Let $g$ be a polynomial automorphism of $\C^2$. We study the Hausdorff dimension and topological dimension of the Julia set of $g$. We show that when $g$ is a hyperbolic mapping, then the Hausdorff dimension of the Julia set is strictly greater than its topological dimension. Moreover, the Julia set cannot be locally connected. We also provide estimates for the dimension of the Julia sets in the general (not necessarily hyperbolic) case.
Given a finitely generated semigroup S of the (normed) set of linear maps of a vector space V into itself, we find sufficient conditions for the exponential growth of the number N(k) of elements of the semigroup contained in the sphere of radius k as k->infinity. We relate the growth rate lim log N(k)/log k to the exponent of a zeta function naturally defined on the semigroup and, in case S is a semigroup of volume-preserving automorpisms, to the Hausdorff and box dimensions of the limit set of the induced semigroup of automorphisms on the corresponding projective space.
We show in prime dimension that for two non-commuting totally irreducible toral automorphisms the set of points that equidistribute under the first map but have non-dense orbit under the second has full Hausdorff dimension. In non-prime dimension the argument fails only if the automorphisms have strong algebraic relations.
We systematically investigate examples of non-hyperbolic dynamical systems having irregular sets of full topological entropy and full Hausdorff dimension. The examples include some partially hyperbolic systems and geometric Lorenz flows. We also pose interesting questions for future research.
Suppose $X_{1}, X_{2}$ are nilmanifolds and $ρ, σ$ are automorphism actions of a discrete group $Γ$ on $X_{1}$ and $X_{2}$ respectively. We show that if $(X_{1},ρ)$ and $(X_{2}, σ)$ satisfy certain additional conditions then every $Γ$-equivariant continuous map from $(X_{1},ρ)$ to $(X_{2},σ)$ is an affine map.
Let ϕ be an automorphism on a connected Lie group G. Through several G-subgroups associated to the dynamics of ϕ we analyze their topological entropy. Assume that G belongs to the class of finite semisimple center Lie groups which admits a ϕ invariant Levi subgroup. Then we prove that the topological entropy information of ϕ is contained in the toral component of the unstable subgroup of ϕ in the radical of G. We specialize the main result in a couple of interesting situations.
We construct suitable metrics for two classes of topological dynamical systems (linear maps on the torus and non-invertible expansive maps on compact spaces) in order to get a lower bound for topological entropy in terms of the resulting Hausdorff dimensions and Lipschitz constants. This reverses an old inequality of Dai, Zhou, and Gheng and leads to a short proof of a well-known theorem on expansive mappings. It also suggests a new invariant of topological conjugacy for dynamical systems on compact metric spaces.
In this paper we study the dynamics of regular polynomial automorphisms of C^n. These maps provide a natural generalization of complex Henon maps in C^2 to higher dimensions. For a given regular polynomial automorphism f we construct a filtration in C^n which has particular escape properties for the orbits of f. In the case when f is hyperbolic we obtain a complete description of its orbits. In the second part of the paper we study the Hausdorff and box dimension of the Julia sets of f. We show that the Julia set J has positive box dimension, and (provided f is not volume preserving) that the …
We extend the definition of topological pressure to locally compact Hausdorff spaces, and we demonstrate a "variational principle" comparing the topological and measure theoretic pressures. Given a continuous $\mathbb{Z}_+^N$-action $T$ over a locally compact Hausdorff space $X$ and a continuous function vanishing at infinity $f \in C_0(X)$, we define topological pressure $P(T,f)$ using open covers of a special type we call "admissible covers". With this topological pressure, we demonstrate that \begin{equation*} P(T,f) = \sup_μ P_μ(T,f), \end{equation*} where the supremum is taken over all $T…
We show that set of points nondense under the $\times n$-map on the circle and dense for the geodesic flow under the induced map on the circle corresponding to the expanding horospherical subgroup has full Haudorff dimension. We also show the analogous result for toral automorphisms on the $2$-torus and a diagonal flow. Our results can be interpreted in number-theoretic terms: the set of well approximable numbers that are nondense under the $\times n$-map has full Hausdorff dimension. Similarly, the set of well approximable $2$-vectors that are nondense under a hyperbolic toral automorphism ha…
We show that if $r\geq 3$ and $α$ is a faithful $Z^r$-Cartan action on a torus $T^d$ by automorphisms, then any closed subset of $(T^d)^2$ which is invariant and topologically transitive under the diagonal $\bZ^r$-action by $α$ is homogeneous, in the sense that it is either the full torus $(T^d)^2$, or a finite set of rational points, or a finite disjoint union of parallel translates of some d-dimensional invariant subtorus. A counterexample is constructed for the rank 2 case.
We provide a sufficient condition for a topological partial action of a Hausdorff group on a metric space is continuous, provide that it is separately continuous.
We combine the pointed Gromov-Hausdorff metric [Ron10] with the locally $C^0$ distance to obtain the pointed $C^0$-Gromov-Hausdorff distance between maps of possibly different non-compact pointed metric spaces. The latter is then combined with Walters's locally topological stability [LNY18] and $GH$-stability from [AMR17] to obtain the notion of topologically $GH$-stable pointed homeomorphism. We give one example to show the difference between the distance when take different base point in a pointed metric space.
