Diffeomorphisms with an anomalous growth of the differential

Slow area-preserving diffeomorphisms of the torus

We construct area-preserving real analytic diffeomorphisms of the torus with unbounded growth sequences of arbitrarily slow growth.

A stretched exponential bound on the rate of growth of the number of periodic points for prevalent diffeomorphisms

Let M be a compact manifold of dimension at least 2, Diff^r(M) be the space of C^r diffeomorphisms of M. Define for any diffeomorphism f in Diff^r(M) number of isolated periodic points of period n by P_n(f)=# {isolated x in M: f^n(x)=x}. Artin--Mazur proved that for a dense set of diffeomorphisms the number of periodic points P_n(f) growth at most exponentially fast in n. The author proved that there is an open set N \subset Diff^r(M) such that for a Baire generic set of diffeomorphisms the number of periodic points P_n(f) growth arbitrarily fast. Arnold posed a problem: Prove that diffeomorph…

Abundance of fast growth of the number of periodic points in 2-dimensional area-preserving dynamics

We prove that there exists an open subset of the set of real-analytic Hamiltonian diffeomorphisms of a closed surface in which diffeomorphisms exhibiting fast growth of the number of periodic points are dense. We also prove that there exists an open subset of the set of smooth area-preserving diffeomorphisms of a closed surface in which typical diffeomorphisms exhibit fast growth of the number of periodic points.

Anomalous partially hyperbolic diffeomorphisms II: stably ergodic examples

We construct examples of robustly transitive and stably ergodic partially hyperbolic diffeomorphisms $f$ on compact $3$-manifolds with fundamental groups of exponential growth such that $f^n$ is not homotopic to identity for all $n>0$. These provide counterexamples to a classification conjecture of Pujals.

On the continuity of topological entropy of certain partially hyperbolic diffeomorphisms

In this paper, we consider certain partially hyperbolic diffeomorphisms with center of arbitrary dimension and obtain continuity properties of the topological entropy under $C^1$ perturbations. The systems considered have subexponential growth in the center direction and uniform exponential growth along the unstable foliation. Our result applies to partially hyperbolic diffeomorphisms which are Lyapunov stable in the center direction. It applies to another important class of systems which do have subexponential growth in the center direction, for which we develop a technique to use exponential…

Anomalous partially hyperbolic diffeomorphisms III: abundance and incoherence

Let $M$ be a closed 3-manifold which admits an Anosov flow. In this paper we develop a technique for constructing partially hyperbolic representatives in many mapping classes of $M$. We apply this technique both in the setting of geodesic flows on closed hyperbolic surfaces and for Anosov flows which admit transverse tori. We emphasize the similarity of both constructions through the concept of $h$-transversality, a tool which allows us to compose different mapping classes while retaining partial hyperbolicity. In the case of the geodesic flow of a closed hyperbolic surface $S$ we build stably…

Growth of groups and diffeomorphisms of the interval

We prove that the so called Grigorchuk-Maki group of intermadiate growth can be seen as a group of $C^1$ diffeomorphisms of the interval. On the other hand, we prove that every group of $C^{1+α}$ diffeomorphisms of the interval having subexponential growth is almost nilpotent.

Generic Diffeomorphisms with Superexponential Growth of Number of Periodic Orbits

Consider a compact manifold M of dimension at least 2 and the space of C^r-smooth diffeomorphisms Diff^r(M). The classical Artin-Mazur theorem says that for a dense subset D of Diff^r(M) the number of isolated periodic points grows at most exponentially fast (call it the A-M property). We extend this result and prove that diffeomorphisms having only hyperbolic periodic points with the A-M property are dense in Diff^r(M). Our proof of this result is much simpler than the original proof of Artin-Mazur. The second main result is that the A-M property is not (Baire) generic. Moreover, in a Newhous…

Distortion growth for iterations of diffeomorphisms of the interval

We obtain several results on the distortion asymptotics for the iterations of diffeomorphisms of the interval extending the recent work of Polterovich and Sodin.

Growth Gap vs. smoothness for diffeomorphisms of the interval

Given a diffeomorphism of the interval, consider the uniform norm of the derivative of its n-th iteration. We get a sequence of real numbers called the growth sequence. Its asymptotic behavior is an invariant which naturally appears both in smooth dynamics and in geometry of the diffeomorphisms groups. We find sharp estimates for the growth sequence of a given diffeomorphism in terms of the modulus of continuity of its derivative. These estimates extend previous results of Polterovich and Sodin, and Borichev.

