actions by automorphisms.
We show that the space of expanding maps contains an open and dense set where smooth conjugacy classes of expanding maps are determined by the values of the Jacobians of return maps at periodic points.
In the moduli space of polynomials of degree 3 with marked critical points c_1 and c_2, let C_{1,n} be the locus of maps for which c_1 has period n and let C_{2,m} be the locus of maps for which c_2 has period m. A consequence of Thurston's rigidity theorem is that the curves C_{1,n} and C_{2,m} intersect transversally. We give a purely algebraic proof that the intersection points are 3-adically integral and use this to prove transversality. We also prove an analogous result when c_1 or c_2 or both are taken to be preperiodic with tail length exactly 1.
Consider homogeneous G/H and G/F, for an S-algebraic group G. A lattice Γ acts on the left strictly conservatively. The following rigidity results are obtained: morphisms, factors and joinings defined apriori only in the measurable category are in fact algebraically constrained. Arguing in an elementary fashion we manage to classify all the measurable Φ commuting with the Γ-action: assuming ergodicity, we find they are algebraically defined.
We prove a geometric criterion on a $\SL$-invariant ergodic probability measure on the moduli space of holomorphic abelian differentials on Riemann surfaces for the non-uniform hyperbolicity of the Kontsevich--Zorich cocycle on the real Hodge bundle. Applications include measures supported on the $\SL$-orbits of all algebraically primitive Veech surfaces (see also \cite{Bouw:Moeller}) and of all Prym eigenforms discovered in \cite{McMullen2}, as well as all canonical absolutely continuous measures on connected components of strata of the moduli space of abelian differentials (see also \cite{Ft…
Let $(\az,F)$ be a bipermutative algebraic cellular automaton. We present conditions which force a probability measure which is invariant for the $\N\times\Z$-action of $F$ and the shift map $\s$ to be the Haar measure on $\gs$, a closed shift-invariant subgroup of the Abelian compact group $\az$. This generalizes simultaneously results of B. Host, A. Maass and S. Mart\'ınez \cite{Host-Maass-Martinez-2003} and M. Pivato \cite{Pivato-2003}. This result is applied to give conditions which also force a $(F,\s)$-invariant probability measure to be the uniform Bernoulli measure when $F$ is a partic…
In several contexts the defining invariant structures of a hyperbolic dynamical system are smooth only in systems of algebraic origin (smooth rigidity), and we prove new results of this type for a class of flows. For a compact Riemannian manifold and a uniformly quasiconformal transversely symplectic Anosov flow we define the longitudinal KAM-cocycle and use it to prove a rigidity result: The joint stable/unstable subbundle is Zygmund-regular, and higher regularity implies vanishing of the longitudinal KAM-cocycle, which in turn implies that the subbundle is Lipschitz-continuous and indeed tha…
We study rational self-maps of $\mathbb{P}^{1}$ whose critical points all have finite forward orbit. Thurston's rigidity theorem states that outside a single well-understood family, there are finitely many such maps over $\mathbb{C}$ of fixed degree and critical orbit length. We provide an algebraic proof of this fact for tamely ramified maps for which at least one of the critical points is periodic. We also produce wildly ramified counterexamples.
We consider partially hyperbolic abelian algebraic high-rank actions on compact homogeneous spaces obtained from simple indefinite orthogonal and unitary groups. In the first part of the paper, we show local differentiable rigidity for such actions. The conclusions are based on progress towards computations of the Schur multipliers of these non-split groups, which is the main aim of the second part.
Let $X$ be a compact metrizable group and $Γ$ a countable group acting on $X$ by continuous group automorphisms. We give sufficient conditions under which the dynamical system $(X,Γ)$ is surjunctive, i.e., every injective continuous map $τ\colon X \to X$ commuting with the action of $Γ$ is surjective.
Let X_1 and X_2 be mixing connected algebraic dynamical systems with the Descending Chain Condition. We show that every equivariant continuous map X_1 to X_2 is affine (that is, X_2 is topologically rigid) if and only if the system X_2 has finite topological entropy.
We study the rigidity properties of a class of algebraic Z^3-actions with entropy rank two. For this class, conditions are found which force an invariant measure to be the Haar measure on an affine subset. This is applied to show isomorphism rigidity for such actions, and to provide examples of non-isomorphic Z^3-actions with all their Z^2-sub-actions isomorphic. The proofs use lexicographic half-space entropies and total ergodicity along critical directions.
We consider the problems of measurable isomorphisms and joinings, measurable centralizers and quotients for certain classes of ergodic group actions on infinite measure spaces. Our main focus is on systems of algebraic origin: actions of lattices and other discrete subgroups on homogeneous spaces G/H where H is a sufficiently rich unimodular subgroup in a semi-simple group G. We also consider actions of discrete subgroups of isometries Isom(X) of a pinched negative curvature space X, acting on the space of horospheres Hor(X). For such systems we prove that the only measurable isomorphisms, joi…
Let $d > 1$, and let $(X,α)$ and $(Y,β)$ be two zero-entropy ${\mathbb{Z}}^d$-actions on compact abelian groups by $d$ commuting automorphisms. We show that if all lower rank subactions of $α$ and $β$ have completely positive entropy, then any measurable equivariant map from $X$ to $Y$ is an affine map. In particular, two such actions are measurably conjugate if and only if they are algebraically conjugate.
We show that for certain classes of actions of Z^d, d >= 2, by automorphisms of the torus any measurable conjugacy has to be affine, hence measurable conjugacy implies algebraic conjugacy; similarly any measurable factor is algebraic, and algebraic and affine centralizers provide invariants of measurable conjugacy. Using the algebraic machinery of dual modules and information about class numbers of algebraic number fields we consruct various examples of Z^d-actions by Bernoulli automorphisms whose measurable orbit structure is rigid, including actions which are weakly isomorphic but not isomor…
We survey some recent developments and applications of the study of the rigidity properties of natural algebraic actions of multidimensional abelian groups.
Let $Γ$ be a weakly irreducible higher rank lattice. In this paper, we will prove various rigidity results for the $Γ$-action following a philosophy of the Zimmer program. We provide new rigidity results including local and global rigidity of the $Γ$-action when $Γ$ does not have Property (T). The new ingredient is a dynamical cocycle super-rigidity theorem. It can be thought of as the generalization of Zimmer's cocycle super-rigidity theorem since it provides almost the same consequences dynamically and algebraically. This allows us to derive various our rigidity results using dynamical super…
We prove the locally differentiable rigidity of generic partially hyperbolic abelian algebraic high-rank actions on compact homogeneous spaces obtained from split symplectic Lie groups. We also gave a non-generic action rigidity example on compact homogeneous spaces obtained from SL(2n,R) or SL(2n,C). The conclusions are based on geometric Katok-Damjanovic way and progress towards computations of the generating relations in these groups.
We prove an operator algebraic superrigidity statement for homomorphisms of irreducible lattices, and also their commensurators, in certain higher-rank groups into unitary groups of finite factors. This extends the authors' previous work regarding non-free measure-preserving actions, and also answers a question of Connes for such groups.