In this article we continue the study of automorphism groups of constant length substitution shifts and also their topological factors. We show that up to conjugacy, all roots of the identity map are letter exchanging maps, and all other nontrivial automorphisms arise from {\em twisted} compressions of another constant length substitution. We characterise the group of roots of the identity in both the measurable and topological setting. Finally, we show that any topological factor of a constant length substitution shift is topologically conjugate to a constant length substitution shift via a l…
We consider a certain two-parameter family of automorphisms of the affine plane over a complete, locally compact non-Archimedean field. Each of these automorphisms admits a chaotic attractor on which it is topologically conjugate to a full two-sided shift map, and the attractor supports a unit Borel measure which describes the distribution of the forward orbit of Haar-almost all points in the basin of attraction. We also compute the Hausdorff dimension of the attractor, which is non-integral.
Let c be a real parameter in the Mandelbrot set, and f_c(z):= z^2 + c. We prove a formula relating the topological entropy of f_c to the Hausdorff dimension of the set of rays landing on the real Julia set, and to the Hausdorff dimension of the set of rays landing on the real section of the Mandelbrot set, to the right of the given parameter c. We then generalize the result by looking at the entropy of Hubbard trees: namely, we relate the Hausdorff dimension of the set of external angles which land on a certain slice of a principal vein in the Mandelbrot set to the topological entropy of the q…
The role of generalized Bowen-Franks groups (BF-groups) as topological conjugacy invariants for $\mathbb{T}^{n}$ automorphisms is studied. Using algebraic number theory, a link is established between equality of BF-groups for different automorphisms ($BF$-equivalence) and an identical position in a finite lattice ($\mathcal{L}$-equivalence). Important cases of equivalence of the two conditions are proved. Finally, a topological interpretation of the classical BF-group $\mathbb{Z}% ^{n}/\mathbb{Z}^{n}(I-A)$ in this context is presented.
We prove that uniform hyperbolicity is invariant under topological conjugacy for dissipative polynomial automorphisms of C^2. Along the way we also show that a sufficient condition for hyperbolicity is that local stable and unstable manifolds of saddle points have uniform geometry.
We study in this paper some connections between the Fraisse theory of amalgamation classes and ultrahomogeneous structures, Ramsey theory, and topological dynamics of automorphism groups of countable structures.
We show that for every linear toral automorphism, especially the non-hyperbolic ones, the entropies of ergodic measures form a dense set on the interval from zero to the topological entropy.
The cookie-cutter-like set is defined as the limit set of a sequence of classical cookie-cutter mappings. For this cookie-cutter set it is shown that the topological pressure function exists, and that the fractal dimensions such as the Hausdorff dimension, the packing dimension and the box-counting dimension are all equal to the unique zero $h$ of the pressure function. Moreover, it is shown that the $h$-dimensional Hausdorff measure and the $h$-dimensional packing measure are finite and positive.
Let $Aut(X,\mathcal{B})$ be the group of all Borel automorphisms of a standard Borel space $(X,\mathcal{B})$. We study topological properties of $Aut(X,\mathcal{B})$ with respect to the uniform and weak topologies, $τ$ and $p$, defined in [Bezuglyi S., Dooley A.H., Kwiatkowski J., Topologies on the group of Borel automorphisms of a standard Borel space, Preprint, 2003]. It is proved that the class of smooth automorphisms is dense in $(Aut(X,\mathcal B),p)$. Let $Ctbl(X)$ denote the group of Borel automorphisms with countable support. It is shown that the topological group $Aut_0(X,\mathcal B)=…
A theoretical approach to computing the Hausdorff dimension of the topological boundary of attractors of iterated function systems is developed. The curve known as the Lévy Dragon is then studied in detail and the Hausdorff dimension of its boundary is computed using the theory developed. The actual computation is a complicated procedure. It involves a great deal of combinatorial topology as well as determining the structure and certain eigenvalues of a $752 \times 752$ matrix. Perron-Frobenius theory plays an important role in analyzing this matrix.
We study topological properties of automorphisms of 4-dimensional torus generated by integer symplectic matrices. The main classifying element is the structure of the topology of a foliation generated by unstable leaves of the automorphism. There two different case, transitive and decomposable ones. For both case the classification is given.