Fast growth of the number of periodic points arising from heterodimensional connections

We consider C^r-diffeomorphisms of a compact smooth manifold having a pair of robust heterodimensional cycles where r is a positive integer or infinity. We prove that if certain conditions about the signatures of non-linearities and Schwarzian derivatives of the transition maps are satisfied, then by giving C^r arbitrarily small perturbation, we can produce a periodic point at which the first return map in the center direction is C^r-flat. As a consequence, we will prove that C^r-generic diffeomorphisms in the neighborhood of the initial diffeomorphism exhibit super-exponential growth of numbe…

仅具有抛物型不动点的区间微分同胚的迭代导数的精确二次增长

Exact quadratic growth for the derivatives of iterates of interval diffeomorphisms with only parabolic fixed points

本文证明仅具抛物型不动点的区间微分同胚，若存在非二次切触且单侧拓扑排斥的不动点，则迭代导数最大增长恰为二次，否则为次二次。

On primes and period growth for Hamiltonian diffeomorphisms

Here we use Vinogradov's prime distribution theorem and a multi-dimensional generalization due to Harman to strengthen some recent results concerning the periodic points of Hamiltonian diffeomorphisms. In particular we establish resonance relations for the mean indices of the fixed points of Hamiltonian diffeomorphisms which do not have periodic points with arbitrarily large periods in $\mathbb{P}^2$, the set of natural numbers greater than one which have at most two prime factors when counted with multiplicity. As an application of these results we partially recover, using only symplectic too…

度量熵与同宿增长速率

Metric entropy and homoclinic growth rate

本文针对紧致黎曼流形上保持遍历双曲测度的C^r微分同胚，证明同宿增长速率以度量熵为下界，推广了Mendoza的曲面结果至高维，并通过Newhouse域的例子检验了估计的尖锐性。

Uniform rigidity sequences for weak mixing diffeomorphisms on $\mathbb{T}^2$

In this paper we will show that if a sequence of natural numbers satisfies a certain growth rate, then there is a weak mixing diffeomorphism on $\mathbb{T}^2$ that is uniformly rigid with respect to that sequence. The proof is based on a quantitative version of the Anosov-Katok-method with explicitly defined conjugation maps and the constructions are done in the $C^{\infty}$-topology as well as in the real-analytic topology.

A consequence of the growth of rotation sets for families of diffeomorphisms of the torus

In this paper we consider $C^\infty $-generic families of area-preserving diffeomorphisms of the torus homotopic to the identity and their rotation sets. Let $f_t:\rm{T^2\rightarrow T^2}$ be such a family, $\widetilde{f}_t:\rm I\negthinspace R^2 \rightarrow \rm I\negthinspace R^2$ be a fixed family of lifts and $ρ(\widetilde{f}_t)$ be their rotation sets, which we assume to have interior for $t$ in a certain open interval $I.$ We also assume that some rational point $(\frac pq,\frac rq)\in \partial ρ(\widetilde{f}_{\overline{t}})$ for a certain parameter $\overline{t}\in I$ and we want to unde…

Growth of maps, distortion in groups and symplectic geometry

In the present paper we study two sequences of real numbers associated to a symplectic diffeomorphism: the uniform norm of the differential of its n-th iteration and the word length of its n-th iteration. In the latter case we assume that our diffeomorphism lies in a finitely generated group of symplectic diffeomorphisms. We find lower bounds on the growth rates of these sequences in a number of situations. These bounds depend on the symplectic geometry of the manifold rather than on the specific choice of a diffeomorphism. They are obtained by using recent results of Schwarz on Floer homology…

Fiberwise volume growth via Lagrangian intersections

We consider Hamiltonian diffeomorphisms $ϕ$ of the unit cotangent bundle over a closed Riemannian manifold $(M,g)$ which extend to Hamiltonian diffeomorphisms of $T^*M$ equal to the time-1-map of the geodesic flow for $|p| \ge 1$. For such diffeomorphisms we establish uniform lower bounds for the fiberwise volume growth of $ϕ$ which were previously known for geodesic flows and which depend only on $(M,g)$ or on the homotopy type of $M$. More precisely, we show that for each $q \in M$ the volume growth of the unit ball in $T_q^*M$ under the iterates of $ϕ$ is at least linear if $M$ is rationall…

Volume growth and topological entropy of certain partially hyperbolic systems

Let $f$ be a $C^{1}$ diffeomorphism on a compact manifold $M$ admitting a partially hyperbolic splitting $TM=E^{s}\oplus_{\prec} E^{1}\oplus_{\prec} E^{2}\cdots \oplus_{\prec}E^{l}\oplus_{\prec} E^{u}$ where $E^{s}$ is uniformly contracting, $E^{u}$ is uniformly expanding and $\dim E^{i}=1,\,1\leq i\leq l.$ We prove an entropy formula w.r.t. the volume growth rate of subspaces in the tangent bundle: $$h_{\rm{top}}(f)=\lim_{n\to+\infty}\frac{1}{n}\log\int\max_{V\subset T_{x}M}|\det Df_{x}^{n}|_{V}|\,d x.$$

不稳定流形的熵与体积增长

Entropies and volume growth of unstable manifolds

本文通过将不稳定叶层熵与覆盖数关联，给出了紧流形上 $C^2$ 微分同胚遍历测度度量熵的新上界，并推广至 $C^{1+α}$ 情形。