For a subshift of finite type and a fixed Hölder continuous function, the zero measure invariant set of points where the Birkhoff averages do not exist is either empty or carries full Hausdorff dimension. Similar statements hold for conformal repellers and two-dimensional horseshoes, and the set of points where the pointwise dimensions, local entropies, Lyapunov exponents, and Birkhoff averages...

1 Describing measurable sets 8 1.1 Families of sets . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.1.1 Semiring of sets . . . . . . . . . . . . . . . . . . . . . . 9 1.1.2 Restricted algebras . . . . . . . . . . . . . . . . . . . . 12 1.1.3 Sigma Algebras . . . . . . . . . . . . . . . . . . . . . . 12 1.1.4 Binary Unions . . . . . . . . . . . . . . . . . . . . . . 14 1.1.5 Initial Sigm...

1 Sigma-Algebra: Describing measurable sets 6 1.1 Families of sets . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.1.1 Semiring of sets . . . . . . . . . . . . . . . . . . . . . . 7 1.1.2 Restricted algebras . . . . . . . . . . . . . . . . . . . . 10 1.1.3 Sigma Algebras . . . . . . . . . . . . . . . . . . . . . . 10 1.1.4 Binary Unions . . . . . . . . . . . . . . . . . . . . . . 12 1.1...

Families of Sets Definition 1.1 (Order properties). A countable1 family {An}n∈N of 1 To make sure we are all on the same page, let us fix some notation and terminology: • ⊆ denotes a subset (not necessarily proper). • A set A is said to be countable if there exists an injection (one-to-one mapping) from A into N. Note that finite sets are also countable. Sets which are not countable are called ...

Algorithms which compute modulo triangular sets must respect the presence of zero-divisors. We present Hensel lifting as a tool for dealing with them. We give an application: a modular algorithm for computing GCDs of univariate polynomials with coefficients modulo a radical triangular set overQ. Our modular algorithm naturally generalizes previous work from algebraic number theory. We have impl...