We consider a lattice of weakly coupled expanding circle maps. We construct, via a cluster expansion of the Perron-Frobenius operator, an invariant measure for these innnite dimensional dynamical systems which exhibits space-time-chaos.

We investigate in a geometrical way the sieving process of 021 β 3 for obtaining the Delone set 0 β 4 051 β 3 of β integers where β is a Perron number in the context of linear asymptotic invariants associated with a canonical inductive system constructed from β . When β is a Pisot number, we exhibit a canonical cut-and-project scheme, a model set associated with 0 β and so prove that it is a Me...

We consider subshifts of nite type on the symbol space generated by incidence matrices over a countably innnite alphabet. We extend the deenition of topological pressure to this context and, as our main result, we construct a new class of Gibbs states of HH older continuous potentials on these symbol spaces. We establish some basic stochastic properties of these Gibbs states: exponential decay ...

Perron-Frobenius operators and their eigendecompositions are increasingly being used as tools of global analysis for higher dimensional systems. The numerical computation of large, isolated eigenvalues and their corresponding eigenfunctions can reveal important persistent structures such as almostinvariant sets, however, often little can be said rigorously about such calculations. We attempt to...

We explore the concept of metastability in random dynamical systems, focusing on connections between random Perron–Frobenius operator cocycles and escape rates of random maps, and on topological entropy of random shifts of finite type. The Lyapunov spectrum of the random Perron–Frobenius cocycle and the random adjacency matrix cocycle is used to decompose the random system into two disjoint ran...

Finite time coherent sets [Froyland et al., 2010] have recently been defined by a measure-based objective function describing the degree that sets hold together, along with a Frobenius–Perron transfer operator method to produce optimally coherent sets. Here, we present an extension to generalize the concept to hierarchically define relatively coherent sets based on adjusting the finite time coh...

The analysis of the Perron eigenspace of a nonnegative matrix A whose symmetric part has rank one is continued. Improved bounds for the Perron root of Levinger’s transformation (1 − α)A+ αAt (α ∈ [0, 1]) and its derivative are obtained. The relative geometry of the corresponding left and right Perron vectors is examined. The results are applied to tournament matrices to obtain a comparison resu...

Invariant measures of dynamical systems play important role in understanding the chaotic nature of dynamical systems. Let (I,B, λ) be a normalized measure space, where I = [0, 1], B is a Borel σ-algebra of subsets of I , λ Lebesgue measure in (I,B). Let τ : (I,B, λ) → (I,B, λ) be a deterministic dynamical system. The Frobenius-Perron operator Pτ of τ is a linear operator Pτ : L (I,B)→ L(I,B) de...