arXiv subjects

Hans Henrik Rugh

Publications and source records attributed to Hans Henrik Rugh.

At least 19 recordsLinked to original sources

arXiv2021

Regularity of characteristic exponents and linear response for transfer operator cocycles

We consider cocycles obtained by composing sequences of transfer operators with positive weights, associated with uniformly expanding maps (possibly having countably many branches) and depending upon parameters. Assuming $C^k$ regularity with respect to coordinates and parameters, we show that when the sequence is picked within a certain uniform family the top characteristic exponent and generator of top Oseledets space of the cocycle are $C^{k-1}$ in parameters. As applications, we obtain a linear response formula for the equivariant measure associated with random products of uniformly expanding maps, and we study the regularity of the Hausdorff dimension of a repeller associated with random compositions of one-dimensional cookie-cutters.

math.DS↗

arXiv2018

Ratio ergodic theorems: From Hopf to Birkhoff and Kingman

Hopf's ratio ergodic theorem has an inherent symmetry which we exploit to provide a simplification of standard proofs of Hopf's and Birkhoff's ergodic theorems. We also present a ratio ergodic theorem for conservative transformations on a $σ$-finite measure space, generalizing Kingman's ergodic theorem for subadditive sequences and generalizing previous results by Akcoglu and Sucheston.

math.DS↗

arXiv2016

Entropy-continuity for interval maps with holes

We study the dependence of the topological entropy of piecewise monotonic maps with holes under perturbations, for example sliding a hole of fixed size at uniform speed or expanding a hole with uniform expansion. We show that under suitable conditions the topological entropy varies locally Hoelder continuously with the local Hoelder exponent depending itself on the value of the topological entropy.

math.DS↗

arXiv2015

The Milnor-Thurston determinant and the Ruelle transfer operator

The topological entropy $h_{\rm top}$ of a continuous piecewise monotone interval map measures the exponential growth in the number of monotonicity intervals for iterates of the map. Milnor and Thurston showed that $\exp(-h_{\rm top})$ is the smallest zero of an analytic function, now coined the Milnor-Thurston determinant, that keeps track of relative positions of forward orbits of critical points. On the other hand $\exp(h_{\rm top})$ equals the spectral radius of a Ruelle transfer operator $L$, associated with the map. Iterates of $L$ keep track of inverse orbits of the map. For no obvious reason, a Fredholm determinant for the transfer operator has not only the same leading zero as the M-T determinant but all peripheral (those lying in the unit disk) zeros are the same. The purpose of this note is to show that on a suitable function space, the dual of the Ruelle transfer operator has a regularized determinant, identical to the Milnor-Thurston determinant, hereby providing a natural explanation for the above puzzle.

math.DS↗

arXiv2014

Kneading with weights

We generalise Milnor-Thurston's kneading theory to the setting of piecewise continuous and monotone interval maps with weights. We define a weighted kneading determinant ${\cal D}(t)$ and establish combinatorially two kneading identities, one with the cutting invariant and one with the dynamical zeta function. For the pressure $\log ρ_1$ of the weighted system, playing the role of entropy, we prove that ${\cal D}(t)$ is non-zero when $|t|<1/ρ_1$ and has a zero at $1/ρ_1$. Furthermore, our map is semi-conjugate to an analytic family $h_t, 0 < t < 1/ρ_1$ of Cantor PL maps converging to an interval PL map $h_{1/ρ_1}$ with equal pressure

math.DS↗

arXiv2012

Micro-canonical thermodynamics: Why does heat flow from hot to cold

We show how to use a central limit approximation for additive co-cycles to describe non-equilibrium and far from equilibrium thermodynamic behavior. We consider first two weakly coupled Hamiltonian dynamical systems initially at different micro-canonical temperatures. We describe a stochastic model where the energy-transfer between the two systems is considered as a random variable satisfying a central limit approximation. We show that fluctuations in energy observables are linearly related to the heat-transfer (dissipation). As a result, on average, heat flows from hot to cold. We also consider the far from equilibrium situation of a non-Hamiltonian thermostatted system as in Evans et al. {\em Phys.\ Rev.\ Lett.} {\bf 71}, 2401 (1993). Applying the same central limit approximation we re-derive their relation for the violation of the 2nd law of thermodynamics. We note that time-reversal symmetry is not used in our derivation.