Given a non-conformal repeller $Λ$ of a $C^{1+γ}$ map, we study the Hausdorff dimension of the repeller and continuity of the sub-additive topological pressure for the sub-additive singular valued potentials. Such a potential always possesses an equilibrium state. We then use a substantially modified version of Katok's approximating argument, to construct a compact invariant set on which the corresponding dynamical quantities (such as Lyapunov exponents and metric entropy) are close to that of the equilibrium measure. This allows us to establish continuity of the sub-additive topological press…
This sequel to our previous paper [MS11b] continues the study of topological contact dynamics and applications to contact dynamics and topological dynamics. We provide further evidence that the topological automorphism groups of a contact structure and a contact form are the appropriate transformation groups of contact dynamical systems. The article includes an examination of the groups of time-one maps of topological contact and strictly contact isotopies, and the construction of a bi-invariant metric on the latter. Moreover, every topological contact or strictly contact dynamical system is a…
The Hausdorff dimension of a product XxY can be strictly greater than that of Y, even when the Hausdorff dimension of X is zero. But when X is countable, the Hausdorff dimensions of Y and XxY are the same. Diagonalizations of covers define a natural hierarchy of properties which are weaker than ``being countable'' and stronger than ``having Hausdorff dimension zero''. Fremlin asked whether it is enough for X to have the strongest property in this hierarchy (namely, being a gamma-set) in order to assure that the Hausdorff dimensions of Y and XxY are the same. We give a negative answer: Assuming…
We show that the existence of a dense set of periodic points for a topologically transitive non-minimal continuous group action on a Hausdorff uniform space with an infinite acting group does not necessarily imply a sensitive dependence to the initial conditions in such a system. This leads to define the chaos in the sense of Devaney for a continuous group action on a Hausdorff uniform spaces with an infinite acting group in the original way, i.e. a non-minimal topologically transitive and sensitive system with a dense set of periodic points is a chaotic system in the sense of Devaney.
This paper considers self-conformal iterated function systems (IFSs) on the real line whose first level cylinders overlap. In the space of self-conformal IFSs, we show that generically (in topological sense) if the attractor of such a system has Hausdorff dimension less than $1$ then it has zero appropriate dimensional Hausdorff measure and its Assouad dimension is equal to $1$. Our main contribution is in showing that if the cylinders intersect then the IFS generically does not satisfy the weak separation property and hence, we may apply a recent result of Angelevska, Käenmäki and Troscheit […
We introduce a notion of topological entropy for continuous actions of compactly generated topological groups on compact Hausdorff spaces. It is shown that any continuous action of a compactly generated topological group on a compact Hausdorff space with vanishing topological entropy is amenable. Given an arbitrary compactly generated locally compact Hausdorff topological group $G$, we consider the canonical action of $G$ on the closed unit ball of $L^{1}(G)' \cong L^{\infty}(G)$ endowed with the corresponding weak-$^{\ast}$ topology. We prove that this action has vanishing topological entropy…
A complex compact surface which carries an automorphism of positive topological entropy has been proved by Cantat to be either a torus, a K3 surface, an Enriques surface or a rational surface. Automorphisms of rational surfaces are quite mysterious and have been recently the object of intensive studies. In this paper, we construct several new examples of automorphisms of rational surfaces with positive topological entropy. We also explain how to define and to count parameters in families of birational maps of the complex projective plane and in families of rational surfaces.
We provide new similarities between regular continued fractions and Lüroth series in terms of topological dynamics and Hausdorff dimension. In particular, we establish a complete analogue for the Lüroth transformation of results by W. Liu, B. Li and W. Liu, S. Wang on the distal, asymptotic and Li-Yorke pairs for the Gauss map.
We use the results of Neshveyev and Stormer to show that for a generic shift on a C*-algebra associated to a bitstream the Voiculescu topological entropy is strictly larger that the supremum of topological entropies of its classical subsystems.
Let $M$ be a simply connected closed $4$-manifold. It is proved that any (possibly finite) compact Lie group acting effectively and homologically trivially on $M$ by homeomorphisms is an abelian group of rank at most two. As applications, let $\mathrm{Aut}(F_{n})$ be the automorphism group of the free group of rank $n.$ We prove that any group action of $\mathrm{Aut}% (F_{n})$ $(n\geq 4)$ on $M\neq S^{4}$ by homologically trivial homeomorphisms factors through $\mathbb{Z}/2.$ Moreover, any action of $% \mathrm{SL}_{n}(\mathbb{Q})$ $(n\geq 4)$ on $M\neq S^{4}$ by homeomorphisms is trivial.
Let f be a holomorphic automorphism of a compact Kahler manifold (X,ω) of dimension k>1. We study the convex cones of positive closed (p,p)-currents T_p, which satisfy a functional relation $f^*(T_p)=λT_p, λ>1,$ and some regularity condition (PB). Under appropriate assumptions on dynamical degrees we introduce closed finite dimensional cones, not reduced to zero, of such currents. In particular, when the topological entropy h(f) of f is positive, then for some m, there is a positive closed (m,m)-current T_m which satisfies the relation $f^* T_m=\exp(h(f)) T_m$. Moreover,every quasi-p.s.h. func…
We introduce a novel quantity for general dynamical systems, which we call the asymptotic uniform complexity. We prove an inequality relating the asymptotic uniform complexity of a dynamical system to its mean topological matching number. Furthermore, we show that the established inequality yields an exact equation for perfect Hausdorff dynamical systems. Utilizing these results, we conclude that a dynamical system is amenable if its asymptotic uniform complexity equals one, and that the converse is true for perfect Hausdorff dynamical systems, such as non-discrete Hausdorff topological groups…
