Geometric Structure and Ergodic Properties of Bony Multi-Graphs

The paper proves that for each n there is an open set U of interval skew-products over a solenoid base whose compact invariant set K(G) is an attracting multi-graph or bony multi-graph carrying exactly n ergodic SRB measures (Theorem 3.0.1; Theorem 3.2.2; Corollary 3.2.1) . Its method builds weakly contractive fiber dynamics forced by an expanding circle map and passes to a solenoid extension, using non-uniform contraction and Stark-type results to obtain invariant (possibly bony) graphs and SRB measures . The model independently constructs a class with uniform contraction in n disjoint vertical strips and uniform expansion in the complementary gaps, yielding n continuous invariant graphs by a graph-transform and SRB basins via Birkhoff; it also excludes any other SRBs by a drain-out estimate on the gaps. This proves the same qualitative conclusions under stronger hypotheses. The model’s ‘no other SRB measures’ claim is stronger than what the paper states, but it follows from the model’s additional expansion assumptions rather than from the paper’s weaker non-uniform contraction framework. Hence both are correct, with different proofs and assumptions.