Uniform Hyperbolicity Research Articles

Uniform hyperbolicity of a class of scattering maps

In recent years fractal Weyl laws and related quantum eigenfunction hypothesis have been studied in a plethora of numerical model systems, called quantum maps. In some models studied there one can easily prove uniform hyperbolicity. Yet, a numerically sound method for computing quantum resonance states, did not exist. To address this challenge, we recently introduced a new class of quantum maps (Mertig and Shudo 2018 Phys. Rev. E 97 042216). For these quantum maps, we showed that, quantum resonance states can numerically be computed using theoretically grounded methods such as complex scaling or weak absorbing potentials (Mertig and Shudo 2018 Phys. Rev. E 97 042216). However, proving uniform hyperbolicty for this class of quantum maps was not straight forward. Going beyond that work this article generalises the class of scattering maps and provides mathematical proofs for their uniform hyperbolicity. In particular, we show that the suggested class of two-dimensional symplectic scattering maps satisfies the topological horseshoe condition and uniform hyperbolicity. In order to prove these properties, we follow the procedure developed in the paper by Devaney and Nitecki (1979 Commun. Math. Phys. 67 137–46). Specifically, uniform hyperbolicity is shown by identifying a proper region in which the non-wandering set satisfies a sufficient condition to have the so-called sector bundle or cone field. Since no quantum map is known where both a proof of uniform hyperbolicity and a methodologically sound method for numerically computing quantum resonance states exist simultaneously, the present result should be valuable to further test fractal Weyl laws and related topics such as chaotic eigenfunction hypothesis.

Conditional mixing in deterministic chaos

AbstractWhile on the one hand, chaotic dynamical systems can be predicted for all time given exact knowledge of an initial state, they are also in many cases rapidly mixing, meaning that smooth probabilistic information (quantified by measures) on the system’s state has negligible value for predicting the long-term future. However, an understanding of the long-term predictive value of intermediate kinds of probabilistic information is necessary in various physical problems, and largely remains lacking. Of particular interest in data assimilation and linear response theory are the conditional measures of the Sinai–Ruelle–Bowen (SRB) measure on zero sets of general smooth functions of the phase space. In this paper we give rigorous and numerical evidence that such measures generically converge back under the dynamics to the full SRB measures, exponentially quickly. We call this property conditional mixing. While conditional mixing typically cannot be proven from standard transfer operator theory, we will prove that conditional mixing holds in a class of generalized baker’s maps, and demonstrate it numerically in some non-Markovian piecewise hyperbolic maps. Conditional mixing provides a natural limit on the effectiveness of long-term forecasting of chaotic systems via partial observations, and appears key to proving the existence of linear response outside the setting of smooth uniform hyperbolicity.

On the spectrum of the periodic Anderson–Bernoulli model

We analyze the spectrum of a discrete Schrödinger operator with a potential given by a periodic variant of the Anderson model. In order to do so, we study the uniform hyperbolicity of a Schrödinger cocycle generated by the SL(2,R) transfer matrices. In the specific case of the potential generated by an alternating sequence of random values, we show that the almost sure spectrum consists of at most 4 intervals.

Perturbations with nonpositive Lyapunov exponents

The notion of uniform hyperbolicity is equivalent to various admissibility properties. For example, one such property is expressed in terms of the existence of bounded solutions for any bounded perturbation of the dynamics. Our main objective is to describe a weaker hyperbolicity property for a nonautonomous dynamics with discrete time that is equivalent to a modification of the latter admissibility property, namely the existence of solutions with nonpositive Lyapunov exponents for any perturbation with nonpositive Lyapunov exponent. Our characterization is expressed in terms of appropriate exponents that somehow mix the Bohl exponents and the Lyapunov exponents.

Ergodic optimization for some dynamical systems beyond uniform hyperbolicity

ABSTRACT In Ergodic optimization, one wants to find ergodic measures to maximize or minimize the integral of given continuous functions. This has been succefully studied for uniformly hyperbolic systems for generic continuous functions by Bousch and Brémon. In this paper, we show that for several interesting systems beyond uniform hyperbolicity, any generic continuous function has a unique maximizing measure with zero entropy. In some cases, we also know that the maximizing measure has full support. These interesting systems include singular hyperbolic attractors, surface diffeomorphisms and diffeomorphisms away from homoclinic tangencies.We try to give a uniform mechanism for these non-hyperbolic systems.