cond-mat.stat-mech↗

arXiv2011

A uniform contraction principle for bounded Apollonian embeddings

Let H denote the standard one-point completion of a real Hilbert space. Given any non-trivial proper sub-set U of H one may define the so-called `Apollonian' metric d_U on U. When U \subset V \subset H are nested proper subsets we show that their associated Apollonian metrics satisfy the following uniform contraction principle: Let D=diam_V (U) \in [0,+\infty] be the diameter of the smaller subsets with respect to the large. Then for every x,y in U we have d_V(x,y) \leq tanh (D/4) d_U(x,y). In dimension one, this contraction principle was established by Birkhoff for the Hilbert metric of finite segments on RP^1. In dimension two it was shown by Dubois for subsets of the Riemann sphere. It is new in the generality stated here.

math.MG↗

arXiv2010

Complexified cones. Spectral gaps and variational principles

We consider contractions of complexified real cones, as recently introduced by Rugh in [Rugh10]. Dubois [Dub09] gave optimal conditions to determine if a matrix contracts a canonical complex cone. First we generalize his results to the case of complex operators on a Banach space and give precise conditions for the contraction and an improved estimate of the size of the associated spectral gap. We then prove a variational formula for the leading eigenvalue similar to the Collatz-Wielandt formula for a real cone contraction. Morally, both cases boil down to the study of suitable collections of 2 by 2 matrices and their contraction properties on the Riemann sphere.

math.FA↗

arXiv2006

Cones and gauges in complex spaces : Spectral gaps and complex Perron-Frobenius theory

We introduce complex cones and associated projective gauges, generalizing a real Birkhoff cone and its Hilbert metric to complex vector spaces. We deduce a variety of spectral gap theorems in complex Banach spaces. We prove a dominated complex cone-contraction Theorem and use it to extend the classical Perron-Frobenius Theorem to complex matrices, Jentzsch's Theorem to complex integral operators, a Krein-Rutman Theorem to compact and quasi-compact complex operators and a Ruelle-Perron-Frobenius Theorem to complex transfer operators in dynamical systems. In the simplest case of a complex n by n matrix A we have the following statement: Suppose that 0 < c < +infinity is such that |Im A_{ij} \bar{A}_{mn}| < c < Re A_{ij} \bar{A}_{mn} for all indices. Then A has a `spectral gap'.

math.DS↗

arXiv2004

On the dimensions of conformal repellers. Randomness and parameter dependency

Bowen's formula relates the Hausdorff dimension of a conformal repeller to the zero of a `pressure' function. We present an elementary, self-contained proof which bypasses measure theory and the Thermodynamic Formalism to show that Bowen's formula holds for $C^1$ conformal repellers. We consider time-dependent conformal repellers obtained as invariant subsets for sequences of conformally expanding maps within a suitable class. We show that Bowen's formula generalizes to such a repeller and that if the sequence is picked at random then the Hausdorff dimension of the repeller almost surely agrees with its upper and lower Box dimensions and is given by a natural generalization of Bowen's formula. For a random uniformly hyperbolic Julia set on the Riemann sphere we show that if the family of maps and the probability law depend real-analytically on parameters then so does its almost sure Hausdorff dimension.

math.DS↗

arXiv2004

Eigenfunctions for smooth expanding circle maps

We construct a real-analytic circle map for which the corresponding Perron-Frobenius operator has a real-analytic eigenfunction with an eigenvalue outside the essential spectral radius when acting upon $C^1$-functions.

math.DS↗

arXiv2002

A Micro-Thermodynamic Formalism

We consider the micro-canonical ensemble of a classical Hamiltonian dynamical system, the Hamiltonian being parameter dependent and in the possible presence of other first integrals. We describe a thermodynamic formalism in which a 1st law of thermodynamics, or fundamental relation, is based upon the bulk-entropy, S. Under an ergodic hypothesis, S is shown to be an adiabatic invariant. Expressions for derivatives and thermodynamic relations are derived within the micro-canonical ensemble itself.

nlin.CD↗

arXiv1998

A Geometric, Dynamical Approach to Thermodynamics

We present a geometric and dynamical approach to the micro-canonical ensemble of classical Hamiltonian systems. We generalize the arguments in \cite{Rugh} and show that the energy-derivative of a micro-canonical average is itself micro-canonically observable. In particular, temperature, specific heat and higher order derivatives of the entropy can be observed dynamically. We give perturbative, asymptotic formulas by which the canonical ensemble itself can be reconstructed from micro-canonical measurements only. In a purely micro-canonical approach we rederive formulas by Lebowitz et al \cite{LPV}, relating e.g. specific heat to fluctuations in the kinetic energy. We show that under natural assumptions on the fluctuations in the kinetic energy the micro-canonical temperature is asymptotically equivalent to the standard canonical definition using the kinetic energy.

