Coupled-Expanding Maps and Matrix shifts

Uniqueness of equilibrium measures for countable Markov shifts and multidimensional piecewise expanding maps

We prove that potentials with summable variations on topologically transitive countable Markov shifts have at most one equilibrium measure. We apply this to multidimensional piecewise expanding maps using their Markov diagrams.

Basic Constructions and Examples

4. Examples. 7 4.1. Rigid rotation of a compact group. 7 4.2. Adding machines. 7 4.3. Interval Exchange Maps. 8 4.4. Full shifts and shifts of finite type. 9 4.5. More examples of subshifts. 10 4.5.1. Prouhet-Thue-Morse 11 4.5.2. Chacon system 11 4.5.3. Sturmian systems 11 4.5.4. Toeplitz systems 11 4.5.5. Sofic systems 11 4.5.6. Context-free systems 12 4.5.7. Coded systems 12 4.6. Smooth expan...

Uniqueness of the Srb Measure for Piecewise Expanding Weakly Coupled Map Lattices in Any Dimension

We prove the existence of a unique SRB measure for a wide range of multidimensional weakly coupled map lattices. These include piecewise expanding maps with diffusive coupling.

Coupled Analytic Maps

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.

Topological Properties of Linearly Coupled Expanding Maps Lattices