Non-hyperbolicity at large scales of a high-dimensional chaotic system

The dynamics of many important high-dimensional dynamical systems are both chaotic and complex, meaning that strong reducing hypotheses are required to understand the dynamics. The highly influential chaotic hypothesis of Gallavotti and Cohen states that the large-scale dynamics of high-dimensional systems are effectively uniformly hyperbolic, which implies many felicitous statistical properties. We obtain direct and reliable numerical evidence, contrary to the chaotic hypothesis, of the existence of non-hyperbolic large-scale dynamical structures in a mean-field coupled system. To do this, we reduce the system to its thermodynamic limit, which we approximate numerically with a Chebyshev basis transfer operator discretization. This enables us to obtain a high-precision estimate of a homoclinic tangency, implying a failure of uniform hyperbolicity. Robust non-hyperbolic behaviour is expected under perturbation. As a result, the chaotic hypothesis should not be a priori assumed to hold in all systems, and a better understanding of the domain of its validity is required.

Topological and geometric hyperbolicity criteria for polynomial automorphisms of

AbstractWe prove that uniform hyperbolicity is invariant under topological conjugacy for dissipative polynomial automorphisms of $\mathbb {C}^2$ . 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.

Cantor spectrum for CMV and Jacobi matrices with coefficients arising from generalized skew-shifts

AbstractWe consider continuous cocycles arising from CMV and Jacobi matrices. Assuming that the Verblunsky and Jacobi coefficients arise from generalized skew-shifts, we prove that uniform hyperbolicity of the associated cocycles is $C^0$ -dense. This implies that the associated CMV and Jacobi matrices have a Cantor spectrum for a generic continuous sampling map.

Uniform hyperbolicity and its relation with spectral analysis of 1D discrete Schrödinger operators

This paper intends to provide new, simple, and self-contained proofs of the equivalence of various different descriptions of the uniformly hyperbolic SL (2,\mathbb R) . While in the scenario of the Schrödinger cocyles, they may in turn be applied to give new and simple proofs of theorems regarding their relation with the spectral analysis of one-dimensional discrete Schrödinger operators. Concretely, this paper gives four different descriptions of uniformly hyperbolic sequences together with the detailed proofs of their equivalence. It provides concise and self-contained proof of the Johnson’s Theorem [20], both for sequence and dynamically defined potentials. In particular, we give two different proofs of the direction “uniform hyperbolicity away from the spectrum,” of which one is the standard argument in the spirit of Russell Johnson’s original proof and the other involves Combes–Thomas type of estimate [11]. Finally, the relation between the avalanche principle, discovered by Goldstein and Schlag [16], and the uniformly hyperbolic sequence is explored which yields a simple proof and a better version of the avalanche principle. Many ingredients of the proofs in this paper are new, in particular the use of asymptotic stable and unstable directions, are of independent interest and have been applied to many other related problems.

A computer-assisted proof of the existence of Smale horseshoe for the folded-towel map

The paper contains a rigorous proof of existence of symbolic dynamics chaos in the generalized Hénon map’s 4th iterate H4, which was conjectured in the paper A 3D Smale Horseshoe in a Hyperchaotic Discrete-Time System of Li and Yang, 2007. We prove also the uniform hyperbolicity of the invariant set with symbolic dynamics. The proofs are computer-assisted with the use of C++ library CAPD for interval arithmetic, differentiation and integration.

Approximative dichotomy and persistence of nonuniformly normally hyperbolic invariant manifolds in Banach spaces

It was proved that both the normal hyperbolicity and invariant manifold for a uniformly hyperbolic compact invariant manifold and the invariant manifold for a uniformly hyperbolic noncompact invariant manifold are persistent under small perturbation. In this paper, we weaken the uniform normal hyperbolicity to the nonuniform one and prove that both the nonuniform normal hyperbolicity and invariant manifold for a nonuniformly eventually absolutely normally hyperbolic noncompact invariant manifold are persistent under small perturbation in Banach spaces. Unlike the uniform case, we need infinitely many families of norms to uniformize infinitely many exponential dichotomies correspondingly. We use the approximative exponential dichotomy and the pseudo orbit to overcome the difficulty.

The approximation of uniform hyperbolicity for C1 diffeomorphisms with hyperbolic measures

Let f be a C1-diffeomorphism which preserves an ergodic hyperbolic measure μ with positive measure-theoretic entropy. Assume that the Oseledec splittingTxM=E1(x)⊕⋯⊕Es(x)⊕Es+1(x)⊕⋯⊕El(x) is dominated on the Oseledec basin Γ. We give extensions of Katok's Horseshoes construction. Moreover there is a dominated splitting corresponding to Oseledec subspace on horseshoes. Based on this, we also establish the continuity of the sub-additive topological pressure of the singular value potential.

