Ruelle Zeta Research Articles

Crossover from Selberg’s type to Ruelle’s type zeta function in classical kinetics

The decay rates of the density-density correlation function are computed for a chaotic billiard with some disorder inside. In the case of the clean system the rates are zeros of Ruelle's zeta function and in the limit of strong disorder they are roots of Selberg's zeta function. We constructed the interpolation formula between two limiting zeta functions by analogy with the case of the integrable billiards. The almost clean limit is discussed in some detail.

Zeta functions for certain non-hyperbolic systems and topological Markov approximations

We study dynamical (Artin–Mazur–Ruelle) zeta functions for piecewise invertible multi-dimensional maps. In particular, we direct our attention to non-hyperbolic systems admitting countable generating definite partitions which are not necessarily Markov but satisfy the finite range structure (FRS) condition. We define a version of Gibbs measure (weak Gibbs measure) and by using it we establish an analogy with thermodynamic formalism for specific cases, i.e. a characterization of the radius of convergence in terms of pressure. The FRS condition leads us to nice countable state symbolic dynamics and allows us to realize it as towers over Markov systems. The Markov approximation method then gives a product formula of zeta functions for certain weighted functions.

Anomalous diffusion in dynamical systems: Transport coefficients of all order.

The theory of Ruelle's zeta function [Thermodynamic Formalism (Addison-Wesley, Reading, MA, 1978)] is extended to describe anomalous transport induced by dynamical chaos. It is shown that P(q) for the generating function of the displacement may not exist for supradiffusive processes, and that the difficulty may be overcome by the introduction of a two-parameter function P (β,q). We present two exactly solvable examples of anomalous diffusion induced by intermittency, to which our method is applied

On the thermodynamic formalism for the Gauss map

We study the generalized transfer operator of the Gauss mapTx=(1/x) mod 1 on the unit interval. This operator, which for β=1 is the familiar Perron-Frobenius operator ofT, can be defined for Re β>1/2 as a nuclear operator either on the Banach spaceA ∞(D) of holomorphic functions over a certain discD or on the Hilbert space of functions belonging to some Hardy class of functions over the half planeH −1/2. The spectra of on the two spaces are identical. On the space is isomorphic to an integral operator with kernel the Bessel function $$\mathfrak{F}_{2\beta - 1} (2\sqrt {st} )$$ and hence to some generalized Hankel transform. This shows that has real spectrum for real β>1/2. On the spaceA ∞(D) the operator can be analytically continued to the entire β-plane with simple poles at $$\beta = \beta _k = (1 - k)/2,k = 0,1,2,...$$ and residue the rank 1 operator $$\mathcal{N}^{(k)} f = \tfrac{1}{2}(1/k!)f^{(k)} (0)$$ . From this similar analyticity properties for the Fredholm determinant of and hence also for Ruelle's zeta function follow. Another application is to the function $$\zeta _M (\beta ) = \sum\limits_{n = 1}^\infty {[n]^\beta } $$ , where [n] denotes the irrational[n]=(n+(n 2+4)1/2)/2. ζM(β) extends to a meromorphic function in the β-plane with the only poles at β=±1 both with residue 1.

On the thermodynamic formalism for the gauss map

We study the generalized transfer operator ℒ β f(z)= $$\sum\limits_{n = 1}^\infty {\left( {\frac{1}{{z + n}}} \right)^{2\beta } \times f\left( {1/\left( {z + n} \right)} \right)} $$ of the Gauss mapTx=(1/x) mod 1 on the unit interval. This operator, which for β=1 is the familiar Perron-Frobenius operator ofT, can be defined for Re β>1/2 as a nuclear operator either on the Banach spaceA ∞(D) of holomorphic functions over a certain discD or on the Hilbert space ℋ Reβ (2) (H -1/2 of functions belonging to some Hardy class of functions over the half planeH −1/2. The spectra of − β on the two spaces are identical. On the space ℋ Reβ (2) (H -1/2 ℒ β is isomorphic to an integral operatorK β with kernel the Bessel function $$\mathfrak{F}_{2\beta - 1} (2\sqrt {st} )$$ and hence to some generalized Hankel transform. This shows that ℒ β has real spectrum for real β>1/2. On the spaceA ∞(D) the operator ℒ β can be analytically continued to the entire β-plane with simple poles atβ=β k =(1-k)/2,k=0, 1, 2,..., and residue the rank 1 operatorN (k) f=1/2(1/K!)f (k)(0) . From this similar analyticity properties for the Fredholm determinant det (1-ℒ β ) of ℒ β and hence also for Ruelle's zeta function follow. Another application is to the function $$\zeta _{\rm M} (\beta ) = \sum\limits_{n = 1}^\infty {\left[ n \right]^\beta } $$ , where [n] denotes the irradional [n]=(n+(n 2+4)1/2)/2.ζ M (β) extends to a meromorphic function in the β-plane with the only poles at β=±1 both with residue 1.

On the location of poles of Ruelle's zeta function for Gauss map

On the location of poles of Ruelle's zeta function for Gauss map

On the location of poles of Ruelle's zeta function

We discuss examples of one-dimensional lattice spin systems of classical statistical mechanics whose generalized zeta function has all its poles and zeros on the real axis. The close relation between certain hyperbolic dynamical systems and these spin systems lets one expect that this is also true for some of the dynamical systems. In fact, we have found several one-dimensional expansive systems, among them the Gauss map whose zeta functions have their zeros, respectively their poles, on the real axis. Such a behaviour is closely related to the spectral properties of the sytems transfer operator which in the cases considered is a positive nuclear operator in a Banach space of holomorphic functions. We formulate a general conjecture concerning the spectrum of this class of operators.