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Proper continued fractions are generalized continued fractions with positive integer numerators $a_i$ and integer denominators with $b_i\geq a_i$. In this paper we study the strength of approximation of irrational numbers to their convergents and classify which pairs of integers $p,q$ yield a convergent $p/q$ to some irrational $x$. Notably, we reduce the problem to finding convergence only of index one and two. We completely classify the possible choices for convergents of odd index and provide a near-complete classification for even index. We furthermore propose a natural two-dimensional generalization of the classical Gauss map as a method for dynamically generating all possible expansions and establish ergodicity of this map.

In this article we study a renormalization scheme with which we find all semi-regular continued fractions of a number in a natural way. We define two maps, T_slow and T_fast: these maps are defined for (x,y) in [0,1], where x is the number for which a semi-regular continued fraction representation is developed by T_slow according to the parameter y. The set of all possible semi-regular continued fraction representations of x are bijectively constructed as the parameter y varies. The map T_fast is a "sped-up" version of the map T_slow, and we show that T_fast is ergodic with respect to a probability measure which is mutually absolutely continuous with Lebesgue measure. In contrast, T_slow preserves no such measure, but does preserve an infinite, sigma-finite measure mutually absolutely continuous with Lebesgue measure. Furthermore, we generate a sequence of substitutions which generate a symbolic coding of the orbit of y. In the last section we highlight how our scheme can be used to generate semi-regular continued fractions explicitly for specific continued fraction algorithms such as Nakada's alpha continued fractions.

Continuing the work in \cite{ergodic-infinite}, we show that within each stratum of translation surfaces, there is a residual set of surfaces for which the geodesic flow in almost every direction is ergodic for almost-every periodic group extension produced using a technique referred to as \emph{cuts}.

We show that generic infinite group extensions of geodesic flows on square tiled translation surfaces are ergodic in almost every direction, subject to certain natural constraints. Recently K. Fr\c{a}czek and C. Ulcigrai have shown that certain concrete staircases, covers of square-tiled surfaces, are not ergodic in almost every direction. In contrast we show the almost sure ergodicity of other concrete staircases. An appendix provides a combinatorial approach for the study of square-tiled surfaces.

We show that for odd-valued piecewise-constant skew products over a certain two parameter family of interval exchanges, the skew product is ergodic for a full-measure choice of parameters.

After relating the notion of $\omega$-recurrence in skew products to the range of values taken by partial ergodic sums and Lyapunov exponents, ergodic $\mathbb{Z}$-valued cocycles over an irrational rotation are presented in detail. First, the generic situation is studied and shown to be $1/n$-recurrent. It is then shown that for any $\omega(n) 1/2$, there are uncountably many infinite staircases (a certain specific cocycle over a rotation) which are \textit{not} $\omega$-recurrent, and therefore have positive Lyapunov exponent. A further section makes brief remarks regarding cocycles over interval exchange transformations of periodic type.

We investigate the set of $x \in S^1$ such that for every positive integer $N$, the first $N$ points in the orbit of $x$ under rotation by irrational $\theta$ contain at least as many values in the interval $[0,1/2]$ as in the complement. By using a renormalization procedure, we show both that the Hausdorff dimension of this set is the same constant (strictly between zero and one) for almost-every $\theta$, and that for every $d \in [0,1]$ there is a dense set of $\theta$ for which the Hausdorff dimension of this set is $d$., Comment: in review

Ergodic properties of a renormalization procedure for studying the 1/2-discrepancy sums driven by rotations are studied, with corresponding implications for almost-sure bounds on the growth rates for these discrepancy sums., Comment: in review

The sequence of 1/2-discrepancy sums of $\{x + i \theta \bmod 1\}$ is realized through a sequence of substitutions on an alphabet of three symbols; particular attention is paid to $x=0$. The first application is to show that any asymptotic growth rate of the discrepancy sums not trivially forbidden may be achieved. A second application is to show that for badly approximable $\theta$ and any $x$ the range of values taken over $i=0,1,...n-1$ is asymptotically similar to $\log(n)$, a stronger conclusion than given by the Denjoy-Koksma inequality., Comment: In review