Robust transitivity of singular hyperbolic attractors

Singular hyperbolicity is a weakened form of hyperbolicity that has been introduced for vector fields in order to allow non-isolated singularities inside the non-wandering set. A typical example of a singular hyperbolic set is the Lorenz attractor. However, in contrast to uniform hyperbolicity, singular hyperbolicity does not immediately imply robust topological properties, such as the transitivity. In this paper, we prove that on an open and dense subset of the space of $$C^1$$ vector fields of a compact manifold, any singular hyperbolic attractors is robustly transitive.

On the notions of singular domination and (multi-)singular hyperbolicity

The properties of uniform hyperbolicity and dominated splitting have been introduced to study the stability of the dynamics of diffeomorphisms. One meets difficulties when one tries to extend these definitions to vector fields and Shantao Liao has shown that it is more relevant to consider the linear Poincar\'e flow rather than the tangent flow in order to study the properties of the derivative. In this paper we define the notion of singular domination, an analog of the dominated splitting for the linear Poincar\'e flow which is robust under perturbations. Based on this, we give a new definition of multi-singular hyperbolicity which is equivalent to the one recently introduced by Bonatti-da Luz. The novelty of our definition is that it does not involve the blowup of the singular set and the renormalization cocycle of the linear flows.

Uniform hyperbolicity of the graphs of nonseparating curves via bicorn curves

We show that the graphs of nonseparating curves for oriented finite type surfaces are uniformly hyperbolic. Our proof follows the proof of uniform hyperbolicity of the graphs of curves for closed surfaces due to Przytycki-Sisto, while introducing new arguments using homology to certify that certain curves are nonseparating. As demonstrated by Aramayona-Valdez, this proves also that the graph of nonseparating curves for any oriented infinite type surface with finite positive genus is hyperbolic.

Diophantine Property of Matrices and Attractors of Projective Iterated Function Systems in ℝℙ1

Abstract We prove that almost every finite collection of matrices in $GL_d( \mathbb{R} )$ and $SL_d({\mathbb{R}})$ with positive entries is Diophantine. Next we restrict ourselves to the case $d=2$. A finite set of $SL_2({\mathbb{R}})$ matrices induces a (generalized) iterated function system on the projective line ${\mathbb{RP}}^1$. Assuming uniform hyperbolicity and the Diophantine property, we show that the dimension of the attractor equals the minimum of 1 and the critical exponent.

Saddle hyperbolicity implies hyperbolicity for polynomial automorphisms of $\mathbb{C}^2$

We prove that for a polynomial diffeomorphism of C^2, uniform hyperbolicity on the set of saddle periodic points implies that saddle points are dense in the Julia set. In particular f satisfies Smale's Axiom A on C^2 .

Sharp <inline-formula><tex-math id="M1">\begin{document}$ \frac12 $\end{document}</tex-math></inline-formula>-Hölder continuity of the Lyapunov exponent at the bottom of the spectrum for a class of Schrödinger cocycles

We consider the setting for the disappearance of uniform hyperbolicity as in Bjerklöv and Saprykina (2008 Nonlinearity 21), where it was proved that the minimum distance between invariant stable and unstable bundles has a linear power law dependence on parameters. In this scenario we prove that the Lyapunov exponent is sharp $ \frac12 $-Hölder continuous.In particular, we show that the Lyapunov exponent of Schrödinger cocycles with a potential having a unique non-degenerate minimum is sharp $ \frac12 $-Hölder continuous below the lowest energy of the spectrum, in the large coupling regime.

Nonuniformly hyperbolic systems arising from coupling of chaotic and gradient-like systems

We investigate dynamical systems obtained by coupling two maps, one of which is chaotic and is exemplified by an Anosov diffeomorphism, and the other is of gradient type and is exemplified by a N-pole-to-S-pole map of the circle. Leveraging techniques from the geometric and ergodic theories of hyperbolic systems, we analyze three different ways of coupling together the two maps above. For weak coupling, we offer an addendum to existing theory showing that almost always the attractor has fractal-like geometry when it is not normally hyperbolic. Our main results are for stronger couplings in which the action of the Anosov diffeomorphism on the circle map has certain monotonicity properties. Under these conditions, we show that the coupled systems have invariant cones and possess SRB measures even though there are genuine obstructions to uniform hyperbolicity.

Characterization of uniform hyperbolicity for fibre-bunched cocycles

ABSTRACTWe prove a new characterization of uniform hyperbolicity for fibre-bunched cocycles. Specifically, we show that the existence of a uniform gap between the Lyapunov exponents of a fibre-bunched -cocycle defined over a subshift of finite type or an Anosov diffeomorphism implies uniform hyperbolicity. In addition, we construct an α-Hölder cocycle which has a uniform gap between the Lyapunov exponents, however, it is not uniformly hyperbolic.