chao-dyn↗

arXiv1997

Transfer Operators for Coupled Analytic Maps

We consider analytic coupled map lattices over $\Z^d$ with exponentially decaying interaction. We introduce Banach spaces for the infinite-dimensional system that include measures with analytic, exponentially bounded finite dimensional marginals. Using residue calculus and `cluster expansion'-like techniques we define transfer operators on these Banach spaces. For these we get a unique probability measure that exhibits exponential decay of correlations.

chao-dyn↗

arXiv1997

A Dynamical Approach to Temperature

We present a new dynamical approach for measuring the temperature of a Hamiltonian dynamical system in the micro canonical ensemble of thermodynamics. We show that under the hypothesis of ergodicity the temperature can be computed as a time-average of the functional, div(grad H/|grad H|^2), on the energy-surface. Our method not only yields an efficient computational approach for determining the temperature it also provides an intrinsic link between dynamical systems theory and the statistical mechanics of Hamiltonian systems.

chao-dyn↗

arXiv1996

Computing Lyapunov spectra with continuous Gram-Schmidt orthonormalization

We present a straightforward and reliable continuous method for computing the full or a partial Lyapunov spectrum associated with a dynamical system specified by a set of differential equations. We do this by introducing a stability parameter beta>0 and augmenting the dynamical system with an orthonormal k-dimensional frame and a Lyapunov vector such that the frame is continuously Gram-Schmidt orthonormalized and at most linear growth of the dynamical variables is involved. We prove that the method is strongly stable when beta > -lambda_k where lambda_k is the k'th Lyapunov exponent in descending order and we show through examples how the method is implemented. It extends many previous results.

chao-dyn↗

arXiv1996

Chaos, Scaling and Existence of a Continuum Limit in Classical Non-Abelian Lattice Gauge Theory

We discuss space-time chaos and scaling properties for classical non-Abelian gauge fields discretized on a spatial lattice. We emphasize that there is a ``no go'' for simulating the original continuum classical gauge fields over a long time span since there is a never ending dynamical cascading towards the ultraviolet. We note that the temporal chaotic properties of the original continuum gauge fields and the lattice gauge system have entirely different scaling properties thereby emphasizing that they are entirely different dynamical systems which have only very little in common. Considered as a statistical system in its own right the lattice gauge system in a situation where it has reached equilibrium comes closest to what could be termed a ``continuum limit'' in the limit of very small energies (weak non-linearities). We discuss the lattice system both in the limit for small energies and in the limit of high energies where we show that there is a saturation of the temporal chaos as a pure lattice artifact. Our discussion focuses not only on the temporal correlations but to a large extent also on the spatial correlations in the lattice system. We argue that various conclusions of physics have been based on monitoring the non-Abelian lattice system in regimes where the fields are correlated over few lattice units only. This is further evidenced by comparison with results for Abelian lattice gauge theory. How the real time simulations of the classical lattice gauge theory may reach contact with the real time evolution of (semi-classical aspects of) the quantum gauge theory (e.g. Q.C.D.) is left as an important question to be further examined.

hep-th↗

arXiv1996

Intermittency and Regularized Fredholm Determinants

We consider real-analytic maps of the interval $I=[0,1]$ which are expanding everywhere except for a neutral fixed point at 0. We show that on a certain function space the spectrum of the associated Perron-Frobenius operator ${\cal M}$ has a decomposition $Sp ({\cal M}) = σ_c \cup σ_p$ where $σ_c=[0,1]$ is the continuous spectrum of ${\cal M}$ and $σ_p$ is the pure point spectrum with no points of accumulation outside 0 and 1. We construct a regularized Fredholm determinant $d(λ)$ which has a holomorphic extension to $λ\in C-σ_c$ and can be analytically continued from each side of $σ_c$ to an open neighborhood of $σ_c-{0,1}$ (on different Riemann sheets). In $C-σ_c$ the zero-set of $d(λ)$ is in one-to-one correspondence with the point spectrum of ${\cal M}$. Through the conformal transformation $λ(z) = 1/(4z) (1+z)^2$ the function $d \circ λ(z)$ extends to a holomorphic function in a domain which contains the unit disc.

chao-dyn↗