We initiate the study of the sets $H(c)$, $0=x-[x]$ stands for the fractional part of $x\in \mathbb R$. We prove that, for rational $c$, the sets $H(c)$ are of positive Hausdorff dimension and, in particular, are uncountable. For integers $m\geq1$, we obtain a surprising characterization of the numbers $\alpha\in H_m= H(\frac1m)$ in terms of their continued fraction expansions: The odd entries (partial quotients) of these expansions are divisible by $m$. The characterization implies that $x\in H_m$ if and only if $\frac 1{mx} \in H_m$, for $x>0$. We are unaware of a direct proof of this equivalence, without making a use of the mentioned characterization of the sets $H_m$. We also introduce the dual sets $\hat H_m$ of reals $y$ for which the sequence of integers $\big([ky]\big)_{k\geq1}$ consistently hits the set $m\mathbb Z$ with the at least expected frequency $\frac1m$ and establish the connection with the sets $H_m$: {2mm} If $xy=m$ for $x,y>0$, then $x\in H_m$ if and only if $y\in \hat H_m$. The motivation for the present study comes from Y. Peres's ergodic lemma., Comment: 9 pages

Given an irrational alpha and an x in the unit interval, the set of balanced times, for which the same number of (k*alpha+x) (modulo one) are less than or equal to one half as are larger than one half, is in general infinite, but sparse in terms of density. We investigate the sparseness of this sequence in terms of summation over reciprocals. Our results are that for the generic pair (alpha,x), the resulting sum diverges, but there are certain exceptional alpha for which the associated sums converge for every x., Comment: 15 Pages

Answering an informal question of K. Park, we show that by fixing some irrational alpha to have a particular standard continued fraction expansion, we may force the associated discrepancy sequences for all x in [0,1), which track the difference between the number of values in the orbit of x under rotation by alpha (modulo one) less than one half versus the number larger than one half, to have maximal values which grow at a prescribed rate., Comment: This paper has been withdrawn by the author. Paper withdrawn - results are replaced and improved in "substitutions and 1/2 discrepancy of {n theta + x}"

Heaviness refers to a sequence of partial sums maintaining a certain lower bound and was recently introduced and studied in "Heaviness: and Extension of a Lemma of Y. Peres." After a review of basic properties to familiarize the reader with the ideas of heaviness, general principles of heaviness in symbolic dynamics are introduced. The classical Morse sequence is used to study a specific example of heaviness in a system with nontrivial rational eigenvalues. To contrast, Sturmian sequences are examined, including a new condition for a sequence to be Sturmian., Comment: 14 Pages, to appear in Discrete and Continuous Dynamical Systems

We are concerned with describing the structure of the set of points in the unit interval which, when subjected to rotation by irrational alpha modulo one, for all finite portions of the orbit contain at least as many points in the bottom half of the interval as in the top half. Specifically, an inductive procedure for describing the set based on the continued fraction expansion of alpha is developed, leading into a discussion of the Hausdorff dimension of this set. Depending on the parameter alpha, all possible dimensions may be achieved, and the essential infimum (with respect to alpha) of this dimension is positive., Comment: This paper has been replaced by the more current "1/2 heavy sequences driven by rotation"

We investigate the dimension of the set of points in the d-torus which have the property that their orbit under rotation by some alpha hits a fixed closed target A more often than expected for all finite initial portions. An upper bound for the lower Minkowski dimension of this "strictly heavy set" H(A,alpha) is found in terms of the upper Minkowski dimension of the boundary of A, as well as k, the Diophantine approximability from below of the Lebesgue measure of A. The proof extends to translations in compact abelian groups more generally than just the torus, most notably the p-adic integers., Comment: 8 pages, in review

We provide an elementary proof of Y. Peres' lemma on the existence in certain dynamical systems of what we term heavy points, points whose ergodic averages consistently dominate the expected value of the ergodic averages. We also derive several generalizations of Peres' lemma by employing techniques from the simplified proof., Comment: 10 Pages, to appear in Houston Journal of Mathematics, one typo corrected